Transcript for:

hello and welcome to pre-algebra lesson one in this video we're going to learn about place value so the lesson objectives for today we want to learn basic definitions of terms relating to place value we want to gain a basic understanding of how our number system works and additionally we want to learn how to construct and use a place value chart all right let's start by talking about the whole numbers so the whole numbers start with zero so that's your smallest whole number and then increase in increments of one indefinitely so I've written here the whole numbers and look how we've listed 0 the smallest whole number first or in the leftmost position and then you're going to put a comma you're going to add one to zero to get to one so you put your next largest whole number there and then you can just keep adding one so then you would go to two then three then four then five there's not a largest whole number you can keep just adding one to get to the next largest whole number so what you'd want to do at some point is stop and put your final comma and then these three dots here and that's just going to tell you that the pattern continues forever so after five comes six then seven then eight then nine and so on and so forth all right so now let's talk about what we call the digits these are the first ten whole numbers and they are used to build numbers so I have here the whole numbers 0 through 9 so the first ten whole numbers are referred to as digits so I've listed them out here so 0 through 9. the whole key is that as we change a digit's position in a number its value is going to change so I have here that our number system relies on the digits so again that's the first ten whole numbers and a place value to build each number so it doesn't matter how large of a number you have or how small of a number you have you're only going to use the digits to build that number the key again is just as you change the position or the placement of a digit in a number its value is going to change that's why we call our number system a place value system so to show this let's take a look at four numbers each containing the digit 9. all right so what we're going to do is look at the four numbers we have here so 49 192 3945 and then 9687. there are some things that we're going to do here that we haven't talked about yet so just take everything as a given and then I'm going to fully explain as we move forward so the first thing is I'm going to write each number in what we call expanded notation and that is going to be covered in the next lesson it's just a way that we can write a number such that the value of each digit is fully shown so for example something like 49 I could write that as 40 plus 9. or something like 192 I would say that's 100 plus 90 plus 2. then something like 3945 that's going to be three thousand plus 900 plus 40 plus 5. and then 9687 that's going to be nine thousand plus six hundred plus eighty plus seven so we'll learn how to do this in the next lesson for right now let's just take it as a given and use it to understand what's going on we have a nine in each number so that digit nine so here here here and here now look at it in the expanded form of each number so it's going to be here here here and then here notice that in the first number 49 it's just a 9. and that's coming from the fact that the 9 is in the ones place there so its value comes from nine times one which is nine then in the next number you have 192. well this 9 is in what we call the tens place so it's 9 times 10 or 90. then when we get to 3945 the 9 is in the hundreds place so it's 9 times 100 or 900 then finally when we get to 9687 then 9 is in the thousands place so it's nine times a thousand or nine thousand the main thing here is to understand that as we change the position of that non that digit in a number it's getting a different value so it's getting its value based on where it is in the number its position or its placement now as a final example let's just think about instead of 49 let's write 94. and let's think about the 9 here let me write this out and say that this is 90 plus 4. well now this 9 is a 90. it's in the tens place it's 9 times 10 or 90. here the 9 is just a nine nine times one which is nine but notice that these numbers are built with the same two digits a four and a nine but this one is 49 which is 40 plus 9 and this is 94 which is 90 plus 4. we know that ninety four dollars would be more than forty nine dollars so obviously these two numbers are not the same even though they contain the same two digits only so the position of a digit in a number is going to determine its value okay let's look at what we call a place value chart so this is a good way to understand how to find the place value for a given digit in a number so to start off I'm just going to pick a number a smaller number let's say something like 1052 and you're just going to feed this into the place value chart so provided you're working with a whole number you can just make your own example if you're working with a decimal then you're going to need an expanded place value chart we'll cover that later on in the course for right now the rightmost digit of your whole number in this case it's a 2 and that's going to be in what we call the ones place I'll talk more about this in a moment but basically for right now let's just feed this in to the rightmost position on that place value chart so that 2 is going to go right there again where it says once okay let me get rid of this highlighting so nobody's lost as I move to the left in my number I'm going to match that movement to the left on my place value chart so this 5 is now going to go here where it says tens I go to the left this zero is now going to go in this position that says hundreds I'm just going to the left then I'm going to go to the left one more time this one is going to go into the thousands so what does it mean to have ones tens hundreds thousands all these things where does that come from and what does it mean well the first thing is this is all based on 10. once you understand that it becomes very easy to come up with these names so I'm just going to start you see that this says once here I'm just gonna write a one then as I move to the left let me just write a 10 let me write 100 and let me write 1000 and I'm just going to stop there and I'm going to show you the pattern so basically what's happening is as I move to the left I'm just multiplying by 10. so 1 times 10 gives me 10. then as I move to the left I'm multiplying by 10. 10 times 10 gives me a hundred as I move to the left I'm just multiplying by 10. so 100 times 10 gives me one thousand this would continue forever and ever and ever as you move to the left now what's also true as you move to the right you're dividing by 10. we can show that real quick too so if I took 1000 and I divide by 10 I get 100 if I take 100 and I divide by 10 I get 10. if I take 10 and I divide by 10 I'm going to get 1. so we see the pattern as we move to the left we multiply by 10 as we go to the right we divide by 10. so we can use that to come up with these names just remember that the rightmost position here is the ones place you can just think about that as a one that's all you have to remember then as you move to the left well hey I multiply by 10 so I have a 10. well now I say tens move to the left multiply by 10 I get 100 so we say hundreds and this just continues so then you'd have thousands ten thousands hundred thousands millions ten millions 100 millions billions ten billions 100 billions trillions you just keep multiplying by ten so you'd have 10 trillions 100 trillions so on and so forth now you can't possibly list everything so at some point you just stop okay so I've stopped here but you can stop wherever you want so now that we understand where these names are coming from and how this pattern Works let's think a little bit about this expanded notation that I showed you earlier essentially we'll get to this in the next lesson in more detail but for right now just think about the digit Stitch multiplied by its corresponding place value so you have a one in the thousands place so basically that's one times a thousand or just one thousand so I'm just going to write that then I'm gonna put plus you have a zero in the hundreds place now the zero there is really a placeholder it's telling me I don't have any hundreds in this particular number but I need that zero there to keep these other numbers in their correct positions otherwise the number would collapse down and you'd have 152. well I don't have 152 I have 1052 so I need that zero there as a placeholder with expanded notation if you have a zero somewhere you could just skip it over then I have five tens so that's five times ten which is fifty and then plus I have two ones so that's two times one which is two so this is the basic idea behind expanded notation that we saw in the last part of the lesson all right so let's just get a little practice writing a number inside of a place value chart it's a very easy exercise so 673 452. let me grab this come down here and paste this in I would advise you to get a piece of paper and a pencil and try this on your own just make a place value chart you don't have to go out to the trillions this number is only going to go to the hundred thousands so you can just cut your paper off there so I'm just going to take the rightmost digit of this whole number here this 2 put that in the ones place that's the rightmost position in this place value chart so I put a 2 here again as I work my way to the left and the number I'm working my way to the left on the place value chart so I'm just matching my movement so this 5 will go into the tens this four would go into the hundreds this three would go into the thousands the seven would go into the ten thousands and the six would go into the hundred thousands okay let's look at a bigger one so now we have seven billion 853 million nine hundred twenty two thousand six hundred seventy three so 7 billion 853 million 922 673 and grab this real quick come down here and paste this in all right so I'm going to take my three which is the rightmost position of this whole number and I'm going to put it in the rightmost position on this place value chart so again that's going into the ones place so I'm going to put a 3 there and I'm gonna work my way to the left in the number and I'm going to match that movement on my place value chart so I'm going to the left and getting that 7 that's going to be in the tens place go to the left I have a six that's in the hundreds place go to the left I have this two here which would be in the thousands place and then go to the left to have this two which is going to be in the ten thousands place and then go to the left I have a nine that's in the hundred thousands place now as I go to the left I have a three that's in the millions Place go to the left I have a five in the ten millions Place go to the left I have an eight in the hundred millions place and then lastly I go to the left and have a seven in the billions place all right let's look at some standard questions now so a typical place value test consists of questions that ask for the place value of of a specified digit so we're going to try a few examples we want to find the place value for each underlined digit so I went ahead and underlined them and also I wrote them in a different color and I'm going to come down here and paste this in we'll do some with the place value chart and then we'll do some without it it's a really easy thing once you do a few examples you sort of start to memorize the pattern and then it's really really easy to figure this out so once again I'm going to start with the rightmost digit of this whole number I'm going to put it in the rightmost position in this place value chart so the one would go here in the ones place the eighth here I'm going to the left so matching that movement on my place value chart the 8 here would go in the tens place the five here would go in the hundredth place the three here would go in the thousands place the seven would go into ten thousands place and then lastly the six would go on the hundred thousands place so to answer our question let me put a little border here we would say for the three that's going to be where that's going to be in the thousands place so this is in the thousands and you could put place if you want I'm just going to put thousands and then the eight here well that's going to be in the tens place so let's put the 8 is going to be in the tens place all right let's take a look at another example so here we have 6 million 358 791. so let me just paste this in here I'm going to start with the rightmost digit of the number I'm going to put that into the rightmost position in our place value chart so this one would go in the ones place and then I'm going to work my way to the left and the number and then match that in my place value chart so I'm going to have a 9 that's going to be in the tens place a seven in the hundreds place in eight in the thousands place of five in the ten thousands place a three in the hundred thousands place and then lastly a six in the millions place now we want to find the place value for the underlined digits let me put a little border here so we have this three and that's going to be in the hundred thousands place so let's just write 100 thousands here and again you can put place if you want that's up to you and then for the other one it's going to be the seven which is going to be right here so that's going to be in the hundreds place so let's put that the 7 is in the hundreds place all right let's look at one that's really big so we have nine trillion 551 billion 72 million six hundred thirty six thousand ten so let me paste that in and we're gonna be looking for the place value of this guy this nine this seven and this three so again I'm just going to feed this into the place value chart so the zero the rightmost digits of that number it's going to go into the rightmost position of the place value chart so that zero is in the ones place just going to work my way to the left here and match that in my place value chart so I'd have a one in the tens place a zero in the hundreds place a six in the thousands place a three in the ten thousands place let me go ahead and highlight that since we have so much going on here and then we're gonna have a six in the hundred thousands place uh two in the millions Place uh seven in the ten millions place so again that's one of the ones that's underlined so let me highlight that real quick and then as we work our way to the left we have a zero in the hundred millions place a one in the billions place of five in the ten billions place and then also a five in the hundred billions place and then lastly we have a nine in the trillions place so let me highlight that in that again because that's underlined okay so really not that bad let me put a little border here I'll slide down just a little bit and let me put this around the number hopefully I can fit everything on the right of this border but maybe I can't we'll see so for the nine that is going to be in the trillions place so I'm just going to write trillions again if you want to write place that is up to you then for the seven that is going to be in the 10 Millions place so 10 Millions again you could put place there and then for the three we are going to say that's in the ten thousands place so let's put 10 thousands and again you could put place all right let's try a few without the place value chart so I'm going to start with 7036. so remember the rightmost position of your whole number that's in the ones place you could just write ones like this and if you're lost again as you move to the left you're just multiplying by 10. so 1 times 10 is 10. so this next position to the left that's in the tens place and we can keep going even though we have our answer multiply 10 by 10 that's 100 so this position right here this is in the hundreds place and then I probably can't fit this here maybe I can just arrow down here like that so going to the left one more time 100 times 10 is a thousand so this would be in the thousands place so the answer we want is actually the tens place let me put the three over here that's what's underlined and we'll say this is in the tens place all right the next example we have sixty four thousand one hundred fourteen I think we probably don't need to write things out at this point so the rightmost digit of that whole number again that's in the ones place as you move to the left is multiple 5x10 so you'd have the Tens The Hundreds the thousands so that's what we're looking for that digit four that's underlined here is going to be in the thousands so the thousands place again not this four this four right here would be in the ones place we're talking about this four specifically that's underlined that's going to be in the thousands place all right let's just look at one more so a pretty big number this is one billion 998 million four thousand five hundred fifty Seven so the guy that's underlined here is the nine now one thing that you could do is by reading the number you immediately say okay well this is one billion well what comes to the right of one billion again you could just divide by 10 that's going to be a hundred million so immediately you know that nine that's underlined is in the hundred millions place but again you could start here and say okay well this is the ones and every time I go to the left that's going to be times ten so tens hundreds thousands ten thousands hundred thousands millions ten millions again hundred millions so however you figure that out and this digit here this nine is going to be in the hundred so the hundred Millions place and again there's two nines here so specifically I'm talking about this one right here that's underlined if you look at the one to the right of it that would be in the 10 Millions place but we're not talking about that one because it's not underlined so this one specifically this nine is in the hundred millions place hello and welcome to pre-algebra lesson two and in this video we're going to be learning about expanded notation so our lesson objectives for today would be to learn how to write a number in expanded notation and we also want to learn the difference between standard and expanded notation so in our first lesson we learned all about place value I taught you how to use this place value chart to find the place value for all the digits in a given number now when we think about expanded notation all we're simply doing is we're writing a number in a way that it emphasizes the place value for each digit in that number so this is kind of just a continuation of the last lesson and in fact in the first lesson I gave you a couple of examples where we wrote a number and expanded notation now let's say I took a number like 1035 and we want to write this in expanded notation well the first thing is we need the place value for each digit in the number so let's feed this into the place value chart so we start with the rightmost digit and we put in the rightmost column of the place value chart so the 5 goes into the ones place and as we move left we move left on the place value chart so the 3 goes in the tens place the zero goes in the hundreds place and the one goes in the thousands place so now that we know the place value for each digit in the number we can write the number to expanded notation all we need to do is form the sum of each digit multiplied by its corresponding place value so starting with this one on the left it's in the thousands place so each digit that one multiplied by its corresponding place value if it's in the thousands place we multiply it by one thousand then plus next we have a zero in the hundreds place now that zero as I've previously explained to you is just a placeholder but to illustrate expanded notation I'm going to go ahead and include it for now although you don't have to so the zero here is multiplied by 100. then plus we have a 3 in the tens place so that digit 3 is multiplied by ten and lastly we have a 5 in the ones place so we'll have plus five that's the digit times one the place value okay let's do a little multiplication now so one times a thousand is a thousand plus zero times a hundred is zero plus three times ten is thirty and then plus five times one that's five so let's just write this as one thousand one thousand plus thirty plus five right we don't need to include that zero so essentially when we write a number in expanded notation you can see that it's emphasizing the place value of each digit this one has a value of one thousand this three has a value of 30. this 5 is still a five the reason this is still a five is because it's in the ones place five times one is just five the reason this is a thirty is because it's in the tens place three times ten is thirty and the reason this one is a thousand is because it's in the thousands place one times a thousand is a thousand so it's a pretty simple concept overall and remember when we talk about this zero here it's a placeholder right it's a placeholder we don't have any hundreds in this case right we don't have any hundreds in this case but our number system will allow us to write 1035 with this zero saying hey I don't have anything there but hold that space otherwise what would you have this one would collapse down and you wouldn't be able to tell the difference between 135 and 1035 so you get all kinds of problems so let's take a look at another one let's say we want to write the number 22 000. 359 in expanded notation well let's get the place value first so I'm just going to feed this into the place value chart and 9 goes in the ones place the 5 goes in the tens place the 3 goes in the hundreds place the 2 goes in the thousands place and then this 2 here is going to go in the ten thousands place so to write the number in expanded notation again we form the sum of each digit multiplied by its corresponding place value so there's a 2 for the digit times its corresponding place value it's in the ten thousands place so just two times ten thousand plus now we have a two in the thousands place so two times one thousand plus now we have a three in the hundreds place so three times one hundred plus we have a five in the tens place so five times ten and then lastly plus we have a nine in the ones place so nine times one let's do some multiplication two times ten thousand is twenty thousand plus two times one thousand is two thousand plus three times one hundred is three hundred plus 5 times 10 is 50 and then plus 9 times 1 that's 9. so again by writing this number and expanding notation we can see the true value of each digit in the number this 2 has a value of 20 000. this 2 let me do in a different color because they're both twos that are side by side this two has a value of two thousand right it's in the thousands place this three has a value of three hundred this 5 has a value of 50 and then this 9 has a value of just nine so each digit gets its value again based on where it is in the number and that's why we call it a place value system let's look at another one okay let's say we had 759 000 275. so let's feed this into the place value chart so the 5 goes on the ones place the 7 goes in the tens place the 2 goes in the hundreds place then 9 goes in the thousands place the five goes into ten thousands place and the Seven goes in the hundred thousands place so to write this in expanded notation again we form the sum of each digit multiplied by its corresponding place value so this digit is a 7 its place value is the hundred thousands place so seven times one hundred thousand seven times one hundred thousand then plus next we have a five and a ten thousands place so five times ten thousand and plus we have a nine in the thousands place so nine times one thousand and plus we have a 2 in the hundreds place so 2 times 100 and plus we have a seven the tens place so seven times ten then lastly we have plus five times one right we have a five in the ones place five times one so we do a little multiplication seven times one hundred thousand is seven hundred thousand plus five times ten thousand that's fifty thousand plus nine times one thousand that's nine thousand plus 2 times 100 that's 200 plus 7 times 10 that's 70. and then lastly plus five times one that's five so by taking this number from standard notation and putting it in expanded notation we can see the true value of each digit in the number the 7 is 700 000 right because it's in the hundred thousands place the five is fifty thousand because it's in the ten thousands place five groups of ten thousand fifty thousand the nine is nine thousand right it's in the thousands place the 2 is 200 it's in the hundreds place the seven is seventy it's in the tens place and lastly the five is still a five because it's in the ones place let's look at another one all right for the next one let's write 36 million 255 000 927 in expanded notation now some of you might be going where's the place value chart at some point you have to be able to do this without a place value chart you have to Simply memorize the place value so if that's something you can't do yet pull out a place value chart write this number into the place value chart so that you can get the place value for each digit in the number and then continue with me but otherwise remember that there's a trick you can use and in case you didn't watch the first video where we talked about place value I'll explain it for you so the rightmost place you should be able to remember is the ones place As you move to the left you're just multiplying by 10. so in other words if you think about this as just one as I move left I multiply by 10 this would be 10. right it's the tens place it's the tens place then as I move left again I multiply by 10. this just keeps going forever and ever now you just keep multiplying by 10. so 10 times 10 is a hundred so this is the hundreds the hundreds place okay then this would be the thousands and let me make that Apostrophe a little better and then this would be the ten thousands and this would be the hundred thousands this would be the Millions and this would be the 10 mountains and again the way you figure this out is the rightmost column is always the ones place so just think about it as just a one As you move to the left you multiply by 10. so 1 times 10 is 10. so this is the tens place now I have a ten multiply by 10 again you get to 100 so now this is the hundreds place multiply by 10 again to get to a thousand that's the thousands and ten thousand then a hundred thousand then a million then 10 million so on and so forth that continues forever and ever and ever so you could have a number as large as you want and provided you had enough time you could sit there and write out the place values for each digit in that number now let's take this number and write it in expanded notation this is kind of a big number so it's going to take a little time we have a 3 that's in the 10 Millions place so we would have 3 times 10 million then plus we have a six in the millions place so 6 times 1 million and plus we have a two and a hundred thousands place so two times one hundred thousand and plus we have a five in the ten thousands place so five times ten thousand [Music] and plus we have another five and that's in the thousands place so five times one thousand and plus we have a nine in the hundreds place so nine times one hundred and plus we have a two in the tens place so two times ten and lastly we have plus seven times one right we have a seven in the ones place so we do a little multiplication 3 times 10 million is 30 million 30 million [Music] and again that's telling me that the value of this 3 here is 30 million then plus we have 6 times 1 million which is 6 million 6 million so the value for this 6 is 6 million then plus we have 2 times 100 000 or 200 000. the two is worth what two hundred thousand then next we have five times ten thousand so plus fifty thousand fifty thousand so this five is fifty thousand and plus we have five times one thousand which is five thousand so five thousand this five is a five thousand and then we have plus nine times a hundred or nine hundred so this nine is worth nine hundred and then plus we have a two in the tens place so two times ten or Twenty so this two is a twenty and then lastly we just have plus seven right seven times one is seven this is just a seven so your number 36 million 255 927 written in expanded notation looks like this 30 million plus 6 million plus two hundred thousand plus fifty thousand plus five thousand plus nine hundred plus twenty plus seven so again just breaking it down and showing you the true value of each digit in that number okay for the last example let's look at one that's really big so we have 10 billion 5 million three hundred twenty seven thousand five now the good thing about this number is I included a lot of zeros and when you have zeros and you write a number in expanded notation you can skip them so in other words I'll have this one here this is in the 10 billions place because essentially this is the ones the tens the hundreds this is the thousands the ten thousands the hundred thousands this is the millions the ten millions the hundred millions then this is the billions and the ten billions okay so this one is in the 10 billions place so I'd have one times ten billion [Music] then plus now all these zeros are basically just placeholders so when we write the number and expanded notation we don't have to put them so I can kind of just skip down to the next non-zero number so that's a five and that's going to be in the millions place so 5 times 1 million then plus next I have a 3 in the hundred thousands place so three times one hundred thousand and plus I have a two in the ten thousands place so two times ten thousand and plus I have a seven the thousands place so seven times one thousand then plus I have a five in the ones place so five times one let's do a little bit of multiplication here one times ten billion is just 10 billion okay then we have plus five times one million is five million [Music] then plus three times one hundred thousand is three hundred thousand and plus two times ten thousand that's twenty thousand and plus seven times one thousand is seven thousand and plus five times one which is five so again all we're doing is we're breaking the number down to see the value for each digit in this number so this one is a ten billion this five is a 5 million this three is a three hundred thousand this two is a twenty thousand this seven is a seven thousand and then lastly the five is just a five because again it's in the ones place five times one is just five hello and welcome to pre-algebra lesson three in this video we're gonna learn about inequalities and the number line all right so the lesson objectives for today we want to learn how to create and work with a number line for the whole numbers and then also we want to learn about inequalities in a few symbols we're going to work with so previously in our course during the lesson on place value we introduced the whole numbers so the whole numbers start with zero so that is the smallest whole number and then they increase in increments of one out indefinitely so after zero comes one then two then three then four then five you just pick a spot because you can't list all of them and you're gonna stop and you can put the three dots there just to show that the pattern contains forever so after five comes six then seven then eight so on and so forth now we can visually represent our whole numbers using a horizontal number line and so the left most Notch here is going to correspond to the smallest whole number which is zero so see this guy would correspond to this guy here and then as we move to the right on the number line each additional Notch is corresponding to the next largest whole number so this going to the right by one unit this would be one the next largest whole number is one then this Notch going to the right would be two the next largest hole number is two so on and so forth you're just increasing by one each time so you go to three then four then five then six so on and so forth now in a similar way to this guy right here where we put the three dots we need to show that the whole numbers continue going out to Infinity so basically they continue forever so what we're going to do here is draw this little arrow and say that hey after 10 comes 11 then 12 then 13 so on and so forth now there's a few things we want to think about when we look at the number line first off numbers increase from left to right so you can pick any two numbers you want and just think about them for a moment let's say I start with something easy like one and two well I know that 2 is is a larger number than one and notice how 2 is to the right of 1 on the number line let's just pick two other ones so let's say we look at six and seven well seven is a larger number than six and notice that seven is to the right of six on the number line so as we move to the right numbers are increasing in value so what you can say is that if a number is to the right of another number on the number line it is a larger number then additionally we can say that numbers decrease from right to left so if I'm going this way numbers are decreasing in value let's take four and five well four is to the left of five on the number line and four is a smaller number so what you can say here is that if a number is to the left of another number on the number line it is a smaller number all right before we move on with the lesson we need to introduce a few new symbols so the first symbol I'm going to talk about is the equality symbol a lot of people just call this the equal sign or equals so let me put that up there I'm sure you've seen that before so this means is equal to so whatever's on the left is equal to whatever's on the right you could also think about this as saying whatever's on the left is the same as whatever's on the right now it's not always going to be very straightforward so let's say I start with something very easy to identify as being equal or the same so I'm going to put a 5 here and a 5 here so obviously 5 is equal to 5 or 5 is the same as 5. if I have five dollars in my wallet and my friend has five dollars in his wallet well we have an equal amount of money or the same amount of money we can each go to the store and buy five dollars worth of stuff now let's say we change up the way this looks a little bit and to give you an example let's say I keep my five dollar bill in my pocket but my friend goes to a bill changer and he exchanges his five dollar bill for five singles or you could say five one dollar bills one way you could write this is by showing this as one plus one plus one plus one plus one so now it does doesn't look as straightforward as this but these two sides represent the same value in the end if he goes through and adds up his singles he still has five dollars right this is one plus one which is two plus one which is three plus one which is four plus one which is five so each of us can go to the store and spend five dollars on stuff we have the same amount of money or an equal amount of money so five is equal to or the same as one plus one plus one plus one plus one at the end of the day the value on each side is five so another symbol we're going to use is the not equal to symbol so here we're going to draw the equal symbol and we're going to put a little slash through it so this means is not equal to so whatever's on the left is not equal to whatever's on the right now it doesn't matter what's on the left and what's on the right as long as they're not equal so you can have a smaller number on the left and the larger number on the right or you can put the larger number on the left and the smaller number on the right it does not matter so what I'm going to do is say something like let's say 5 is not equal to 7. so let's say my friend got two additional dollars and so now he has two dollars more than me to spend he could spend seven dollars but I can only spend five dollars so the amount of money we have is not the same it's not equal again this can also be more complicated so let's say that I have a 5 here and let's say over here keeping with what we did a moment ago let's say we have one plus one plus one plus one plus one so that's my five ones and let's add an additional two here so here we have one plus one which is two plus one which is three plus one which is four plus one which is five plus one which is six plus one which is seven so five is not equal to at the end of the day seven so it doesn't matter if it's straightforward like this you're going to see a lot of stuff that's not straightforward and you need to go through and think about well what is the value of this once I perform this operation well this would be 7 and this is five and five is not the same as seven so here we can show that with the not equal to symbol all right now the next two symbols we'll use for the rest of this lesson these are called the strict inequalities we'll talk about non-strict inequalities later on but for right now when you talk about a strict inequality you're talking about a less than symbol or a greater than simple so let's start with a less than symbol so this right here means is less than so whatever's on the left is less than whatever's on the right so the way you remember this when you're looking at these symbols this guy right here is always pointing to the smaller number so if I grab two numbers from the number line let's say 2 and 3 again 2 is smaller than 3 because 2 is to the left of 3 on the number line in order to make this a true statement I need to put that smaller number on the left and the larger number on the right so it's very important that you do it this way if you put here that 3 is less than 2 that's a false statement okay so with this one you want to make sure that the smaller number is on the left and the larger number is on the right so you can say that in this case 2 or whatever that smaller number is is less than the 3 or whatever that larger number is then the other guy is going to be the greater than symbol so this means is greater than so now the number on the left is larger than the number on the right so here the number on the right needs to be smaller the symbol always points to the smaller number here it points to 2 because 2 is smaller than 3. here it's going to point to you let's just pick two numbers let's say we go with eight nine so here we would point to eight because Eight's a smaller number it's to the left of 9 on the number line 9 needs to go over here because it's the larger number so we would say 9 is greater than eight so whatever the bigger number is goes on the left and that's greater than whatever the smaller number is that's going to go on the right all right let's go through some examples I think you're going to find this very very easy but something you'll definitely see in your textbook so we want to replace each question mark with the inequality symbol less than or greater than all right for the first example we have five question mark two so let me write this over here five and then we have a question mark and then a two so first let's Circle five and then let's Circle two so five is to the right of 2 on the number line so 5 is a larger number than two so here I would want to use a greater than symbol again if you can't remember the symbol always points to the smaller number 2 is a smaller number than 5 so just point the symbol to the smaller number so here you would say 5 is greater than 2. now if we flip this around and let's say the problem was 2 question mark five then now we want to use a less than symbol always point the symbol to the smaller number so this guy right here would Point towards two two is less than five here five is greater than two with a greater than the larger numbers on the left with a less than the smaller numbers on the left all right for the next problem we'll look at three question mark zero here's three here's zero so three is to the right of zero on the number line so 3 is going to be a larger number so we would say that 3 is greater than zero again that symbol is going to point let me use a different color here is going to point to that smaller number zero is the smaller number if you flip this around and let's say you got zero and then question mark three well now I want to use a less than 0 which is the smaller number is less than three the symbol always points to the smaller number what about the next one we have four question mark eight so four question mark eight here's four here's eight so four lies to the left of eight on the number line so it's a smaller number so here we would want to use the less than symbol again the symbol is going to point to the smaller number if we had gotten eight question mark four well now we want to use the greater than symbol 8's a bigger number it lies to the right of four in the number line so we we would want to say 8 is greater than 4. what about 7 question mark three so seven question mark three here's seven here's three seven lies to the right of three on the number line so with a bigger number so you wanna use a greater than there again the symbol always points to the smaller number so the symbol would point to three if you had gotten three question mark seven well now you want to use your less than symbol to say three is less than seven symbol always points to the smaller number what about one question mark two here's one here's two so one is to the left of two on the number line so you would say one is less than two symbol always points to the smaller number if you got two question mark one well then you want to use the greater than symbol 2 is a larger number than one the symbol always points to the smaller number the smaller number here is one hello and welcome to pre-algebra Lesson Four and in this video we're going to learn about rounding whole numbers so our lesson objective for today is simply to learn how to round whole numbers so before we get into the procedure that we use to round a whole number let's talk a little bit about why we would round whole numbers so in many situations an exact value is not needed in these cases an approximation is easier to use and remember and when we talk about approximations we're talking about a value that is close to the original value but not exactly there all right so rounding a whole number the first thing you want to do is locate the digit in the round off place so for example if we're rounding to the nearest ten you would start by locating the digit in the tens place when you first start out I would Circle that digit or highlight it or do something to say hey this is my round off place so kind of step two is to look at the digit to the right of the round off place so if the digit to the right of the round off place is and you have two different scenarios the first one is four or less you want to leave the digit in the round off Place unchanged then the second scenario that could occur is that it's five or larger in this case we increase the digit in the round off place by one okay increase the digit in the round off place by one and then your third and final step is pretty easy you just replace each digit to the right of the round off place with a zero okay here we want to round 631 to the nearest ten so again I'm looking for the tens place and that's going to be this three here and then in step two what I want to do is I want to look at the digit to the right and that's going to be this one now does this one fall in the category of four or less or does it fall in the category of five or larger well it's clearly in the category of four or less so when that happens I want to leave the digit in the round off place which is the 3 right it's in the tens place unchanged so I would have six and then the 3 is going to be unchanged and then in step three I'm going to replace each digit to the right of the round off place with a zero so the one is to the right of the round off place I'm going to replace that with a zero and so 631 rounded to the nearest ten is 630. okay next we want to round 7952 to the nearest thousand so seven thousand nine hundred fifty two and we're looking for the digit in the thousands place so that's going to be this 7 here and then we look for the digit immediately to the right of that so that's this 9. and obviously the 9 falls in the category of 5 or larger so when this occurs we increase the digit in the Randolph Place by one so I'm going to increase that seven by one seven plus one is eight and then I replace each digit to the right of the round off place with a zero so I'm going to replace the nine with the zero the five with the zero and the two with a zero and so 7952 rounded to the nearest thousand is eight thousand okay next we want to round eight hundred thirteen thousand two hundred seventy-five to the nearest ten thousand all right so we have 813 275 and the ten thousands place is right here so I'm looking at the digit that follows that and that's this 3 here three falls in the category of four or less so when this occurs we're going to leave this digit in the round off Place unchanged unchanged so the eight and the one are going to stay unchanged I don't need to do anything to those but I'm going to replace each digit to the right of the round off place with a zero so the 3 is to the right of the one so I'm going to replace that with a zero and I'm going to replace the 2 the 7 and the five and so 813 275 rounded to the nearest ten thousand is eight hundred ten thousand okay for the last problem we want around 61 million 5387 to the nearest million so 61 million 5387 to the nearest million so the millions place is right here that's the one and we want to look at the digits to the right of the one so I'm looking at this digit here which is a zero zero is obviously in the category of 4 or less so we're going to leave the digit in the round off Place unchanged unchanged so basically the six and the one will be unchanged and then everything that follows the one which is the digit in the round off place will just be a zero so I'll have 0 0 0 comma zero zero zero so 61 million 5387 rounded to the nearest million is 61 million hello and welcome to pre-algebra lesson five and in this video we're going to learn about the properties of addition okay so for the lesson objectives we want to learn about the identity property of zero we want to learn about the commutative property of addition we want to learn about the associative property of addition and then also we want to learn to identify the parts of an addition problem okay so let's begin the lesson by thinking about what addition actually is so addition is the operation that allows us to group items together so if we think about this in a very simple way kind of the way that you would think about it if you're in grammar school we have this picture here and we have three boxes and we're adding to that two more boxes so we can just go through and count how many boxes we have we can just say okay I have one two three boxes right from up here and then I added to that two more so then four and then five and at this point we all know that three plus two is five but this just visually represents what's going on when we add we're trying to get a total for all the amounts that we've combined together right I've combined this group of three boxes with this group of two boxes and it gave me a total of five boxes okay so let's talk about a little bit of vocabulary so in an addition problem the numbers being added together are called add-ins okay add-ins and then the result of an addition operation is known as the sum okay the sum so let's look at a quick problem we want to identify each part of the addition problems below okay so we're going to start with 5 plus 6 equals 11. and all we're going to do is just label so we know that the numbers being combined together or added together are called add-ins so 5 and 6 would be add-ins and then 11 which is the result of the addition would be called the sum okay let's look at the next one we have 12 plus 8 equals 20. so 12 and 8 are the numbers being added together those are called add-ins so these are the add-ins and again the result of the addition is called the sum so 20 is the result of 12 plus 8. so this is the sum okay so the first property we're going to learn about today is called the identity property of zero and it's really pretty simple it just tells us that adding 0 to any number leaves the number unchanged okay so I can add 0 to any number and it will remain unchanged so let's take a look at some examples here so 5 plus 0 equals what well adding 0 to 5 leaves five unchanged so 5 plus 0 equals five seven plus zero equals what adding zero to seven leaves seven unchanged so seven plus zero equals seven and then one million 365 371 plus zero again adding zero to anything leaves it unchanged so it would just be one million three hundred sixty five thousand three hundred seventy one as the answer okay the next property is also very simple it's called the commutative property of addition and the commutative property of addition tells us we can add numbers in any order without changing the sum so the order that you add does not matter whatsoever and kind of to prove this I have some examples so we have six plus three equals three plus six basically the order is changed so 6 is on the left here 6 is on the right over here it doesn't matter you're going to get nine either way right if I say six plus three what does that equal it equals 9 if I say 3 plus 6 what does that equal it equals 9. so changing the order of the add-ins does not change the sum again as another example 7 plus 9 equals nine plus seven so changing the order here of the add-ins does not change the sum 7 plus 9 is 16 and 9 plus 7 is 16 as well and as a final example we have 12 plus 5 equals 5 plus 12. again switching the order or changing the order of the add-ins does not affect the sum 12 plus 5 is equal to 17. 5 plus 12 is also equal to 17. okay so the last property we're going to talk about today is another easy one it's known as the associative property of addition and the associative property of addition tells us that the addition of three or more numbers can be grouped in any order without changing the sum so the way that we group numbers when we're adding does not change the sum so let's look at some examples of this we have three plus six here on the left and that's inside of parentheses and then we're adding to that two and we're saying that's equal to 3 plus and here we have six plus two inside of parentheses on the right now the associative property is telling us that since we're adding the same three numbers three plus six plus two it doesn't matter what we put parentheses around we're going to get the same result we haven't done a lesson on the order of operations yet but the order of operations tells us that if you have operations inside of parentheses you have to do them first so if we were to follow that rule we would say three plus six is nine we would do that operation first and then we would add 2 to that so 9 plus 2 gives me 11. kind of the left side here gives me a result of 11. on the right I would do 6 plus 2 first because that's inside of parentheses 6 plus 2 is 8 so we would have 3 plus 8 and that equals 11 as well so it didn't matter that we had parentheses here on the left around 3 plus 6 and parentheses on the right around six plus two we got the same result either way and that's exactly what the associative property of addition is telling us okay for another example we have seven plus five plus two and five plus two is inside of parentheses and this is equal to seven plus five now seven plus five is inside of parentheses plus two so seven plus and a lot of times you'll hear that people say the quantity when something's inside of parentheses so seven plus the quantity five plus two five plus two is done first five plus two is seven so we get seven plus seven and that gives us fourteen so the left side here is 14. what's the right side going to be it's going to be 14 as well the quantity seven plus five seven plus five is twelve again we're doing that first because it's inside of parentheses and then plus 2 gives us 14. so again we group this different in each case we have 5 plus 2 inside of parentheses on the left we have 7 plus 5 inside of parentheses on the right but it didn't matter right because we got 14 either way okay as our last example we have 15 plus 9 inside of parentheses so the quantity 15 plus 9 plus 2 and this is equal to 15 plus now we have parentheses around nine plus two so 15 plus the quantity nine plus two so on the left here we start with 15 plus 9 because that's in parentheses so the quantity 15 plus 9 is 24. 24. then plus 2 would give me 26. on the right I'm going to do 9 plus 2 first so the quantity 9 plus 2 is 11 so we'd have 15 plus 11 and of course that gives us 26 as well so again that's what the associative property of addition is telling us we have the same three numbers being added 15 9 and 2. we're going to get 26 no matter how we group the addition we can put parentheses around 15 plus 9 we can put parentheses around nine plus two it doesn't matter we get 26 as a result either way hello and welcome to pre-algebra lesson six and in this video we're going to be learning about adding whole numbers using a number line so the lesson objective for today will be to just learn how to add whole numbers again using a number line so during a previous lesson we talked about using a number line to visually represent a group of numbers so let's look at a number line that shows the whole numbers and again I've talked about the whole numbers in previous lessons the whole numbers are a group of numbers that start with zero so zero is your smallest whole number and then they increase in increments of one so after zero you get one then two then three then four again just increasing in increments of one and this continues forever and that's why we have the three dots here when you see the three dots in this case it means that the pattern which we're increasing in increments of one is going to continue forever so after 4 comes five then six then seven so on and so forth so if we look at the whole numbers using a number line again this is a a number line This Is A visual representation of the whole numbers so you'll see the leftmost whole number is zero that's your smallest whole number and you see you have a little Notch here now every time you have another notch As you move to the right you're going to the next largest whole number so you go to one then you're going to two and you're going to three and you go into four so on and so forth so it's impossible to list all the whole numbers so basically what we do is at some point we just stop and it can be basically anywhere I could have stopped this at five you just draw an arrow to say hey this is going to continue forever okay so we can use the number line to visually show the addition of whole numbers so this may seem like a trivial exercise but it will help us a lot when we start working with integers so this is going to seem kind of like it's a waste of time to you at first but when you start working with integers especially for the first few times that you work with integers with addition and subtraction it's a lot easier to start out doing it on a number line and if you've already had that experience doing it with whole numbers it'll make it a little bit easier for you when you do it with integers so how do we add whole numbers on a number line well we start at the leftmost number so for example if you have three plus seven you would start out at three on the number line and then you move to the right by the number of units being added so we're adding seven to three so we would just move to the right by seven units it's actually very very very simple so let's take a look at some examples now we want to add two plus six using the number line so the leftmost number the two is where we're going to start on the number line so we're just going to start right here and then we're adding six okay we're adding 6. so I'm just going to go 6 units to the right and I'll be at my answer so I'm going to go one two three four five six units to the right I'm at eight so that is my answer and all of us know at this point that two plus six is eight so two plus six equals eight it's just to get the practice of doing it on the number line okay now we want to add four plus seven using a number line and so I have a number of lines a little bit larger it's a little bit more crowded we can still find everything we're adding four plus seven so we're going to start at the leftmost number or the four and that's going to be right here this is your four and then you're adding seven so we're just going to move seven places to the right so each Notch is a place or increase by one so one two three four five six seven so that's going to put us right here at 11. okay at 11. and again we all know that four plus seven is eleven again it's just to get the practice of doing it on the number line so four plus seven equals eleven okay now we want to add 6 plus 5 plus 1 using a number line although we're adding three numbers the process is basically the same we're starting at this leftmost number so we're starting at six so that's going to be right here and then we're going to go five places to the right because we're adding five so one two three four five so that puts us at 11. and we all know that six plus five is eleven now we're adding one more so we just go one more place to the right so one more place to the right that puts us at 12. and so our answer for 6 plus 5 plus 1 is of course equal to 12. okay now we want to add three plus two plus one plus seven again using a number line so let's find three that's where we're going to start the leftmost number and that's right here that's this guy and we're going to move two places to the right because we're adding two so that's going to go one two places to the right and now we're adding one so we're gonna go one more place to the right and then we're adding seven so we're going to go seven places to the right one two three four five six seven so we end up here at 13. so 3 plus 2 plus 1 plus 7 equals thirteen and I kind of checked this since we can all do addition three plus two is five five plus one is six six plus seven is in fact thirteen essentially you're just starting at the leftmost number of the addition problem every time you add something you just move that number of places to the right on the number line so we started at three we moved two places to the right to get to five then we moved another place to the right to get the six and then we moved seven places to the right to get to our final answer of 13. okay let's look at the final one we want to add 2 plus 3 plus 5 plus 4 plus 7 plus 1 using a number line so again we're going to start at the leftmost number so that's two so we'll start out here on the number line and every time I add something I just move that number of places to the right so I'm adding three so I just want to go three places to the right so one two three places to the right so that put me at five then I'm adding five so I'm going five places to the right let me just change color so one two three four five places to the right so that would put me at ten then I'm adding four so now I'm going to go four places to the right so one two three four places to the right that's gonna put me at 14. I'm adding seven to that so that's going to put me seven places to the right one two three four five six seven places to the right that's going to put me at twenty one and then lastly I'm going to add one more to that so that is going to put me at 22. so that's my final answer here it's 22 so we have two plus three plus five plus four plus seven plus one and the answer is 22. and to check that just go through and do your addition two plus three is five five plus five is ten ten plus four is fourteen fourteen plus seven is twenty one twenty one plus one is twenty-two so again adding whole numbers on a number line seems pretty trivial it's actually very very easy but it's going to help us when we get to adding integers and again what we do is we start at the leftmost number of the addition problem I want the number line and then every time we add a number we move to the right on the number line by that number of units hello and welcome to pre-algebra lesson seven and in this video we're going to learn about vertical addition so the lesson objectives for today would be to learn how to properly set up and perform vertical addition and also to learn the process of carrying okay so when are we going to use vertical addition well vertical Edition is a process we use when we're adding multi-digit numbers together in a multi-digit number is any number that is larger than one digit so to perform the vertical addition the first step is to arrange the numbers to be added vertically okay vertically and by place value so when I say vertically I mean we're going to stack the numbers on top of each other let me just show you real quick with an example let's say we're adding 13 to 24. so 13 plus 24 is the problem in this first step we're just going to put the numbers on top of each other doesn't matter which numbers on top of which numbers on bottom so 13 and 24 we're going to write like this and there's more to it but we'll get to those in the next steps now notice how when I stack these numbers on top of each other the digits in the ones place for each number line up and the digits in the tens place for each number line up this is your ones and this is your tens let me just make this a little cleaner this is the ones okay so that is the first step stack the numbers on top of each other and buy place value so don't do something like this where you have 13 and 24 over there that won't work again vertically and then by place value all right so let's take a look at the Second Step so let me just rewrite this we have 13 and we have 24 written like this at this point and for step two we want to draw a horizontal line underneath the bottom number so let's do that and place a plus symbol okay this is the plus symbol to the left of the bottom number so to the left of this bottom number I'm just going to put a plus symbol and now we have our addition problem set up and in step three we're gonna learn how to actually perform the addition okay so for step three let me just rewrite this we have 13 plus 24. we want to add the numbers in each column starting with the ones column and moving left so that means we're going to start in this column you always work right to left when you're doing vertical addition now we want to place the result directly below the horizontal line so I'm going to start with the numbers in this column here we're starting in the ones column so what is 3 plus 4 3 plus 4 is 7. so all I'm going to do is I'm going to put that directly below this horizontal line and I got to stay in line basically this 7 is going to be in the ones place for the answer and then I just move left so next I'm going to do the addition in the tens column so I'm doing the addition here so now I'm doing 1 plus 2 and that gives me 3. so when we add 13 and 24 together we get 37. okay so let's take a look at some examples here we're going to start with some very easy problems and then we'll progress to some that are a little bit more challenging so we have 27 plus 12. again we're going to begin by stacking the numbers on top of each other so 27 can go on top 12 can go on the bottom and again you can reverse that order you can put 12 on the top and 27 on the bottom it really doesn't matter now we're going to draw a horizontal line beneath the bottom number and we're going to put a plus symbol off to the left of that bottom number now we start by adding the numbers in the ones column so we have 7 plus 2 and that gives us 9. put your answer down in the ones column just keep following straight down then we move to the left so now we're going to look at adding the numbers in the tens column so 2 plus 1 is 3. so 27 plus 12 is 39. let's kind of break this addition problem down a little bit further if we think about 27 plus 12 again we did it like this 27 plus 12. we added in the ones column seven plus two is nine we add in the tens column two plus one is three we got an answer of 39. but if we think about 27 it's what it's two tens or 20 plus seven ones or seven if we think about 12 it's 110 or just 10. plus two ones or two so we think about adding these numbers together just looking at them like this it makes sense to have okay we want seven ones plus two ones to know the amount of ones that we're going to have in our answer so that's why those are lined up so seven plus two is not you get nine once and it makes sense to think about the tens together as well we have two tens plus one ten or we have basically 20 plus 10 or 30. so we have three tens and when we think about this this is basically 30 plus nine or Thirty naught so very important to translate this longer process into this shorter process here this is just a shorthand for this over there now what if you didn't stack the numbers up by place value you'd run into a mess you'd never get the right answer if we think about 27 plus 12. let's say I did it like this let's say I put 27 here and I don't know let's say I put 12 over here would I get the right answer well no because now I think I only have seven ones so this would just come straight down and now I think I have two tens plus another two tens or four tens or 40 and this would come down so I get an answer of 147 which is completely incorrect right completely incorrect so that's why it's important to line up your numbers by place value it's going to tell you how much of each you're going to have how many ones how many tens how many hundreds that you're going to have in your answer okay let's look at the next one we have 405 Plus 314. so again we're going to line these up vertically and by place value so everything is lined up we got the ones the tens and now the hundreds so we work right to left so we're going to start by adding the numbers in the ones column what is 5 plus 4 that is nine so you're going to write your nine in the ones column for the answer five ones plus four ones going to give me nine ones then we're just working our way to the left so now we're going into the tens column 0 plus 1 is going to give me one and we worked our way to the left again now we're in the hundreds column four plus three is going to give me seven and So my answer is 719. and again just thinking about this we had five ones plus four ones that gave us nine ones we had zero tens plus one ten which gave us one ten then we had four hundredths plus three hundreds which gave us seven hundredths so essentially we have seven hundred plus ten plus nine or the number 719 as our result okay now we have two thousand fifty Seven plus fourteen thousand one hundred twelve so again we're going to stack these numbers on top of each other and buy place value but one thing you're going to notice here is that one of the numbers has fewer digits than the other typically when this happens you put the number with more digits on top so typically you're going to see 14 112 on top and 2057 on the bottom but I'm going to do this addition problem twice and show you that it doesn't actually matter this is just a little bit easier so we're going to start in the ones column two plus seven is nine and we're going to the tens column one plus five is six and we're going to the hundreds column one plus zero is one and then I'm going to put comma there and now I'm going into the thousands column so four plus two is six and then I'm going to the ten thousands column and you'll notice how there's nothing here to add with one typically we'll teach this as okay just bring the number down and you'll put a one there so your answer is 16169. let's say I change what's on top let's put 2057 on top and let's put fourteen thousand one hundred twelve on the bottom kind of move that over a little bit and it's the same thing we start in the ones column seven plus two is nine then we move to the left five plus one is six zero plus one is one two plus four is six and then bring down the one we get sixteen thousand one hundred sixty nine either way because typically you're gonna see this setup because the number with more digits is generally going to be on top okay now we want to look at a problem that's going to involve something known as carrying so I'm going to work through this problem and then I'm going to go back and explain how it works so 27 plus 35 is what we're going to work on so 27 plus 35. so what happens is when we try to add the numbers in the ones column seven plus five you get 12. I can't fit a 12 in this space right here I can only fit numbers 0 through 9. I can only fit a digit in this space so 12 is too big so what I'm going to do I'm going to break 12 apart I'm going to think about the left number and I'm going to think about the right number so the right number is going to stay it's going to stay right here in the answer column just like you normally would do and then the left number is going to get carried into the next column okay so we're going to carry the one into the next column and now I'm just going to include it with the addition of that column so 1 plus 2 plus 3 1 plus 2 is 3 3 plus 3 is 6. so we end up with 62. but let's think about what we actually did here let me just erase this real quick and think about what we actually did when we look at 27 and we break it down we have two tens plus seven ones okay so this is twenty plus seven and when we think about 35 we have what we have three tens or we have Thirty plus we have five ones or just five so when we add the numbers in the ones column we have seven seven ones and we have five ones seven plus five is twelve so we would have twelve once basically all we're doing is rewriting 12 ones 12 ones as 110 plus two ones okay 110 plus two ones so when we think about the result here if I give you a ten dollar bill and two ones that's the same as if I handed you twelve ones it's the same so all I'm doing here is I'm just going to scratch this out instead of putting 12 ones I'm going to put two ones and then I'm going to go into the next column into the tens column and say hey I have another 10. I have another 10. so now the result of the addition we have 10 20 and 30 that we're adding 10 plus 20 is 30 30 plus 30 is 60. so we'll end up with 6 tens or 60 plus two ones which is 62. that's what we got so this is essentially what we're doing when we're using the carrying process it's just a shorthand way to kind of go through and do all this and evaluate place values let's take a look at another one okay next we have 242 Plus 977. so 242 Plus 977. so again we're going to start by adding the digits in the ones column so 2 plus 7 is 9. and then we work our way to the left so next we're looking at the tens column so we have 4 plus 7 but that's 11 and I can't fit I cannot fit an 11 right there you cannot fit anything larger than a 9 in this space so what we need to do is evaluate the place values and figure out what we need to do so we use carrying to do that so if I have 11 I take the rightmost digit of 11 and that's a 1 and I put that in the answer column so that goes there then I carry the leftmost digit which is also a 1 into the next column so that's going to go right here and I'm just going to add that when I add this column together so 1 plus 2 is 3 and then 3 plus 9 is 12. now when you don't have any more numbers to the left here I can just simply write a multi-digit number into the answer and the reason for that is because I'm still maintaining place value I'm still having a 2 in the hundreds column and this one is just pushed out into the thousands column because essentially if there are more numbers over here but one would have got carried over but there are none so it just comes straight down so we end up with our answer of 1219 and to think about this carrying action just a little bit more when we look at the result of this column here which was four tens four tens plus seven tens really we got 11 tens right we got 11 tens which essentially all we did was we broke it down into 100 plus one ten if I gave you 11 10 bills you would have a hundred ten dollars if I gave you a hundred dollar bill and a ten dollar bill you'd still have a hundred ten dollars so it's the same either way so this is exactly what we did we put 110 right here and then we carried 100 into the next column to be added with the hundreds okay for the next one we have six thousand ninety nine plus fifteen thousand seven hundred sixty three and as I told you earlier typically you're going to take the number with more digits and put that on top so we'll take fifteen thousand seven hundred sixty three and put that on top we'll take 6099 and we'll put that on the bottom and let me just make this a little cleaner so again we're going to start with the digits in the ones column 3 plus 9 is 12. 12 is too big to fit in the answer column so I keep the rightmost digit which is a 2 right we think about 12 the rightmost digit is a 2 and then I carry the leftmost digit which is a 1 into the next column now I add 1 plus 6 is 7 7 plus 9 is 16. so again I have to carry so I'm going to put a 6 down right think about 16 the rightmost digit is a six and again I'm going to be carrying a one and now I'm going to add 1 plus 7 is 8 8 plus 0 is still eight and just keep moving on to the left now we have five plus six that's 11. so you think about 11 you have a one that's going to go into the answer column the rightmost digit and then you have a one that's going to get carried over into the next column so now we add this column one plus one is two and we just bring that down so we get 21 862 as our answer hello and welcome to pre-algebra lesson eight and in this video we're going to learn about subtracting whole numbers using a number line so for the lesson objectives we want to learn how to identify the parts of a subtraction problem you'll also want to learn how to subtract whole numbers using a number line so what is subtraction well subtraction is the mathematical operation that allows us to take away the main idea behind subtraction is to find out how much is left over so suppose you have three one dollar bills and you go to a fast food chain and you spend two dollars of your three dollars on a hamburger how much would you have left so you can kind of picture these three one dollar bills with two crossed out now because you've spent those well you just have one left and this is basically the subtraction problem three which is what the amount you start with minus 2 which is the amount you're going to spend or the amount that's going to be taken away and that's going to be equal to 1 which is the amount that you're going to have left over after you've spent the money so again subtraction is the operation that allows us to take away in the example 2 was taken away from a starting amount of three again we started with three dollars we spent two that's what's taken away and then the result was one that's what we had left over we just had a buck left okay so to be more formal let's give names to the parts of a subtraction problem so we're going to start out with this called the menu end okay the menu end and this is the whole amount or starting value so in the problem 3 minus two equals one the whole amount of starting value is three because I started out with three dollars so this is the menu end this is the menu end then the next part of the subtraction problem is called the subtrahend okay the subtrahend so this is the amount being taken or subtracted away so in the problem three minus two equals one two is the subtrahend this is the subtra and okay that's the amount that we took away to buy the hamburger then finally we have the difference okay the difference this is the result of the subtraction operation basically what we have left over so three minus two equals one the one here the result or what we have left over is the difference okay so let's just label these parts too again this is the subtrahend and this is the menu end okay let's look at a few examples we want to name the parts of the following subtraction problems so we have 13 minus 7 equals 6. so 13 is your menu end your menu end okay this is the number from which another is subtracted from or your starting amount or the whole amount that you have however you want to think about this this number 13 is your menu Land then we're subtracting away 7 from 13. 7 is what we're taking away so this is the subtrahend this is the sub trehend and then finally 6 is the result of the subtraction operation here and so it's going to be the difference okay so 6 is the difference and again you can also think about the differences just being the amount left over when you're finished performing the subtraction I started with 13 that's my menu end I took away seven that's my subtrahend and I have six left over after that operation that's my difference okay now we'll look at eight minus three equals five the amount we're starting with or your whole amount from which another subtracted is eight That's the menu end then we have three that's what we're subtracting away so eight minus three okay we're taking away three that's the subtrahend and then lastly we have this 5 here that's the result of our subtraction operation we start with eight we take away three the result is five so five is the difference okay the next thing we want to talk about if you remember in addition we have something called the commutative property and the commutative property tells us that we can add in any order that we'd like and we don't change the sum so in other words three plus nine is equal to nine plus three right three plus nine equals twelve nine plus three equals twelve changing the order around doesn't affect what's going on with my answer now what I want to draw your attention to is that it's very important to note that the commutative property that is associated with addition does not work with subtraction does not work with subtraction so in most cases if you change the order of your subtraction you're not going to get the same answer so I have some examples of this and I started out with the addition one just to show you that it does work for addition 3 plus 6 equals nine six plus three equals nine this works out 6 minus three is three three minus six is negative three and we haven't covered negative numbers yet or integers or anything just take my word for it it's negative three so these two values are not the same so the way that we subtract here made a difference if I change the order meaning I put I go from having three lasts to three first I don't get the same result same thing over here nine minus two is seven two minus nine is negative seven so changing the order did change the result then lastly we have seven minus five is not equal to five minus seven seven minus five is two five minus seven is negative two so again changing the order does change the result when you're subtracting in almost all the cases there are some special cases where it's not but we want to have things as a general rule so as a general rule when you're adding you can change the order around as much as you want as a general rule when you're subtracting you cannot do that okay so now let's talk a little bit about subtracting whole numbers on the number line this exercise will help us greatly when we start working with integers so we already practiced adding whole numbers on a number line yeah it was a trivial thing but it's going to help us when we get to integers and it's going to be the same thing for subtracting we need to get a foundation going so that when we get to integers things flow more smoothly okay so now we want to subtract each using a number line and to perform subtraction on a number line it's essentially the same as performing addition on a number line it's just a little bit of a change on the second step so let's just learn through an example we have 7 minus 4. the first thing you're going to do is you're going to start at the leftmost number on the number line so if you recall from adding on a number line that step is the same you always start at the leftmost number so that's going to be right here that's 7 on the number line now when we add so in other words if we're adding 4 we move to the right by four units on the number line but now since we're subtracting 4 okay since we're doing subtraction we're just going to the left by four units so the only thing that's changing is instead of going to the right we're going to the left other than that the procedure is the same so we're subtracting 4 we're just going to go one two three four units to the left so that's going to put us at three and obviously we know seven minus four is three but it's just cool to do it on a number line so this is our result so we have seven minus four equals three okay now we have 12 minus five so we're going to start out at the leftmost number so we're starting out at 12 on the number line that's right here and then we're subtracting away five so we're going to move five units to the left so I'm gonna go one two three four five units to the left and I end up at seven and of course we all know twelve minus five is seven but we just did it on another line so 12 minus five equals seven okay now we have eight minus seven so again we're starting out at the leftmost number on the number line so we're starting out at eight so that guy's right here and we're subtracting away seven so we're moving seven units to the left so we're gonna go one two three four five six and then finally seven units to the left and we end up at one all right we end up at one and of course we know that eight minus seven eight minus seven is equal to one okay we'll look at one last problem overall this is pretty simple right if you're adding on a number line you start out at the leftmost number you go to the right by the number of units you're adding if you're subtracting on a number line start at the leftmost number you go to the left now by the number of units that you're subtracting away so we have five minus five five minus five so that means I'm going to start out at 5 on the number line that's right here and then if I'm taking away five we know that we're going to end up at zero right because a number minus itself is zero so let's go five units to the left one two three four and five okay so we end up at zero right here and five minus five equals zero hello and welcome to pre-algebra lesson nine and in this lesson we're going to learn about vertical subtraction so our lesson objectives for today would be to learn how to subtract multi-digit numbers and also we want to learn the borrowing procedure so in a previous lesson I taught you how to do vertical addition and vertical addition is a convenient way to add multi-digit numbers together and again when I say multi-digit numbers I mean numbers that are larger than one digit so vertical subtraction is essentially the same thing it's a convenient way to subtract multi-digit numbers but there's some differences that we need to cover so let's explain through an example here the first step for vertical subtraction we want to line up the subtraction problem vertically and by place value so you'll recall with vertical addition it's the same first step so let's say that we have I don't know 47 plus 13. for the addition problem and let's say that we have 47 minus 13 for the subtraction problem so here's the main thing that you need to understand when we do addition addition is commutative so the order that we add in does not matter so in other words when I go through and I stack these numbers on top of each other it doesn't matter what I put on the bottom and what I put on the top so I can do 47 plus 13 like this or I could do 13 plus 47 like this either way I'm going to get the same answer so we can go through and add 7 plus 3 is 10 and we know we need to carry here so the one goes into the next column one plus four is five five plus one is six so I get 60. if I come over here I'm going to get the same thing 3 plus 7 is 10. carry the one one plus one is two two plus four six I get sixty now with subtraction we don't have the commutative property so what goes on top and what goes on the bottom is going to matter so you want to take the leftmost number for your subtraction problem and remember this is called the menu end the menu end and you want to put that on top so when we're stacking these numbers that guy has to go on top and then you want to take your subtrahend that's 13 in this case your subtrahend and that's the value that's being taken away or subtracted away and you want to put that guy on the bottom and again notice how we're stacking the numbers they have to be lined up by place value so all the numbers in the ones place have to line up all the values or the numbers in the tens place okay they have to line up as well and if we had hundreds and thousands and so on and so forth those would have to be lined up so that's your first step and again you have to make sure that the correct number is on top and the correct numbers on bottom that way you get the right answer okay for the second step we want to draw a horizontal line directly underneath the bottom number just like we did with vertical addition and then place a minus symbol okay the minus symbol to the left of the bottom number so essentially it's the same thing as when you're doing vertical addition you're just putting a minus symbol in instead of a plus symbol so we had 47 and we have 13 stacked on top of each other so again draw a horizontal line underneath the bottom number and put a minus symbol or a subtraction symbol to the left of the bottom number and we're ready to move to the next step so we have 47 minus 13 like that and for step three we subtract in the ones column and place the result directly below the line just like we did with vertical addition so if I do seven minus 3 that gives me 4 and I just put that directly below in the ones column you just think about this as a ones column that just goes all the way down now in The Next Step we're just going to move to the left just like we did in vertical addition so in the next step I would just move to the left and I would say okay what is 4 minus 1 well that's 3. and then I've finished the problem you just keep working right to left so you start all the way in the rightmost column and you just keep going to the left so we end up with 34 as our answer for 47 minus 13. okay let's look at some examples now so we're going to start out with 96 minus 35 so stack the numbers on top of each other 96 35 and remember this 96 here this is your menu in that's got to go on top and then 35 that's your subtrahend that's got to go on the bottom we'll throw our subtraction symbol out there and put our horizontal line underneath and we're ready to subtract remember we're going to start in the ones column the rightmost column and work our way to the left so 6 minus five is one and then we go to the next column the tens column and we have 9 minus 3 and that's 6. so 96 minus 35 is 61. let's look at 87 minus 23 okay we're going to stack these numbers on top of each other and make sure they're lined up by place value so remember 87 your menu end has to go on top 23 the subtrahend has the goal on the bottom so we're going to subtract beginning in the ones column so we have 7 minus 3 that's 4 and then we work our way to the left now we're in the tens column eight minus two that's six so our answer here is 64. okay now we have 25 375 minus 10 164. again we're going to stack these numbers on top of each other this number the leftmost number the 25 375 the menu end has to go on top the subtrahend 10 164 has to go on the bottom okay so we're going to start on the ones column and we're going to do five minus four that's one we're just going to work our way to the left now we're in the tens column seven minus six that's one now we're in the hundreds column three minus one that's two now we're in the thousands column five minus zero that's five and now we're in the ten thousands column two minus one is one so we end up with fifteen thousand two hundred eleven so the examples that we've looked at so far are not very challenging at all they're actually very very simple but in the next few examples we're going to see some things that are going to kind of challenge us a little bit first so in many cases the lower digit of a column is larger than the upper digit when this occurs we use a procedure called borrowing and it's best for me to explain this through an example so let's look at 43 minus 17. so we start the problem off just like we would any other so we'd have 43 on top minus 17 on the bottom and what happens is when we try to subtract in the ones column we see that 3 is smaller than 7. so what happens when I try to do 3 minus 7. we haven't gotten to integers yet so we don't know how to do this essentially we only know how to subtract when the left number is bigger than the right number right when the menu end or the amount that you're starting with is larger than the subtrahend or the amount you're trying to take away so what we're going to do in this case is we're going to use this procedure called borrowing okay we're going to borrow and how this works is I go to the next digit to the left and I'm going to borrow one okay so I'm going to cross this 4 out and I'm going to subtract away 1 and that would give me 3. so now this 4 is now a 3. and I'm borrowing it and I'm sending it over to the column to the right and I just put a 1 in front of the digit in that column so now this 3 is a 13 and I'm able to subtract so I can say 13 minus 7 is 6 and then I just move to the column to the left and I say 3 minus 1 is 2. so I end up with 26. now let me explain a little bit further as far as why this works if we think about the numbers in terms of the place value of the digits involved 43 is what it's 40. it's 40 plus 3. and 17 is 10 plus 7. so if we think about what we just did when I crossed out this 4 made it a 3 you think about this I cross this 4 out and making it a 3 I basically made it a 30. right I made it a 30. and then I put a 1 in front of that 3. so if I put a 1 in front of that 3 it's a 13. so 30 plus 13 is still 43. I just changed the way it looked right I didn't change the value of the number so I didn't do anything illegal here so now when you kind of look through the columns you can see that okay I have 13 minus 7 that gives me 6 and I have 30 minus 10 that gives me 20 or a 2 basically because it's in the tens place right two groups of 10 is 20. so that's essentially what we're doing when we're borrowing we're not changing the value of the number we're just changing the way it looks temporarily so that we can perform the subtraction in that column okay let's take a look at another one so we have 512 minus 395. so 512 goes on top minus 395 which is on the bottom and when we try to do our subtraction in the ones column we have 2 minus 5. we know that 2 is a smaller number so we need to borrow so I'm going to go to the next column over to the left and I'm going to take one away so 1 minus 1 is 0. so now that's 0 and then I'm going to put a 1 in front of this digit 2. okay so I'm putting a 1 in front of that I end up with a 12. so 12 minus 5 is 7. now I have another problem because when I get to this column I have a 0 and I'm trying to subtract away a 9. so I need to borrow again so I'm going to the next column over and I'm going to take 1 away from this so 5 minus 1 is 4 and I'm going to put a 1 in front of this so now this 0 becomes a 10. so then 10 minus 9 is 1 okay 10 minus 9 is 1 and then 4 minus 3 is 1. so we end up with 117. so again let's look at these two numbers in expanded notation and see if we can figure out what we just did so 512 is 500 plus 10 plus 2. 395 is 300 plus 90 Plus 5. so basically when we started out we rewrote the number by taking this 10 and sending it over here so this became 0 and this became 12. right just think about adding 10 to 2 so now I have 500 plus 12 that's still 512. haven't changed the value of the number so we did our subtraction we did 12 minus 5 we got 7. but now when I went to the tens I had a 0 here and I was trying to take away 90. so what I did was I went over to the hundredths place and I borrowed a hundred so I took this and made it 400 right I subtracted away a hundred and I sent it over here and I made this 100. so then 100 minus 90 gave me 10 okay gave me 10 and now I was just left with 400 minus 300 and that gave me a hundred and so now if I look at this number this is exactly what we got 100 plus 10 plus 7 which is 117. so again that's what borrowing is doing it's just temporarily changing what these columns look like so that you can do your subtraction but in order to really understand it just go through as you're doing it for the first few times write the numbers in expanded notation and look at what you're actually doing okay for the final problem we have 190 000 5 minus 79 877. so we'll start out again stack the numbers on top of each other 190 05 goes on top seventy nine thousand eight hundred seventy seven goes on the bottom and of course when you start trying to subtract here you notice that there's a problem five minus seven we need to borrow but we'll notice when we look at our neighbor to the left you have a zero there you can't borrow from that zero so basically when this happens you have to go to the left until you find a non-zero number so we have to go all the way to this guy over here to this 9 and borrow from that but it's not as straightforward as you think take one away from the nine you get eight I have to send it to this guy here okay I can't just send it express to the five so we got to go through this long tedious process so this zero becomes a ten and then I'm going to borrow from that so this 10 will become a nine and then this zero become a ten and this 10 will become a nine and the zero will become a ten then this 10 will become a nine and this 5 will become a 15. so all that worked just so I could subtract here so now I can finally subtract in the ones column 15 minus seven is eight and we move to the left this is now a nine nine minus seven is two move to the left nine minus eight is one move to the left nine minus nine is zero move to the left eight minus seven is one and then we have this one here we're not subtracting anything away from it so it just comes down and we end up with 110 128 as our answer so for one final time let's write these numbers in expanded notation and see if we can figure out what happened scroll down a little bit get a little room so one hundred ninety thousand 5 is what it's one hundred thousand plus ninety thousand plus five and then our other number is seventy nine thousand eight hundred seventy seven and that's going to be seventy thousand plus nine thousand plus eight hundred plus seventy plus seven okay now this isn't really lined up so let me try to line this up for you so one hundred thousand plus ninety thousand plus you'd have a blank spot here for the thousands place you'd have a blank spot here for the hundredths place you'd have a blank spot here for the tens place and then finally you'd have a five in the ones place okay for this number you'd put 70 000 here plus nine thousand plus eight hundred plus seventy plus seven so when we start our subtraction problem we immediately see that five is smaller than seven so we need to borrow but when we look at our neighbors here to the left they don't have anything that we can take so we got to go all the way to the ten thousands place so in the problem if we go back up here you remember we cross this out and we subtract away one we got eight but that 8 is in the ten thousands place so it's really eighty thousand so if I cross this out I'm subtracting away ten thousand I'm going to write this as eighty thousand now that's the new value and I'm going to send 10 000 to my neighbor to the right so now he has 10 000 there and I haven't changed the value of the number because a hundred thousand plus eighty thousand plus ten thousand plus five is so one hundred ninety thousand five now I need to keep going until I get to that five there so I'm going to keep borrowing so I'm going to cross this out and basically since I'm in the thousands place now I'm gonna take one thousand away so this is going to become nine thousand so now I have nine thousand and I'm going to send that thousand that I took away over here so now this is a thousand and I need to keep doing this so I'm going to take now I'm in the hundreds place I'm going to take 100 away so this is going to become 900 and I'm going to throw 100 over here in the tens place now I'm going to do it one last time so that I can reach this 5 over here so I'm in the tens place so I'm going to take 10 away so this will become 90 and this will become 5 plus 10 or 15. okay think about that as 15. now when we do our subtraction 15 minus 7 is 8. 90 minus 70 is 20. 900 minus 800 is 100. nine thousand minus nine thousand is zero and eighty thousand minus seventy thousand is ten thousand and this I'm not subtracting anything away from it so one hundred thousand is going to stay there so when we look at this right here this is 100 000 plus ten thousand plus one hundred plus twenty plus eight or one hundred ten thousand 128 which is exactly what we got right here hello and welcome to pre-algebra lesson 10. and in this video we're going to learn about the properties of multiplication so for the lesson objectives we want to learn the basic parts of a multiplication problem we want to learn the commutative and associative property of multiplication we want to learn about the identity property of one the multiplication property of zero and the distributive property of multiplication okay so I want to start out the lesson by just briefly thinking about what's going on when we're multiplying whole numbers together so back in elementary school when you first started thinking about multiplying whole numbers you probably started out with learning the times table and your teacher probably at the same time put something like this up on the board so here we have 7 plus 7 plus 7 plus 7 is the same as four times seven and essentially if we look at an addition problem where the same whole number is being added so in this case that whole number is seven right we have seven plus seven plus seven plus seven we can use multiplication as a shortcut so instead of writing this out each time and going through and saying okay well seven plus seven is fourteen fourteen plus seven is twenty one twenty one plus seven is Twenty Eight we can do it more quickly with this multiplication we have four groups right we have one two three four four groups of seven so we use that fact to say that okay this is 4 times 7 or 28. so it's a more quick way to perform a repeated addition if we look at this example here we have 9 plus 9 plus 9 is the same as three times nine right because I have one two three groups of nine so we know that 3 times 9 is 27 and if we went back and did the addition we get the same result 9 plus 9 is 18 18 plus 9 is 27. okay let's look at two quick problems we wanna rewrite each as a multiplication problem and solve so what I have here is I have 2 plus 2 plus 2 plus 2 plus 2. so essentially I have one I have one two three four five groups of two so this would represent five times two if we do it using multiplication and we know that five times two is ten and if we go back and verify through addition two plus two is four four plus two is six six plus two is eight and finally eight plus two is in fact ten okay now we have four plus four plus four plus four plus four plus four so if I count these out I have one two three four five six groups of four so this represents six times four and we know that that's equal to twenty four and again we can go through and verify this through addition four plus four is eight eight plus four is twelve twelve plus four is sixteen sixteen plus four is twenty and then lastly 20 plus 4 is 24. okay now let's talk about a little bit of basic terminology so numbers being multiplied together are called factors okay they're called factors then the result of the multiplication is called the product so if we look at six times five equals 30 6 and 5 here are being multiplied together so those are going to be the factors and then 30 is the result of the multiplication process so this is the product okay the product and a lot of times you'll hear people say 30 is the product of 6 and 5. okay now we have 9 times 7 times 2 equals 126. so these three numbers here these three numbers here are the factors for this problem nine seven and two that's what we're multiplying together and then 126 is the result of the multiplication so this is the product okay the product okay now let's talk about some properties of multiplication and a lot of these are extremely straightforward some of them you've used before you just might not have known the official name for them so we're going to start out with the commutative property of multiplication and essentially this just tells us that we can multiply in any order and not change the product and we've already seen the commutative property of addition remember we could add in any order and it didn't change the sum here again we can multiply in any order we can multiply in any order and not change the product so if you have 3 times 6 which 3 in this case is first 6 is in this case is you can think of as last we put this as equal to or the same as 6 going first and three going last doesn't matter how you multiply you're going to get the same answer either way 3 times 6 is 18 and 6 times 3 is 18 as well okay as another example of the commutative property we have 12 times 9 equals 9 times 12. and again you can prove this to yourself if you multiply 12 times 9 you get 108. if you multiply 9 times 12 again you get 108 so changing the order does not affect your product okay let's look at an example real quick where three factors are involved so we have seven times five times four is equal to 5 times 7 times 4 and that's also equal to four times seven times five so the same three numbers are being multiplied seven five and four we're just changing the order around and no matter how you do this you'll get the same answer seven times five times four seven times five is thirty-five thirty five times four is one hundred forty now if I do it this way 5 times 7 times 4 . well we already know that 5 times 7 and 7 times 5 are the same so we know we would get 5 times 7 is 35 and then 35 times 4 is 140. and then if I do it this way 4 times 7 times 5 well I could do 4 times 7 that's 28 and then multiply times five that's also 140. so again again it's very important for you to understand that when you're doing multiplication and there's only multiplication involved you can do it however you want the order is never going to matter it's never going to change your end result okay now let's talk about the associative property of multiplication so with this property we can group the multiplication of three or more numbers in any order without changing the product notice that where we have six times three times one we have parentheses here around 6 times 3. and we're saying this is equal to or the same as six times and now I have parentheses around three times one so when we say grouping we're talking about putting parentheses around different numbers so if you just have multiplication involved you can put parentheses around different numbers or take the parentheses away it's not going to matter at all you will get the same answer so if I look at 6 times 3 times 1 where 6 times 3 is inside of parentheses we would do this first and I know we haven't gone through order of operations yet but whenever you have operations inside of parentheses you start with that so we would do this first 6 times 3 is 18 then 18 times 1 is 18. and just thinking about this over here now we have three times one inside a parentheses so we'd start with that first so you would have 6 times the result of this which is three right three times one is three and six times three is eighteen as well so either way you do it you end up with 18 as the answer all right let's take a look at a few examples here we have 4 times 2 inside of parentheses then times eight and we're saying this is equal to four times and then we have two times eight inside of parentheses so again you want to start out with what's inside a parentheses first so on the left here we have 4 times 2 inside of parentheses that gives us 8. so we would have 8 times 8 and that equals 64. over here on the right I have 2 times 8 inside of parentheses so I do that first so we would have 4 times 2 times 8 is 16 and 4 times 16 is also 64. so again changing what we had parentheses around or changing our grouping did not change the product it didn't matter that we switched what we had parentheses around we got 64 either way all right let's look at one last problem we have nine times and then we have 7 times 5 inside of parentheses and we're saying this is equal to here we have 9 times 7 inside of parentheses then times five so again start with the parentheses on the left I have 7 times 5 inside of parentheses so we would have 9 times 35 and 9 times 35 is 315 on the right I have 9 times 7 inside of parentheses so I would do that first 9 times 7 is 63 and then times 5 is 315 as well so again just to kind of wrap this up I know this is a very easy topic but essentially when you think about the associative property of multiplication it's really just telling you that you can put parentheses around whatever numbers you want when you're multiplying three or more numbers it will not in any case change the product or the result that you get from your multiplication okay now we're going to talk about two very simple properties the first one is going to be called the identity property of one and then we're going to talk about the multiplication property of zero so the identity property of one states that any number multiplied by one remains unchanged and I think we all know this fact because when you think about the times tables it's always about okay one times one is one two times one is two three times one is three four times one is four five times one is five so you kind of learn that fact immediately because that's the easiest one to do so when we look at 15 times 1 we know we just get 15. or if we look at kind of some bigger numbers what's 97 times 1 well it's just 97. what's 3 million eight hundred sixty six thousand one hundred fifty one times one well it's just 3 million eight hundred sixty six thousand one hundred fifty one so a very easy property and a lot of times kind of in a pre-algebra test when you're first starting out so throw this in there just to kind of trip you up oh hey this is a big number how am I going to do this multiplication without a calculator well use the simple fact that a number times one is just itself okay so probably just as easy the multiplication property of zero tells us that any number multiplied by zero gives us a result of zero so when we start out with 30 times 0 what's the result it's 0. when we think about 25 times 0 well the result is zero when we think about 2 577 313 times zero well the result is zero and a lot of times they'll put this question on a test but they won't put it straight forward like that they'll give you these numbers that are being multiplied together so let's say something like 14 times 10 times 5 times 0 times 30. and a lot of students will see this and they'll go oh how am I going to do that without a calculator many of them could do it but it would just take a long time well you use the fact that zero times any number is always zero it doesn't matter what the result of the multiplication is through here because whatever number that you get as a result when you multiply it by zero you get zero then when you multiply 0 times 30 you get zero so when you see something like this where you have a lot of numbers being multiplied together and there's only multiplication involved and you have zero as a factor you know that the result of that problem is going to be zero okay now let's talk about a very important property the distributive property of multiplication tells us that multiplication is distributive over addition or subtraction and I have as a side note here that we use this property constantly in algebra so now is a great time to learn and understand this property because when you get to algebra you're literally going to use it all the time okay so I'm going to explain the distributive property by looking at a few examples so we're going to start with four times and then we have 3 plus 6 inside of a set of parentheses or we can think about this as four times the quantity okay the quantity three plus six one of the things I'm going to draw your attention to and this is kind of a side note is that up to this point most of you have only used this to imply multiplication so if I said what is 4 times 3 you'd probably write that problem out like this and you'd solve it and say okay 4 times 3 is 12. as we move higher in math this symbol is going to go away in terms of multiplication you won't use that anymore and there's a variety of different ways that we can apply multiplication and we'll kind of get to those as we move on but for right now I just need you to know that when we put a number outside of parentheses it implies multiplication so I could write four times three like this okay I have 4 and it's directly next to a 3 that's inside of parentheses so that's why I say 4 times this quantity 3 plus 6 because 4 is next to that set of parentheses okay now let's talk a little bit about the distributive property and before I even get into that let's think about solving this problem without using distributive property so I know we haven't done the order of operations yet but that tells us before we do anything else to perform operations inside of parentheses so we have three plus six inside of parentheses 3 plus 6 is 9. so this would be 4 times 9 or 36. now when we Use the distributive property we have an alternative way to get this answer so 4 times this quantity three plus six essentially what we're going to do is we're going to take this 4 and we're going to distribute it to each number inside of the parentheses so we would say 4 times 3 plus okay plus 4 times 6 4 times 6. so you have to remember to multiply by each number inside of the parentheses and everywhere there's a plus symbol or in some cases it's going to be a minus symbol because this works with subtraction 2. you have to write that okay you can't skip that over now if we go through and we solve this we're going to do the multiplication first because that's what the order of operations tells us so we're going to treat these as separate problems before we add so 4 times 3 is 12. then plus 4 times 6 which is 24. so 12 plus 24 is also 36. so either way we're getting the same result okay 36 either way now I'm going to erase all of this real quick and we're just going to rewrite this and then I'm going to kind of do this a long way so 4 times 3 plus 4 times 6 and you can leave it like this or you can put parentheses around this like this it won't change your answer if you know the order of operations I just put the parentheses around it so that you know to do 4 times 3 separately from 4 times 6. I don't want you to get confused at this point now if I think about the addition problem that we did earlier nine plus nine plus nine this is three groups of nine so we said this is three times nine if I apply that same logic here I have four groups I have four groups of three plus six so what does this translate to three plus six plus three plus six plus three plus six plus three plus six now I know that the associative property in the commutative property allow me to reorder and regroup this Edition however I want so what I'm going to do is I'm going to write inside of parentheses 3 plus 3 plus 3 plus 3. I have one two three four of those then plus inside of another set of parentheses I'm going to put the sixes so I have one two three four of those so six plus six plus six plus six now if I look at it this way it is clear to me how I arrived at this result I have one two three four groups of three which is known as four times three I have one two three four groups of six which is four times six so you can see that this directly matches this using the distributive property was just a shorter way to get there right because I know in the end I'm going to have four groups of three and I'm going to add to that four groups of six all right let's look at another one and we won't go through that long tedious process again we have seven times the quantity nine plus two so we're going to do this two different ways we're going to do it the first way where we do nine plus two first because that's what's inside a parentheses so this is 7 times 11 and that's 77 and then let's do it using distributive property so remember I have seven groups of nine or seven times nine plus I have seven groups of two or seven times two and I'm going to put parentheses around each here so we know to do that first before we add and again you won't need that but we haven't done the order of operations yet so I don't want to confuse you so 7 times 9 is 63. and then plus 7 times 2 that's 14. and so 63 plus 14 is 77. so either way using the distributive property or adding what's inside a parentheses first we get 77. okay now let's look at one that involves subtraction because this does also work with subtraction so we have a times the quantity four minus 3. so working inside of parentheses first I'd get one four minus 3 is 1. so we'd have eight times one and that just equals eight if I Use the distributive property again I want to distribute this so eight times four eight times four and then we're subtracting we're subtracting eight times three eight times three so again I'm going to put parentheses around the multiplication problem so that you don't get confused all right so 8 times 4 is 32 and then we're subtracting away 8 times 3 which is 24. and 32 minus 24 is 8. so again either way we get the same result okay let's take a look at one last problem so we have 5 times the quantity nine minus four again doing this where we do 9 minus four first nine minus four is five you'd have five times five that is 25. using the distributive property we're going to distribute the 5 so 5 times 9 5 times 9 then subtract away five times four five times four and again I am going to put these multiplication operations inside of parentheses just so that you don't get confused and when we get to the order of operations which will be a few lessons from now you'll learn that you need to perform multiplication before subtraction but for right now we know that we gotta perform operations inside of parentheses first so let's just put the parentheses around it and work with that rule so 5 times 9 is 45. then minus 5 times 4 that's 20. and then we do this subtraction 45 minus 20 is 25. so again either way we get the same result either by doing what's inside of the parentheses first or by using our distributive property where we distribute our multiplication to each number inside of the parentheses hello and welcome to pre-algebra lesson 11. and in this video we're going to learn about vertical multiplication so for the lesson objective we want to learn how to multiply multi-digit numbers together okay so up to this lesson we have reviewed how to do addition of whole numbers and how to do subtraction of whole numbers and we saw that when we needed to add multi-digit numbers or when we needed to subtract multi-digit numbers we did it in a vertical format now the same is going to be true when we think about multiplying whole numbers now before we dig into that I just want to make you aware that in order to be successful with vertical multiplication you have to at minimum know your times tables and those are the tables that you learned back in elementary school where you memorized you know one times one is one one times two is two one times three is three so on and so forth you need to at least know up through nine times nine so if you haven't memorized that you probably need to go back make some flash cards and memorize that so that you can be successful with vertical multiplication so essentially you can think about it this way what do we do when we want to multiply large numbers together are we really expected to memorize the multiplication for every number that we may come across well no of course not when we multiply multi-digit numbers together we use a process called vertical multiplication all right so I prefer to teach through examples so let's start out by just looking at 26 times 6. now with vertical multiplication it's going to be the same as vertical addition or vertical subtraction where we want to Stack the numbers on top of each other and line them up by place value now you'll recall with addition it didn't matter what went on the top and what went on the bottom but it was easier or I would say more common also to put the number with more digits on top and the number with fewer digits on the bottom that's going to be the same thing with multiplication multiplication is commutative I can do 26 times 6 or I could do 6 times 26 I get the same answer it will make your life easier if you take the number with more digits which in this case is 26 and you put it on top so I'm going to put 26 on top and I'm going to put 6 on the bottom okay now just like with vertical addition we want to draw a horizontal line underneath the bottom number and we want to put an operator symbol which in this case is the multiplication symbol out to the left of the bottom number so performing this operation is very very simple and once you've practiced it enough it'll be very very easy for you okay so to begin our vertical multiplication we're going to start with the rightmost digit of the bottom number and we're going to multiply by the rightmost digit of the top number and we're going to continue working right to left so we would begin by doing six times six and we know that equals 36 but 36 is too big to put in the ones place for the answer so we're going to do a thing called carrying which is similar what we did with addition but with a slight twist so this part is the same you take the rightmost digit which is a six and you write it into the answer column so I'm going to write a 6 right there I'm going to then take the left digit which is a 3 and I'm going to write that in the next column to the left okay so it's going in the tens column if you think about 36 again thinking about place value this is 30 plus 6. so all I've done is said that hey we have six ones and I take three tens and I put it into the tens column so that's why that's a 3 there again that's three tens now it's important to understand that that's three tens this is 30. because that 30 is going to be added to the result of the next multiplication that we do it's not going to be multiplied we will simply add it on to the answer that we get so now we're going to move to the left and we're going to do we're going to do 6 times 2. so 6 times 2 and once we have our result we're going to attack that 3 on so plus 3 okay so 6 times 2 plus 3. so 6 times 2 is 12 and then if we add 3 to that we get 15 we get 15. so we're going to write that into our answer we would write a 5 and normally we carry a 1 but there's nothing else over here so that one can just be written right there so we get an answer of 156 and really it's just that simple so if I think about six times I'm going to break 26 up into 20. Plus 6. so if I think about this using the distributive property I'd have 6 times 20 6 times 20. then plus I'd have 6 times 6. 6 6 times 6. this right here is exactly what we did here and I'm going to show you that in a second so I'm going to use some parentheses here to separate my multiplication operations from this addition and you normally do not have to do that but we at this point have not studied the order of operations so I want it to be clear that we're going to do 6 times 20. and then 6 times 6 and then add the results I don't want you to get confused on what operations we're doing so 6 times 20 is 120 plus 6 times 6 that's 36 and when we add here we get 156 which is the same thing that we got there so looking at this we can basically see that this over here this vertical multiplication is just a condensed version of the distributive property that's all it is I started out by doing 6 times 6. that's right here 6 times 6 gave me 36. then I did 6 times 20. it looks like 6 times 2 but that 2 is in the tens place so it's really a twenty so I did 6 times 20 and that gave me 120. now I added 120 and 36 and I got 156. the only thing that's a little different here is we had to carry a 3 because we couldn't fit the three in the ones place but essentially all I did was I said 6 times 20 is 120. plus this 3 here which was really a 30. plus the 6 which was the number from the ones place when we did six times six right that 36 right there so plus 6 and that still gives us 156. so just letting you see this in a different light essentially the vertical multiplication process that we use is just a condensed version of the distributive property that's all it is okay let's look at another one so we're going to look at 613 times 5. so again I'm going to take 613 and I'm going to put that on top because it has more divots and then I'm going to put 5 on the bottom and we multiply from right to left so I start out with this digit on the right and soon you'll have digits over here too so you start out on the right and you multiply by the rightmost digit of that top number so we're doing 3 times 5. so 3 times 5 equals 15. so I'm going to take the 5 and put it down and then I'm going to carry this one now remember I'm not going to multiply the one I'm going to only use that one once I'm done with the multiplication so now I'm going to multiply five times the one there so five times one and then when I'm done with that I'm going to add this one I'm going to add that one so five times one is five then plus one is six okay and again think about what you're saying here this 5 has a value of five this one has a value of ten because it's in the tens place so 5 times 10 is 50 and then I'm adding another ten that's a one in the tens place then I'm adding another 10 to get to 60. so I have a 6 that's in the tens place that's a 60. so that's what we just did now the last thing I'm going to do is I'm going to multiply 5 times 6 and that gives me thirty five times six equals Thirty now normally I'd have to put the zero down and carry the 3 but there's nothing else over here so I can just write that 3 there I end up with 3065 as my answer all right for the next problem we have 1752 times 8. again I'm going to take this number here this 1752 and I'm going to put that on top and I'm going to put 8 on the bottom again you could reverse that if you chose to I just find this to be a little easier so we're going to begin by taking the rightmost digit of the bottom number which in this case is just an 8 so that's all there is and multiplying that by the rightmost digit of the top number so we're going to do 2 times 8 to begin and that is 16. so for 16 we're going to need to carry so I put a 6 down and I carry a 1 into the next column and remember that's not going to be multiplied that's going to be added after we do our multiplication so we're working our way left now we're multiplying 8 times 5. so 8 times 5 and then once we're done with that we're going to add this 1 in we're going to add this one and we're basically saying hey whatever you get add 10 to that because that one's in the tens place add 10 to that all right so 8 times 5 is 40 then plus 1 is 41. so I need to carry again I'm going to take the 1 the rightmost digit and I'm going to write that there and I'm going to carry the 4 the leftmost digit into the next column now we're ready to multiply in the next column 8 times 7 is what we're doing next so we'd have 8 times 7 then plus what we carried over that 4 is in the hundreds place so I'm saying hey whatever you get for the result here I want you to add 400 to it okay so I'm just writing this as a 4 because it's in the hundreds place so 8 times 7 is 56 plus 4 is 60. again we need to carry so I'm going to take the zero I'm going to put that in the answer column and I'm going to carry the 6 into the next column to the left and we have one more multiplication left to do okay so now we want to do 8 times 1 so 8 times 1 and we're going to add to that the 6 that we carried over so again whatever we get for this we're adding 6 more and that 6 is in the thousands place so we're basically tacking on another 6 000 to that answer so 8 times 1 is 8 then plus 6 is 14. now because there's no more numbers to the left I can just write 14 into my answer so we get 14 016 as our result all right let's take a look at another easy one so we have 15 319 times 7. and again I put the number with more digits on top the number with fewer digits on the bottom now let's just crank this out we're going to start at the rightmost digit in this case it's just it's just a 7. of the bottom number we're going to multiply it by the rightmost digit of the top number and we're working our way to the left so 7 times 9 is 63. so I would put a 3 down and I would carry a 6 into the next column next I would do 7 times 1. 7 times 1 is 7 then plus six is thirteen so I put a 3 down and I'd carry a one then I'd have 7 times 3 that's 21 then plus 1 that's 22. so I put a 2 down and carry a 2. now I have 7 times 5 that's 35 then plus 2 that's 37. so I'd put a 7 down and I'd carry a 3. and lastly I have 7 times 1 that's 7 then plus three that's ten so I just write a 10 here and we end up with one hundred seven thousand two hundred thirty three okay so up to this point it's been pretty easy and the reason for that is the number on the bottom has been a single digit number when we start multiplying a multi-digit number by another multi-digit number the work gets more tedious so let's start out by thinking about 315 times 12. so again I'm going to write the number with more digits on top and the number with fewer digits on the bottom and again I want to start out at the rightmost digit of the bottom number and multiply it by the rightmost digit of the top number so 2 times 5 would give me 10. now I'm going to put a 0 down and I'm going to carry a 1. next I'm moving to the left so I'm doing 2 times 1. so 2 times 1 is 2 then add 1 so that's three okay working my way to the left now I have 2 times 3 2 times 3 is 6. so the result here is 630 but I'm not done okay I'm not done I still have this other number over here to the left and there's something that you have to do when that occurs if you move to the left to multiply you have to move to the left when you start your answer so we're going on a whole nother row here a whole other row and we're starting right here okay right here and the reason we're doing that is because this one is not a one it's actually a 10 because again it's in the tens place so what I'm doing is I'm multiplying 10 times 5 and I'm getting 50. so I would write 50 by putting a 5 in the tens place okay by putting a 5 in the tens place if I was to just multiply 1 times 5 and put 5 here I didn't get the right answer because this isn't a one it's a 10. so I really need to do 10 times 5. so to correct for that we just move one place to the left and we start our answer in the tens place so let's begin there so we're going to do one times five one times five as I just said is 5. Now we move to the left I'm going to do 1 times 1 and that's one and we move to the left one final time we're going to do one times three and one times three is three now you might say what are we going to do now well essentially this 630 and this what looks like 315 it's really 3150 because of where it's positioned those are called partial products okay partial products what we actually have to do is we have to find the sum of these two rows to get the product for 315 times 12. this zero here can either come down or you can write a zero in it doesn't really matter so zero plus zero is zero then three plus five is eight then six plus one is seven and then you can just bring that 3 down so you get three thousand seven hundred eighty so again if we want to just think about this a little further again Use the distributive property take this number 315 and multiply it by ten plus two that quantity again this is 12 I just broke it up so doing this we'd have 315 times 10 315 times 10. plus 315 times 2. 315 times 2. and that's exactly what we just did here we did the rightmost digit of this bottom number 2 times 315. the result of that was 630. so if I think about this I would have plus 630 over here and we also did 315 times 10. because this one is in the tens place I multiply 10 times 315 and that gave us 3150. 3150. so then we did some addition just like we did over here 3150 plus 630 which gives us 3700 e okay let's try another one so we have 765 times 23. so 765. I'm going to put that on top times 23. I'm going to put that on the bottom so again we start out at the rightmost digit of the bottom number and we multiply it by the rightmost digit of the top number and we're just going to work left so 3 times 5 is 15. put a 5 down carry a 1. 3 times 6 is 18 plus the one that we carried is 19. I'm going to put a 9 down and carry the 1. 3 times 7 is 21 plus the one that we carried is 22. and again I can just write 22 because there's no more numbers out here now we're going to the left and the bottom number we have to remember to start our answer to the left right here because again this isn't a 2 times 5 this is a 20 times 5. so I've got to start my answer right there so then 2 times 5 is 10. so I'm going to put a 0 down and I'm going to carry a 1. 2 times 6 is 12 plus the one I carried is 13. put a 3 down carry a 1. 2 times 7 is 14 plus 1 is 15. and because I don't have any more numbers out here I can just write 15. now really because of where this zero starts this number is fifteen thousand three hundred I can put a zero in there if I want or I can leave it out it doesn't really matter we're going to do some addition so we add these two partial products up again 2295 is a result of doing 3 times 765 and then 15 300 is a result of doing 20. again that 2 is in the tens place so it's a 20 times 765. so we find the sum here five plus zero is five nine plus zero is nine two plus three is five two plus five is seven and then this one can just come down and we end up with seventeen thousand five hundred ninety five okay let's look at one final problem and as your numbers get larger your work doesn't get harder it gets more tedious and you have to pay very close attention to what you're doing because it becomes very easy to make a mistake especially if you're on like your 11th or 12th problem and you're just exhausted you'll start making these mental mistakes so you have to pay close attention to what you're doing okay so 212 050 times 1075. I'm going to take this number two hundred twelve thousand fifty and I'm gonna put that on top and I'm going to take 1075 and I'm gonna put that on the bottom okay so again we start out the rightmost digit of the bottom number times the rightmost digit of the top number and we're just working our way to the left so we're going to have 5 times 0. so 5 times 0 is 0. then we're going to have 5 times 5. 5 times 5 is 25 so I'm putting a 5 down and I am carrying a 2 into the next column now I have 5 times 0. that is 0 plus 2 is 2. now I have 5 times 2 that is ten so I'm going to put a 0 down and I'm going to carry a 1. now I have 5 times 1 that's 5 plus 1 is 6. lastly I have five times two that's ten no more numbers out here so I can just write that in I go through and put some commas so that's one million sixty thousand two hundred fifty and that's basically the result of five times two hundred twelve thousand fifty okay now we're moving to the left so what happens when I move to the left I've got to move to the left where I start my answer again this is not a seven it's a seventy okay it's a seventy so we're going to start out with seven times zero seven times zero is zero seven times five is thirty-five so I put a five down I carry a three seven times zero is zero plus three is three then seven times two is fourteen so I'm going to put a four down and I'm going to carry a one now I have seven times one that's seven plus one is eight and then lastly I have seven times two and that is fourteen okay put in the commas so I get 14 million eight hundred forty three thousand five hundred you might be saying well there's no zero over here well because I started this over here there's really a zero here there's really a zero there but when we add we don't have to put the zero in because this can come just straight down and you're going to get the same answer so I'm going to go ahead and just write a 0 there just so it's not confusing for you okay so now we're going to continue and we're moving left again so I'm moving left and when I do that I've got to move left so I'm not going to start in the tens anymore I'm going to have to start in the hundreds I'm going to have to start in the hundreds and the reason for that is I'm starting in the hundreds here so a lot of people like to just put the zeros in you can do that and start your answer there or again just as I told you you can leave it blank let's go ahead and put the zeros in so that you're not confused all right so we're going to start with zero times zero that is zero zero times five that is zero zero times zero that is zero so you're starting to notice something if you have a zero and you're doing your multiplication this whole row here will be zero so essentially I'm just going to include one zero there just so I can maintain my place I don't really need to write anything else because if I add 0 when I'm doing the addition it's not going to change the value of the number so I'm just going to move to the left move to the left and look at this one here so now I've got to start my answer here in the thousands place okay I've moved all the way to the thousands place notice how this one is in the thousands place okay let's start so we have one times zero that's zero we have one times five that's five we have one times zero that's zero we have one times two that's two we have one times one that's one and we have one times two that's two and another cool trick you can think about is if you're multiplying one times this it's just going to be itself remember one times anything leaves it unchanged so I could have just wrote 212 050 and been done with it as long as I know where to start that's the important part so if I want to I can go ahead and write three zeros here just so that we don't get confused okay so we're going to go ahead and add let me make these zeros a little better so they line up so I'm going to come straight down I have a 0 everywhere and I can put zeros in here if I want it doesn't matter this is just 0 in the end so coming straight down I have a zero then I have 5 plus 0 plus 0 plus 0 that's 5. then I have two plus five that's seven then plus zero plus zero that's seven then I have zero plus three plus zero plus zero that's three then I have six plus four that's ten plus zero is still ten plus five is fifteen so put a 5 down carry the one now we have one plus zero which is one plus eight which is nine plus zero plus zero which is still nine then I have one plus four which is five plus zero which is still five plus two which is seven then I have one plus zero which is one plus one which is two and then I can just bring this two down and end up with two hundred twenty seven million nine hundred fifty three thousand seven hundred fifty hello and welcome to pre-algebra lesson 12. and in this video we're going to learn about multiplication with trailing zeros so our lesson objective is just to learn how to multiply numbers with trailing zeros and you might be asking yourself what in the world is a trailing zero well a trailing zero is just a zero at the end of the number or you might say trailing zeros if there's multiple zeros at the end of a number so let's take the number 600. 600 has two zeros at the very end of it so this number has two trailing zeros or you could look at the number 17 000. this number has three zeros at the end of it so it has three trailing zeros where people get confused is you might have a number like I don't know let's say 100 four thousand Thirty this number has three zeros in it but they're not all trailing zeros you're only looking at zeros at the end of the number it can't be interrupted by something that's not zero so this one right here at the very end that's a trailing zero but because there's a three here you can't say that this is a trailing zero also yeah it's close to the end of the number but it's not at the end it's interrupted by this three here so this number would have one trailing zero let's think a little bit about a simple multiplication fact that you might not have ever thought of when you multiply let's say something like 5 times 10. we know that this is 50. but another way to attain this result is to multiply the non-zero parts so five times one that would give you five and then just attach the trailing 0 to the end of the result so in other words I can do five times one and then I can attach a zero to the end that would give me 50 and I would have the result for 5 times 10. and this works no matter how large your numbers are let's say I would have said what is five times a hundred well now I would do five times one that's five and then I would attach two zeros to the end two zeros to the end to get 500. let's say I did five times ten thousand I do five times one that's five and I'd attach one two three four zeros to the end one two three four zeros and I'd get fifty thousand okay so it's a simple multiplication fact but once we learn it and understand how to use it it can greatly speed up our multiplication you'll come across a lot of problems with trailing zeros and you might not have a calculator to use so this is a trick that can speed you up significantly so just to look at the official procedure essentially you just want to leave off any trailing zeros and multiply the numbers that result and then next you want to attach the total number of trailing zeros and this is between all factors to the right of your product so again super super easy to do let's just look at some examples all right so let's start out with three times sixty so we want to multiply and leave off the trailing zero in this case what would three times six be that's 18. and then all we need to do is attach the total number of trailing zeros between all factors in this case just one to the end of that number so we end up with 180 and of course that's correct 3 times 60 is 180. what about 17 times 1000 Well we'd multiply 17 times 1 right you would leave this part off these are the trailing zeros 17 times 1 is 17 and then you would just attach them to the end of the number I have one two three trailing zeros so I'll write one two three zeros there and so 17 times 1000 is seventeen thousand all right let's look at an example where both numbers have trailing zeros here we have 70 times 300. so 70 has one trailing zero 300 has two trailing zeros and essentially just leave those off leave those off and you would do seven times three seven times three is twenty one and then I'm going to attach one two three zeros to the end because that's how many trailing zeros there are between all the factors so one two three trailing zeros are attached and so Seventy Times three hundred is twenty one thousand okay what about 153 times nine thousand so a lot of us cannot do 153 times 9 in our head I'm going to do 153 times 9 in vertical multiplication format and then when I'm done I'm going to take this result bring it up here and attach three zeros to it all right so 9 times 3 is 27 put a 7 down carry the 2. 9 times 5 is 45 plus 2 is 47 put a 7 down carry the 4. 9 times 1 is 9 plus 4 is 13. so I get 1 377 so 1 377 and I'm just going to attach three zeros to the end of that one two three comma comma and so our result is one million three hundred seventy seven thousand okay let's look at another one we have 216 000 times 108 000. so we have three trailing zeros here and we have three trailing zeros here so I'm just going to multiply 216 times 108 216 times 108 and when I'm done I'm going to attach one two three four five six six zeros to the end of whatever the result is from that so 8 times 6 is 48 put an eight down carry of four a times one is eight plus four is twelve put a two down carry a one eight times two is sixteen plus one is seventeen zero times 216 is zero so I can put a zero here or I can leave it off doesn't really matter I'm just going to leave it off and next we move to this one here I have 1 times 216. remember the one is in the hundreds place so I have to start my answer in the hundreds place and one times anything is just itself so one times six is six one times one is one and then one times two is two so we find our sum bring down the eight bring down the two seven plus six is thirteen put a 3 down and carry the one one plus one is two two plus one is three and then bring down this two and we get 23 328 now I'm going to bring this guy up here twenty three thousand 328 and I'm going to attach six zeros to it so let me actually write it a little bit lower 20 23 328 one two three four five six so this is our answer for 216 000 times 108 000 we get the result of 23 billion 328 million for the final problem we have five million 300 000 times 1 615 000. so I just need to do 53 times 1 615 right these trailing zeros I'm going to cut off and I'm going to cut these off as well we have a total of one two three four five six seven eight between the factors so eight zeros so I want to do 1615 times 53. 3 times 5 is 15. put a five down carry the one three times one is three plus one is four first time 6 is 18 put an 8 down carry the one three times one is three plus one is four okay now we're moving left to the five remember the five is in the tens place so we start our answer here 5 times 5 is 25 put a five down carry the two five times one is five plus two is seven five times six is thirty put a zero down carry the three five times one is five plus three is eight and now we add bring down the five four plus five is nine eight plus seven is fifteen put a five down carry the one one plus four is five bring down the eight we get eighty five thousand five hundred ninety five so now we're going to take this result here 85 595 and we're going to attach to the end of that eight trailing zeros all right because that's what we had between the factors so one two three four five six seven eight comma comma comma comma so we end up with eight trillion 559 billion 500 million as the answer hello and welcome to pre-algebra lesson 13 and in this video we're going to talk about the properties of division okay so for our lesson objectives we want to learn how to identify the parts of a division problem we want to learn the properties of division and also we want to learn how to solve division problems with remainders okay so let's begin our lesson by thinking a little bit about what happens when we divide one whole number by another and I'm going to start by just taking a look at 6 divided by 3. and you'll see I have an illustration that I drew and it's a very simple illustration but it'll get the point across basically I have six boxes here let's label them one two three four five and then six and if you look at the division problem 6 divided by 3 this amount on the left 6 is what we start with we have this 6 and that's what I have in the illustration I have six items six boxes however you want to think about it then we're dividing by three okay what does it mean when you say the word divide it means you're cutting something up or you're splitting it up so to divide by three means I'm going to ask the question how many equal groups of three can I make out of this number six that's all six divided by three is same if I just look at the illustration it's very easy to see that I'd have one two three so this would be one group and then you could say four five six we know that's three I could erase this and make it more clear by saying okay this is another group of one two and then three so we would end up with two equal groups of three that can be made out of the number six so then six divided by three is equal to two now another way you can think about this is by using subtraction so I could take and count how many times we can subtract 3 away from six until the result is zero okay so if I started with six and I said okay I'm going to subtract away three what's the result well it's going to be 3. okay so now I'm going to take 3 and subtract away another group of three and now the result is zero so I was able to do this twice so again that tells me that 6 divided by 3 is 2. now I want you to look at how closely Division and multiplication are related to each other forget about the fact that we have a problem six divided by 3. so let's just look at this and think back to multiplication when we saw this we had one two groups of three so what is that that's basically three plus three or two times three and we know this equals 6. well if I kind of look at 2 times 3 equals 6 and 6 divided by three equals two I kind of see a little pattern that's going on here and I'm going to show you that pattern real quick so if I have 6 divided by 3 equals 2 then that means that this number right here the result which we're going to learn later on is called the quotient times this right here the 3 which we're going to learn later on is called the divisor should give me this back this 6 which we're going to learn later on is called the dividend so if 6 divided by 3 equals 2 then that means that 2 times 3 equals 6. you can kind of think about this as going backwards right I take this number and I go this way so we'd have 2 times 3 now and that should equal 6. and you can use this to solve a lot of basic division problems if I have something like I don't know let's say 15 divided by 3. what does that equal or how would I figure that out well I could say well what number what number times 3 equals 15. right what number times 3 equals 15. well we know our times tables pretty well at this point we know that 5 times 3 equals 15. so the answer to 15 divided by 3 would be five and you can check this by using the subtraction method that I just did or you could draw 15 boxes and you could split them up into equal groups of three you're going to see the way that you get 5 as a result okay let's take a look at another one so here we have 8 divided by two and I'm just going to do the same thing basically I have made another illustration here and we have one two three four five six seven eight what are we asking for when we say eight divided by two again it's how many equal groups of two can be made from the number eight well and I'm just going to go ahead and re-label this and I know I keep doing that but I'm just trying to make things clear so here's one two that's one group here's one two that's another group here's one two that's a third group here's one two that's a fourth and final group so we have one group of two second group of two third group of two and a fourth and final group of two so that tells me that eight divided by two is equal to four and again if I want to use subtraction I can do it that way as well so I can say okay how many times can I subtract 2 away from 8 before the result is 0. so 8 minus two would give me six then six minus 2 would give me four then four minus 2 would give me 2 and then 2 minus 2 would give me 0. so you can see I did one two three four operations to get that so the answer would be four and then lastly probably the method you're going to use most often until you have everything memorized you would say Okay 8 divided by 2 equals what okay equals what well remember I can go backwards I can say what times 2 equals eight right go backwards what times 2 equals eight again we know our times tables by heart at this point so we know that this answer will be four right we know this answer would be four four times two gives us 8. okay so now that we've talked a little bit about the basics for dividing whole numbers let's talk a little bit about some terminology so there are three parts to a division problem the first part the leftmost number the number that we're going to split up or caught up is called the dividend okay again that's the left number this is the number that is being divided or split into equal groups so when we looked at six divided by three equals two six was the dividend okay 6 was the dividend okay the next part of a division problem is called the divisor and this is the number we are dividing by so where six divided by three equals two this three is the divisor this is the divisor and then lastly you have a quotient so this is the result of the division operation so in this case it would be two right because 6 divided by 3 gives us a result of two so that is the quotient okay so let's do a real quick exercise we want to label the parts of each division problem so we have 10 divided by 5 equals 2. so 10 the number on the left the number or the amount that we start with is called the dividend this is the dividend five the amount that we're dividing by is called the divisor and then two the result of the division operation again is called the quotient this is the quotient okay here we have 20 divided by 10 equals 2. again this number on the left this 20 the amount we start with is the dividend then we're dividing by 10. so that's your divisor and then lastly the result from this division is 2 so that is your quotient okay let's talk about a few important properties of division the first thing is that division is not commutative meaning the order is important when you divide and we've seen this before with subtraction we saw that when we subtracted and we did six minus 3 it was not equal to 3 minus six right those did not produce the same result so with division if I do 6 divided by 3 I will not get the same result if I flip this and put 3 divided by 6. the order does matter when you divide okay the next fact is that 0 divided by any non-zero any non-zero number is always zero we're going to cover why I said non-zero in a second but if I have 0 divided by whatever it is as long as it's not zero I'm going to get 0 as a result so let's say it's 17. and think about why that's the case remember if I go this way and make a related multiplication statement 0 times 17 equals what it equals zero because zero times any number is always zero okay so this is a very important rule that you need to remember we can never divide by zero ever when we see a problem and zero is the divisor we write the answer as undefined undefined so for example if I were to write a problem like 6 divided by zero you don't try to solve that you just say it's undefined undefined that's all you need to put next to it and you might think a little bit about why this is the case and I don't want to get into algebra or anything like that but just think about the fact that if you have 6 divided by zero it equals what just just try to come up with a solution there well do the related multiplication so something times 0 equals 6. well you come across a bit of a conundrum there because we know that when we multiply by zero we get zero so what number can you think of that when you multiply by zero produces six you can't find any so this is why we just say that it's undefined we just stop because we end up with sort of nonsensical statements like this all right the next thing is when we divide a number by itself the result is one and I put accept zero okay you can't divide 0 by itself because you can't divide by zero so if I have something like I don't know let's do an easy one let's say 4 divided by 4 this is one and think about why that's the case how many equal groups of four can I make out of four well only one right if I were to draw one two three four boxes well I can only make one group of four and it's the same no matter what you do if I was to say if I was to say 132 divided by 132 I'd still get one because if I drew out 132 boxes and I said well how many equal groups of 132 can I get out of that well just one all right so that works for any number as long as it's not zero okay so another important rule when we divide a number by one okay by one the result is itself so for example let's say you have five divided by one you get five okay five divided by one you get five and think about that in terms of the related multiplication statement if you think about the five there just say question mark what times one equals five okay what times one equals five well we know the answer to that is five and another way to think about it is again using your groups so if I have five if I have five of something let's say they're boxes and I split it up into equal groups of one well I'm always going to get that number back so one two three four five right and that works with any number and you can even use that one with zero because you can divide 0 by a number I could divide 0 by 1. there's no problem there zero divided by one just equals zero I just can't divide by zero okay now let's talk a little bit about division problems where you have a remainder so this basically means that you're going to have some amount left over and the way we talk about remainders when we're in grammar school is different from how we're going to talk about it when we kind of move forward in math and we're using you know calculators and we've learned how to work with fractions and decimals and so on and so forth so what is 7 divided by three well I have an illustration here with seven boxes and think about splitting up the seven boxes into equal groups of three what do you get well I Can See Clearly this is one two three here here's one two three so those are two groups but I have this one over here and I have nothing else to kind of pair with it to make another group so that's a left over amount I can't do anything with it that's considered a remainder so the answer for seven divided by three would be two with a remainder of one okay two with a remainder of one and when we check our division and you have a remainder you have to do it a little bit differently so now I'm going to multiply this part right here which is still called the quotient okay still called a quotient but then when I'm done so I'll do 2 times 3 2 times 3 and once I'm done with that operation I add my remainder to get back to the original number the dividend okay so 2 times 3 plus 1 would give me 7. so a little bit different to check when you have a remainder involved so another way again to think about this would be to use subtraction and if we wanted to subtract we would do 7 minus 3 this would kind of be the first time that would give us four then the second time we would do four minus 3 and that would leave us at one now I can't subtract another three away in other words if I do one minus 3 I'm going to end up with a negative value which I know we haven't talked about yet but basically I'm going to pass zero so I don't want to do that I want it to stop when I got to zero so we're going to have to stop here at 1. so that's my remainder that's my remainder okay so before I move on I want to talk a little bit real quick about the difference you might see when you do this on a calculator so all of us have done division on a calculator so you punch in 7 divided by three and you get something that says something like 2.3333 and basically this 3 is going to repeat forever so I'm just going to stop right there I'm going to put one two three dots this repeats forever when we did our division we said that the answer was two with a remainder of 1. so what gives why is there a discrepancy in the answer well we haven't gotten to fractions yet we haven't gotten to decimals yet but essentially your calculator is taking this one the remainder and it's dividing it by three and this part right here that comes after that decimal point is the result and again we're going to cover that later so for right now until we feel comfortable with that until we've covered that we're going to write this as 7 divided by 3 is equal to two with a remainder of one okay let's look at another example and again we're going to have a remainder here so we have 19 divided by 8. so how would we figure that out well again we could use subtraction we could count out how many times we can subtract away 8 from 19 until we get to zero so let's do it that way so kind of the first time 19 minus 8 would give us 11. and then the second time 11 minus 8 would give us three and kind of when we go to do a third time we have 3 minus eight and you think about this as we're going to pass zero in the wrong direction I don't have enough to subtract away eight so we're not going to be able to make a third group of eight with the number 19. so we've made two full groups and then we're going to have three left over okay two full groups so the answer is two with three left over and does that make sense just think about what that means two groups of eight two groups of eight two times eight that's sixteen then plus three left over plus three left over that's nineteen that's what we started with so 19 split up into equal groups of eight would give me two equal groups of eight with three kind of left over that I can't make into another group now if you didn't do this with subtraction if you wanted to start this out kind of with multiplication you would start out by just kind of guessing and I know you kind of think about that what do you mean you were guessing Well we'd have to say well what number times eight would give us 19. and you kind of go through your times tables and say well I don't know of any whole numbers that multiply with eight and give us 19. so you try to get as close as you could you'd say okay well 1 times 8 equals eight two times eight gives me 16. 3 times 8 gives me 24. and when you get to this point this number is too big so you'd stop you'd say okay it's not that number so we'd use two because that's as close as we can get so if we think about 2 times 8 gives me 16 2 times 8 again gives us 16. to get to 1993 more so that means I have a remainder of 3. and I know a lot of this when you first start hearing it if you've never heard it before it can seem a little bit complex but really it's not let's look at another one make it a little bit more clear okay so we have 21 divided by 4. so if we have 21 divided by 4 again we can use the subtraction method so kind of the first time we do this 21 minus 4 is equal to 17. the second time we do this 17 minus 4 is equal to 13. the third time we do this 13 minus 4 is equal to 9. the fourth time we do this nine minus 4 is equal to 5. and the fifth time we do this 5 minus 4 is equal to 1. now when I go to try to make a sixth group and I would do 1 minus four I can't pull 4 away from one not without going into a negative value which again we haven't covered yet so I have to stop here I can only make five equal groups of four with the number 21 and then I have one left over okay so the answer here is going to be five with a remainder of one and again just thinking about if that's logical to you if I take the number five okay five and multiply it by four I'm saying that I had five equal groups of four so that gives me 20. okay and then my remainder was one so I add twenty and one together to get the original amount the dividend of Twenty One and again if you don't want to use subtraction you can use multiplication you can use multiplication so 21 divided by 4 equals question mark again work backwards so question mark times four equals 21. so do we know a whole number that when we multiply by 4 we get 21. now so you can kind of start out and say well I know that I know that 4 times 4 16. I know that 4 times 5 is 20. and I know that 4 times 6 is 24. well 24 is too big too big and we know four four times four is not going to work that's too small because we could have made another group so I go with four times five and once I see my result is 20 all I have to do is say okay well the difference between 21 and 20 is just one so that's going to be my remainder so this would be this would be five okay that right there with a remainder of one because when I get to 20 I need one more to get to 21. all right let's look at one final problem so we have 37 divided by 3 and we're not going to use subtraction here because realistically you're going to find these answers by using multiplication right it's just quicker so I'm going to put a question mark here I'm going to put equals question mark and we're going to work backward using multiplication so question mark times 3 equals 37. so just thinking about our times tables here just think about multiplying by three what's going to get me close to 37 without going over it well I have I know 10 times 3 is 30. 11 times 3 is 33 . and I can keep going I can keep making more groups of three so 12 times 3 is 36 13 times 3 is 39. too big and you could have stopped when you got to 36 because you know multiplying the next number right when we got the 13 by 3 would add 3 or 36 and get you to 39 and we just needed to go up by one more to get the 37. so we would stop there and say okay well the results or the quotients going to be 12. but that's not the full answer because I kind of have one left over right I've made 12 equal groups of three but I have one left over so that's my remainder and again just to review this we start with 37 our dividend we split it up into equal groups of three our divisor we get 12. so we get 12 equal groups of three but there's a problem we have one guy kind of left over and so that's our remainder and again when you want to check something like this and there's a remainder involved you've got to multiply the quotient times the divisor and then when you're done add the remainder that's going to give you your dividend back and check this 12 times 3 is 36 36 plus 1 is 37. hello and welcome to pre-algebra lesson 14. and in this video we're going to learn about long division the lesson objective for today is to learn how to divide multi-digit numbers using long division okay so so far in our pre-algebra course we have reviewed how to do addition with whole numbers how to do subtraction with whole numbers and how to do multiplication with whole numbers now what we saw in each case is that when we had large numbers that we were working with like let's say with addition if we wanted to add two large numbers together we didn't do it in our head we used a process called vertical addition and what vertical addition does is it breaks it down into a series of easier steps so for example if I want to add 24 322 and I don't know let's say 308 I didn't just go okay this is the answer I would write 24 322 on top 308 on the bottom and I would go column by column until I got an answer so essentially what we're doing again is breaking this down into a series of easier steps now we saw the same thing when we were subtracting large whole numbers we used a vertical subtraction also when we were multiplying large whole numbers we used a vertical multiplication this is what long division is going to do for us in the last lesson we learned how to do some basic division but we didn't see any division with you know large whole numbers so if I asked you to do something like 20 divided by 4 pretty much everybody can do that without a long division so 20 divided by 4 is what you would use a related multiplication statement to find the answer what times 4 equals 20 right and from your knowledge of the times tables you know that that question mark would be replaced with a 5 and so 20 divided by 4 is 5. but what if I asked you what is 2 million six hundred forty two thousand three hundred eleven divided by 1622 most people cannot do that in their head I definitely cannot now I've met some people in college that could but the majority of us cannot so we would use a long division to solve a problem like that and all long division is going to do is break this problem up into a series of manageable steps let's start out by looking at a very simple problem one that we can do without long division I'm going to go through the steps here and we're just going to work a lot of problems I know most of you have done long division before and this is going to just be a review but even if you haven't you can pick it up from this lesson so let's try something like 30 divided by 6. that's an easy one so to set up a long division you'll have this new symbol and it looks like this and there's some variations of this sometimes you'll see your teacher go straight down however you want to do it I don't think it matters you're just trying to get a result here so whatever you're comfortable with I try to make mine look like that now you take your dividend remember that's the left number of the division problem that's the amount you start with this is the dividend that's going to go underneath the symbol and a lot of teachers will call this the house so it goes underneath the house you might hear so the 30 goes there this is for the dividend now your divisor which is 6 in this case this is your divisor is going to go to the left of this symbol so it sits out here this is for your divisor okay now once you've set up the division problem like this you go through a series of steps now I'm going to write those steps down I want you to copy them down on your paper you have to memorize them and there's a little trick I'm going to give you to do that so the first step is to divide and you're not going to divide the whole thing you're going to basically take the dividend on one digit at a time okay and I'll show you how to do that in a minute so you divide then you multiply you multiply you subtract you bring down and then kind of this next step is repeat or remainder and this will kind of make more sense once we get into the problem but basically what this is telling you is that again you have a series of steps to get your answer so you're gonna have to repeat the process sometimes when you come to the end of that there's no more numbers to say bring down you're going to have a remainder and in some cases you're not going to have a remainder the remainder might be zero it doesn't matter this is basically telling you hey stop the problem you have your answer all right now how do we remember something like this I know in a lot of places we'll have kind of tricks to remember stuff when you get to the order of operations you're going to hear something like please excuse my dear Aunt Sally for parentheses exponents multiply divide addition and subtraction well the trick for the long division is going to be dad mom sister brother Rover okay you think about your family so the D and dad for divide the m and Mom for multiply the essence sister for subtract the B and brother for bring down and then the RN Rover for repeat or remainder so again that's a little trick you can use to remember the steps all right now having said that let's kind of move into our actual division so what we're going to do is we're going to start by asking a question we take the divisor here and we ask how many times does it go into the leftmost digit of the dividend okay we're going to attack that dividend one digit at a time so how many times how many times will 6 go into 3. now a lot of people don't know what this terminology means when I say how many times will six go into three all that's asking is what is 3 divided by 6. what is 3 divided by six these two things are the same if your teacher asks you how many times will six go into three and what is three divided by 6 she asks you do the exact same question just a different way to put it so 6 goes into three zero times and I want you to think about y so I'm going to erase this real quick and I want to think about three boxes one two and then three remember if I say what is 3 divided by 6 I'm taking 3 and I'm splitting it up into equal groups of six now we are not using fractions or decimals some of you are going to go out oh that's 0.5 or oh it's one half well if we're only using whole numbers which is what we're doing here the answer is zero and the reason is because I cannot make a group of six having only three items if I have three dollars in my wallet and I need six dollars to purchase a hamburger I'm not going to get the hamburger okay unless somebody loans me another three dollars so the answer would be zero with a remainder of three okay zero with a remainder of three I made zero groups and I had three left over now with long division you never think about this part until the very end so I'm never going to include this in my answer that goes up here so once I ask how many times will six go into three I take this quotient part and just this part not the remainder and I put it directly over what I was working with okay you have to think about place value when you're working with the long division problem because you're lining this up with the three that's what I was working with I've done my first step here which was to divide now I'm going to multiply and what I multiply is the answer that I just had times the divisor so 0 times 6. we know that zero times any number is always zero so zero times six is zero so that step is done and then we subtract now notice that I wrote the zero directly underneath what I was working with we're always going to do that line everything up by place value with what you're working with so I'm going to subtract here 3 minus zero that's going to give me three and then I'm going to bring down we're always going to be bringing down the next digit to the right next digit to the right so now this is going to be a 30. and now we have the stuff that says repeat or remainder so I just brought down this zero to make a new number 30. so I'm going to repeat this process and go back up and I'm going to start by dividing again so now I'm going to ask how many times will 6 go into 30. so I'm basically saying what is 30 divided by 6. okay remember that's the same thing if I say how many times will 6 go into 30 or what is 30 divided by 6 I've asked you the same question so 30 divided by 6 equals what we can solve this using our times tables remember you just work backwards what times 6 equals 30 well 5 does so the answer would be 5. now I write 5 directly to the right of my previous answer remember I was working with this 30 here whereas the ones place for the 30 it's right here so I'm going to line up that 5 right there now I go through my steps again I'm going to multiply I am going to multiply so 5 which is the answer we just got times 6 the divisor gives me 30. write it underneath what you have that's what we're working with we got to line everything up based on that then we're going to subtract 30 minus 30 is 0. and then we want to bring down but there's nothing else over here to bring down if I highlight this section over here see no more numbers to the right so we're done bringing stuff down so when we get to this step where it says repeat or remainder if I had nothing else to bring down I don't repeat I go to the remainder part now because this is zero I know we say remainder there but because this is zero there basically is no remainder right so we think about 30 divided by 6 it's 5 with no remainder so to think about this more clearly if I had 30 boxes and I split it up into equal groups of six I would end up with exactly five of those groups right I wouldn't have anything left over now I know that for some of you you'll be like what step goes where as you work more problems you will get this down you memorize this just like you memorize anything else over time this will not be complicated at all now I want to show you a little trick that you're going to use and some of you will already be screaming and saying why did you put a zero right here my teacher doesn't show me that well the other way is kind of a shortcut so let me erase this real quick and I'm going to show you some other way recall that if I have a 0 next to this 5 this is still five right five is the same as a zero and then a five or if I put zero zero zero five that's the same as 5. these zeros to the left of five they don't add any value to the number whatsoever now let me start this problem over and show you what we could have done so for the first step divide we could ask okay how many times does 6 go into 3. if you know you're going to get a result of zero what you can do is just expand your selection to include the next digit to the right okay so now I can just ask how many times does 6 go into 30. realize that if I would have done this the first way how many times does 6 go into three zero zero times six is zero subtract and I get three bring down the zero I'm expanding my selection here anyway I'm doing the exact same thing it's just taking me longer because I went through all these steps to do it so the shortcut is just start out by saying okay how many times does 6 go into three zero times okay so I can just expand this and say now how many times does 6 go into 30. we know the answer is 5 already and the main thing is maintain your place value by putting the five directly over that zero think about that number 30 where's the ones place for thirty well it's where the zero is so I'm putting the 5 there and alternatively if that's hard for you to remember each time you kind of expand your selection you would have had a zero so I passed up that three so I would have had a zero there I just think about it that way either way you do it you're going to end up with five so then you would multiply 5 times 6 is 30. we'd subtract and get zero nothing to bring down over here and so where it says repeat we don't do that we didn't bring anything down our remainder will be zero meaning we don't have a remainder and so 30 divided by 6 is 5. okay let's look at a slightly more challenging problem but one that's still easy okay I want to take a look at a hundred divided by five so 100 divided by 5. what is that going to be again we're going to set this up in long division format so we're going to have this symbol that we're going to use remember your dividend which is 100 is going to go underneath and the 5 is going to go off to the left that's your divisor now what are our steps let me try to paste this and I'll put that right there so here's our steps remember you can think about your trick dad mom sister brother and then Rover now we're going to start with divide so take your divisor and then take on your dividend one digit at a time so 5 goes into one how many times so basically I'm asking what is 1 divided by 5. that would be zero with a remainder of one right if I had one box and I said split it up into equal groups of five using whole numbers again not using fractions or decimals I'm not going to be able to do that right I can't even make one group because I don't I only have one I need four more to make just one group so the answer would be zero and again we don't think about the remainder until the end so I can put a zero here or I can use my shortcut remember if I put 0 here and I go through and I multiply 0 times 5 is 0 I subtract I get one I bring down I have 10 and then I would repeat I could have gotten this 10 here just by expanding and not putting a zero there right so this is the shortcut so I can say how many times will five go into 10 5 will go into 10 twice right if you think about what is 10 divided by 5 go backwards what times 5 equals 10 we all know from the times tables that that is 2. now the main thing again is to maintain the place value I don't want to put a 2 right here if I think about that relative to this number 100 I'm saying that that 2 is in the hundreds place and it's not okay that 2 needs to be relative to this zero think about it in terms of this I was working with the number 10. where's the ones place for the number 10 it's where the zero is okay it's right there so I want to put the 2 directly in line with the ones place from 10. and again alternatively think about the fact that we would have had a zero here if we would have done it the slow way so zero then two okay now we multiply two times five is ten and again you want to line that up with the 10 and then subtract 10 minus 10 is 0 then bring down so it's always the next digit to the right and then repeat or remainder now 0 0 is the same thing as zero okay I don't need two zeros there I can just use one so when we repeat we just go back up to the top and we divide what is five into zero again think about this using the properties of division so I'm basically asking zero divided by five equals what remember if you take zero and you divide it by a non-zero number the answer is always zero okay and if you reverse that you can see why what times five equals zero well when you multiply zero times anything is always zero so the answer is zero okay so I'll write my zero directly to the right of that too and then I'm going to multiply 0 times 5 is 0 subtract get zero and then there's nothing else over here to bring down so when we get to this step where it says repeat we don't repeat because there's nothing else to repeat with we have our remainder or our termination step now this remainder is 0 meaning we don't have a remainder so 100 divided by 5 is 20 no remainder meaning if I had a hundred of something let's say boxes I split them into equal groups of five I would get exactly 20. all right let's take a look at another one so we have 315 divided by nine so again we're just going to set up our long division 315 divided by nine again your dividend goes underneath your divisor goes to the left okay so again we want to divide multiply subtract bring down then repeat or remainder dad mom sister brother and then Rover okay so we start with divide so we have our divisor here and again we're going to take on the leftmost digit of that dividend so how many times will 9 go into 3 meaning what is 3 divided by 9. well you can't do 3 divided by 9 with whole numbers right 3 divided by 9 would just be 0 remainder 3. so we can just expand our selection and say okay well it's not going to go into three how many times would it go into 31. how many times would it go into 31. so 31 divided by 9 equals what again work backwards what times 9 equals 31. we're using our times tables we know that there's no number or whole number that multiplies with 9 to give you exactly 31. so we have a situation where we want to get as close to 31 as possible without going over so if I think about 3 times 9 that's 27 3 times 9 equals 27. 4 times 9 would give me 36. this is too big 36 is bigger than 31 so I have to go with 3 that's as close as I can get so let's erase this and we'll put 3 down as our answer and again make sure to line that up with the ones place of 31 which we were just working with so the 3 is going to go here and again alternatively I would have had a 0 here if I went through the long way right when I said 9 goes into 3 0 times so that's where your 3 is going to go now we need to multiply and let me kind of fix this a little bit we need to multiply we're going to do 3 times 9 okay that's going to give me 27 and write that underneath the 31 right just underneath what we were working with then we're going to subtract so I'm going to borrow here and then 11 minus 7 is 4 2 minus 2 is 0. so I just leave that blank I don't need to put that and then I bring down always the next number to the right and I'm going to repeat right I brought something down I have something new to work with and I'm going to divide again so 9 goes into 45 how many times 45 divided by 9 equals what well this one's simple we can use our knowledge of the times tables to do this what times 9 equals 45 most of you will know that 5 times 9 is 45 so the answer would be 5. so I put a 5 directly next to that 3. so now we'd multiply 5 times 9 is 45 and then subtract we will get 0. nothing else to bring down over here I don't have any more digits to the right so when we get to this step we're not going to repeat we just say hey we're at our stepper we have our remainder now the remainder is 0 meaning we just don't have one it just tells us to stop right we've got our answer so 315 divided by 9 is 35 and let's go ahead and check this and particularly when you start out you want to check everything just make sure you get the right result and if you didn't go back and see what you did wrong it's probably some silly mistake as long as you understand how to do the steps and you get the wrong answer go back and check it and make sure that you didn't just make a silly mistake the answer here is 35 and working backwards 35 times 9 should give us 315. 9 times 5 is 45 9 times 3 is 27 plus 4 is 31. so yes we do in fact get our dividend back so we have the correct answer okay let's take a look at 942 divided by 30. so 942 divided by three and let's paste our steps okay so again we have divide multiply subtract bring down and then repeat our remainder so we start out by asking how many times will three the divisor go into the leftmost digit of that dividend so in this case it's nine now 9 is bigger than 3. so we're not going to get a result of zero here we're basically saying what is 9 divided by three I think all of us know that off the top of our head it's three right but if you didn't you could just say okay well what times 3 gives me 9 from your knowledge of the times tables you know that would be three so put your answer directly above what you're working with I'm working with that nine so I'm going to put that 3 right there then I'm going to multiply 3 times 3 is 9 right that directly underneath the nine and then subtract I minus 9 is 0. and then we bring down so I'm going to bring down this 4 here so now we repeat the process and we want to divide again so how many times will 3 go into 4. well just once right if we think about what is 4 divided by 3 I would get 1 with a remainder of 1. and again we don't think about the remains until the very end so we just put this part so the answer would be one and notice how I write it directly to the right of that 3. okay maintaining place value now I multiply 1 times 3 is 3 then I subtract 4 minus 3 is 1 then I bring down so I'm bringing down the next digit to the right which is a 2. and then I repeat go back up and we're going to divide again and again as you do this more and more you see that these steps become very very easy to go through it just becomes very mechanical for you right divide multiply subtract bring down repeat divide multiply subtract bring down repeat and then at some point you run out of numbers to bring down and you end up with a remainder which in some cases would be zero meaning you don't have a remainder okay so we divide again 3 goes into 12 how many times I'm asking what is 12 divided by three and again I can use a related multiplication statement and just ask okay what times 3 would equal 12. and for my knowledge of the times tables we know that that would be 4. so I'm going to write a 4 directly next to that one and now I'm going to multiply 4 times 3 is 12. then we subtract 12 minus 12 is 0. nothing else to bring down and so we're not going to use this repeat step we go to the remainder right and the remainder is 0 here meaning we don't have one so 942 divided by 3 is exactly 314. and let's check it just going to work backwards and say okay well 314 times 3 should be 942. 3 times 4 is 12. 3 times 1 is 3 plus 1 is 4. 3 times 3 is 9. 942. again that's our dividend that we started with all right let's take a look at one that will have a remainder so 461 divided by 2 and again our steps for the division divide multiply subtract bring down and then repeat or remainder all right so we're going to start with divide how many times will 2 go into 4 the leftmost digit of this dividend well 4 divided by 2 is 2. I think all of us know that by now so I'm going to put a 2 right here directly over that 4 that I was working with and let me move this down just a little bit and now I can multiply 2 times 2 is 4 and put that directly underneath the four that you're working with and subtract 4 minus 4 is 0. now we bring down the six the next digit to the right and we repeat right we go back up when we divide so 2 goes into 6 how many times Well three times right because 6 divided by two is three right three times two is six so six divided by 2 is 3. so now we multiply 3 times 2 is 6. subtract 6 minus six is zero bring down bringing down a one here and then I'm going to repeat I want to divide again how many times will 2 go into one well it's not if your divisor is smaller then your dividend that you're working with here then you're not going to be able to do it using whole numbers you're going to end up with a fraction which we're going to cover later on but for right now 2 is not going to go into 1 right you get a result of 0 remainder one which we don't think about the remainder until the end so 2 goes into one zero times then you multiply 0 times 2 is 0. and then we subtract so let me kind of scroll down 1 minus 0 will be one and nothing else to bring down over here so we end up with repeat or remainder we're not going to repeat we didn't bring anything down we actually have a remainder that's not zero now so the remainder is one and for most teachers they want you to drag this back up here and write it next to the quotient so I would write R1 like that so 230 with a remainder of 1. now I want you to remember what you do to check something with a remainder it's a little bit more tedious so you will multiply the quotient which is 230 times the divisor to start so 230 times 2. a lot of you can do that in your head because of the trailing zero trick right 2 times 23 is 46 put a zero at the end you get 460. but the Long Way 2 times 0 is 0 2 times 3 is 6 2 times 2 is 4. now when I'm done with that I would I would add to that the remainder the amount that was left over so 460 plus 1 would be 461. and think again why this is the case I started out with 461 boxes I split them into equal groups of two I was able to make 230 of those groups but then I had one left over right I only used 460 out of the 461. I've got one that's just hanging out by itself it doesn't have another box to make into another group of two so that is my remainder okay let's take a look at another one we have 193 divided by 9. so 193 divided by 9. and again where's the steps so we start out with divide 9 goes into the leftmost edge of that dividend one how many times if this is smaller than this the result is going to be zero okay so I could put a 0 there or again I can just expand the selection and say how many times will 9 go into 19. well 19 is bigger than 9 so I know that that's going to go in at least once so 19 divided by 9 equals what what times 9 equals 19 well there's no whole number that multiplies with 9 that gives you 19 but 2 times 9 will give you 18. that's as close as you can get without going over so I'm going to put a 2 as my answer and again line that guy up if I'm working with 19 where's the ones place for 19 it's right here so that's where my 2 is going to go alternatively as I keep saying and I know you'll get tired of this but I want to make sure that you're clear on the place value part this would have been a zero so the two would just go to the right of that however you can remember that place value is important in long division now we're going to multiply 2 times 9 is 18. put that underneath the 19 and subtract 19 minus 18 is 1. now we bring down bringing down a 3 there and then we repeat right we go back up to the Divide so how many times will 9 go into 13 just once right because we found that 9 times 2 was 18. that's too big so it has to only go in just once now we multiply 1 times 9 is 9 and we subtract 13 minus 9 is 4. and nothing else to bring down over here so we're not going to repeat this is our remainder okay 4 is our remainder so drag it back up to the top and just push r four right remainder four and again to check something like this just go backwards so if this is 21 remainder 4 take your quotient multiply by your divisor 21 times 9 9 times 1 is 9 9 times 2 is 18 and then add your remainder back in So 189 plus 4 nine plus four is thirteen one plus eight is nine bring down the one you do in fact get 193 back right your dividend okay let's take a look at one with a two digit divisor now so I have a hundred twenty six thousand three hundred twenty three divided by 15. so nothing is going to change here the process is the same when you start getting into bigger divisors your guesswork becomes a little harder and I'll show you what I mean by that so here's our steps here and we have 126 323. okay so we start out by asking how many times will the divisor go into the leftmost digit of the dividend right The Divide step well it's not going to go in there again if the divisor which in this case is 15 is bigger than what you're trying to go into right kind of think about it as your temporary dividend or your step-by-step dividend in this case it's one it's not going to work right you don't have enough to make a group of 15 with one using whole numbers again so you would expand your selection or you could put zero whatever you want to do and ask how many times will 15 go into 12. well again 12 is smaller than 15 so it's not going to work so again you can put 0 or you can expand and say well how many times will 15 go into 126. well I know that I'll go in there but I just don't know how many times right we don't work with 15 and 126 very often so 126 divided by 15 equals what so what times 15 equals 126. here's where we can kind of do some estimating okay now I can think about 126 is just 120. make a little easier for us and a lot of times you'll be able to get close and see where you need to go and you can kind of fine tune it with some extra multiplication so I know that 10 times 15 is 150 using my trick for trailing zeros that's too big so what about 8 times 15 15 times 8 so 8 times 5 is 40. a times 1 is 8 plus 4 is 12. so that's exactly what we need to get to 120. but what we needed was 126. so let's see if that would work out as our answer so 15 times 8 we know is 120. we need to know what 15 times 9 is well you know how to add and multiply you know you would just add 15 to 120 that would give you 135 right no need to go through and do the multiplication that's too big it's larger than 126. so 8 is going to be what we're going to use that's as close as we can get without going over so we're going to line up that eight on top of the six which is the ones place for that number 126. and then we are going to multiply so 8 times 15 as I just said was 120. and then we subtract 126 minus 120 is six bring down always the next digits to the right so we bring down that 3 and then we're going to repeat all right so we go back up and we divide so how many times will 15 go into 63. so 63 divided by 15 equals what so what times 15 equals now let's round this down to 60. because I think all of us probably know that 4 times 15 would give you 60. and think about that's as close as you're going to get to 63 without going over because 5 times 15 would be 15 more than 60 or 75. so I'm going to use a 4 as my answer so I'm going to put a 4 here and then I'm going to multiply 4 times 15 as I just said was 60 and then we subtract 63 minus 60 is 3 and we bring down so I'm bringing down this 2 here and now I'm going to repeat I want to divide how many times will 15 go into 32 and this one's kind of easy because if you just think about doubling 15 what is 15 times 2 it's 30 right that's as close to 32 as you're going to get so we're going to put the answer of 2. and then multiply 2 times 15 is 30. and we're going to subtract 32 minus 30 is 2 and then we're going to bring down so bring down this 3 here and now we're going to repeat how many times will 15 go into 23 well we know the answer is 1 there because 15 times 2 as we just said was 30 that's bigger than 23 so we have to go with 1. okay so now we multiply 1 times 15 is 15. we're going to subtract 23 minus 15. we're going to borrow here 13 minus 5 is 8 and then we're going to bring down but there's nothing else over here so we're at our last step we're not going to repeat we didn't bring anything down so we're at the step where we say okay what's our remainder so our remainder is 8 and we could just drag that back up to the top and say okay the remainder is 8. now I'm not going to go through and check this one because of the amount of work that would be involved but if you want to check it at home I advise you to do so just to get the practice take this number here 8421 multiply it by 15. when you're done add 8 and you will get this number back our dividend 126 323. all right so let's look at one final problem we have 163 971 divided by a hundred fifty so again as your divisor gets larger in terms of how many digits it has your guess work on each step is going to be harder so let's write this in long division format 163 971 again we're dividing this by 150 and let me paste the steps here and so we begin with divide so how many times will 150 go into the leftmost digit of that dividend it's not going to go right 150 is bigger than one so we expand that to 16. it's bigger than 16 so we expand that to 163. so 150 will go into 163 one time and again put the 1 directly over that 3. I can't stress this enough because I just have a lot of students that'll go okay it's once and they put it over here in terms of this number here this if you think about the whole number you know when you have an answer this looks like it's in the hundred thousands place but really it needs to be here in the thousands place right so it's a completely different answer if you put that one over there so make sure you line that up on top of the three and again alternatively you pass this number up so it would have been a zero you pass this number up so it would have been a zero until you got there so that's where you would have lined it up to all right so now we're going to multiply 1 times 150 is 150 again lining that up with 163 and then we're going to subtract 163 minus 150 3 minus 0 is 3 6 minus 5 is 1. bring down okay we're going to bring down the nine can 150 go into 139 no it cannot so if I have 139 boxes and using only whole numbers I tell you to make a group of 150 you won't be able to all right you'd have 0 with a remainder of 139. so we put 0 here okay we put 0 here and we multiply 0 times 150 is zero and some of you at this point might say okay well and over here we just kind of skipped it over when you have a zero in the middle of a number it changes the value of the number you can't use that shortcut in this case so if I had 1005 these two zeros do add value to the number but if I had the number you know 17 and I put zeros off to the left this is not changing the value of the number so that's why we can skip that there once you get going you can't skip that anymore so if your result is zero you have to write it and go through the steps so zero times 150 is zero and then we subtract so we get 139 now we bring down okay we're going to bring down that seven and now we're going to repeat so 150 goes into 1397 how many times so let's do this down here because this is going to be a tough one so 1 397 divided by 150 equals what so this is where it gets complicated we don't know the times tables for 150. so we have to get somewhat creative Now using my trick for trailing zeros 150 times 10 equals 1 500 or 1500 however you want to think about that number now that's clearly too big but I can mentally subtract away 150 from 1500 it's pretty easy to do it's 1 350. that looks like that's going to work out so it looks like we need 150 times 9. all right again take 150 away from this you get 1 350. that's as close as you can get to this without going over we go back up to the top here and we put an answer of 9 and we multiply 9 times 150 as I just said was 1 350. then we subtract 7 minus zero seven I minus five is four and then three minus three is zero and one minus one is zero again if you have zeros over here it wouldn't change the value of the number so I just don't put them right so we have 47. so now we bring down this one and we have 471. so now we want to divide how many times will 150 go into 471. well if I mentally think about 150 times 2. I think everybody can kind of do that in our head that's going to be 300. 15 times 2 is 30. put a zero at the end so what about times 3 well that's going to be 450. might I just add 150 to that pretty easy to do in your head so that's as close as we can get without going over because if I went times four it would be 600 right adding it on 150 and that's too big so we're going to use three okay we're going to use three so I write a 3 here and then I multiply 3 times 150 is 450. and I know it's cut off right now but the next step will be to subtract let me scroll back down 1 minus zero is one seven minus five is two and then we want to bring down but when we come up back up here nothing else to bring down so we can't repeat now we're at our step where we have a remainder right this is what we have left over and that is 21. so if I drag that back up here I can just put R 21. so again if you want to check this answer 1093 with a remainder of 21 multiply 1093 which is your quotient by 150. when you get that result add 21 and you should get this number back the dividend 163 971. hello and welcome to pre-algebra lesson 15. in this video we're going to learn about exponents okay for the lesson objectives we want to learn the terminology associated with exponents we want to learn how to write a repeated multiplication in exponent form and we want to learn to solve problems that contain exponents so what exactly is an exponent well let me show you through an example let's say we had something like 3 times 3 times 3 times 3 times 3 times three times three that takes up a lot of room on my sheet I wish there was an easier way to write that well in fact there is using an exponent essentially what I would do to write this in exponent form is I would count the number of Threes that I have there I have one two three four five six seven so I would take this number that I'm multiplying by itself this 3 and I would write it down so I would have my 3 there and then I would put an exponent which is a small little number that goes up in the top right corner just telling me how many factors of 3 that I have this number right here is called a base it's called a base when you write a number in exponent form the base is the number that is being multiplied by itself in this case it's three right I have 3 times 3 times 3 times 3 times 3 times 3 times 3. so 3 is the base the seven again this little small number in the top right corner is the exponent the exponent and again this tells us how many factors of 3 we have let's think about another one let's say I had two times two times two if I want to write this in exponent form I have this number 2 that's going to be my base 2 is going to be my base again it's the number that we're multiplying by itself and then I'm doing it three times one two three I have three factors so that's the exponent 3 is the exponent so for the purposes of pre-algebra you can just think about an exponent as telling you how many factors of the base you're going to have because the only exponents we're going to look at at this point will be whole numbers usually one or larger okay and in a lot of cases just going to be two or larger we have some special rules when we start dealing with exponents that are 0 and 1. so essentially for pre-algebra you're just going to think about it using whole number exponents and it's going to tell you how many factors of the base that you have and again when you think about 0 and 1 you'll have some special rules for those so any number raised to the power of 1 is just itself so in other words if I have 6 and it's raised to the power of one and I say well what is 6 to the power of one well it's just itself it's just six or I could flip that and say okay I want to write 12 in exponent form how would I do that well I'd take 12 that would be the base and then I would take 1 and write that for my exponent so if you have a number that's just hanging out by itself let's say 13. and again you want to write in an exponent form you just write the number as the base and put a 1 as the exponent so any number raised to the power of 1 is just itself now we have a rule for a number raised to the power of zero but we're going to cover that later on because that's something we need to go through and explain so for right now we're just going to work with again whole number exponents that are generally going to be two or larger and we know the rule for one so we're safe there all right so let's talk a little bit about some language that you're going to hear when you work with exponents so when we raise something to a power we generally say the number to the and then that power so for example if I have 14 and my exponent is let's say 7 this would be the number which is 14 and then to the in this case it's the seventh power right because it's a seven so 14 to the seventh power if I have 8 and my exponent is 4 this would be 8 to the fourth power if I have 9 and my exponent is 13 it would be 9 to the 13th power so on and so forth okay so there's two special cases that I want you to know oh and the first one is when we raise the number to the second power okay meaning we have an exponent of two we say the number is squared if you don't know this at this point take out a pen and a piece of paper and write it down you need to memorize that because moving forward you're going to hear that all the time you're going to have somebody say what is 6 squared what is 4 squared what is 8 squared and you might be going I don't know what that means when we say squared we're meaning raised to the power of 2. so I could say this as 6 to the power of 2 I could say 6 to the second power or I could say 6 squared they all mean the same thing so very important that you understand that definition and another scenario that's going to come up when we raise a number to the third power we say the number is cubed so that means you have an exponent of 3. again write this down it's going to come up a lot so if I have 4 to the third power we generally will say that's 4 cubed now you're not wrong if you say 4 to the third power or if you have 4 squared you say 4 to the second power it's just you need to know both in case your teacher asks you it might say what is 4 cubed and you go well I don't know what that is you'll be in trouble so again memorize that when someone says something is cubed it means they have raised it to the third power when something is squared they have raised it to the second power okay let's do a little basic practice here we want to write each in exponent form another thing that's going to be too complicated so I have the number seven and it's being multiplied by itself so I have 7 times 7 times 7 times 7 times 7. so the base is the number that is being multiplied by itself so obviously that is 7. again this is the base and then my exponent tells me how many factors of 7 I have so it's one two three four five 5 is the exponent okay so this is 7 to the fifth power here we have six times six times six times six times six times six times six so the base is obviously six right that's the number that's being multiplied by itself and then what's the exponent what is the exponent let me just label this real quick as the base the exponent again tells us how many factors of 6 we're going to have one two three four five six seven so this would be to the seventh power this is my exponent what about 9 times 9 well the base here again is the number that's multiplied by itself and that's 9. so this is my base and the exponent tells me how many factors I have so I have one two of those so the exponent is two this is my exponent very very easy what about four times four times four again my base is going to be the number that's multiplying by itself so that's going to be 4. and then my exponent tells me how many factors I have so I have one two three so three is my exponent okay so I have here that we also need to be able to reverse this process so what does that mean well basically what I'm saying is that when you see a problem with exponents in it you need to be able to reverse it and write it as a repeated multiplication so in other words if I have 10 to the fourth power write this using multiplication so I know the base is 10. so that's the number that's being multiplied by itself and the exponent is 4 telling me I have four factors of 10. so this is 10 times 10 times 10 times 10. all right I just did the thing I was doing before only backwards right I took it from where I had an exponent and wrote it with multiplication whereas before I was taking with multiplication and writing it where it had an exponent what about 9 to the sixth power well my base is nine so that's the number I'm multiplying by itself and 6 is the exponent that's how many factors I'm going to have so 9 times 9 times 9 times 9 times 9 times 9. I have one two three four five six six factors of 9. what about twelve to the tenth power well I'm gonna have ten factors of 12 right my base is twelve again that's the number that's being multiplied by itself and 10 is the exponent tells me how many of them I have so this is 12 times 12 times 12 times 12 times 12 times 12 times 12 and I lost count how many is that one two three four five six seven so I need three more so times 12 times 12 times 12. so now we have 10 factors of 12. then the last one we'll look at is 5 cubed or again you could say 5 to the third power or 5 to the power of three doesn't matter as long as you know that this is three factors of five right the exponent is a three and the base is five so that means I have five times five times five okay so the next thing you need to be able to do is evaluate when you see an exponent so if I see something like 2 to the seventh power and your teacher says what's the value for that or what's it equal to well the first thing I'm going to do is I'm going to write this using multiplication so in other words I have 2 as my base and I have 7 factors of 2. so just exactly what we just did so 2 times 2 times 2 times 2 times 2 times 2 times 2. so 7 factors of 2. and then I just go through a multiply and a lot of times especially for these scenarios that come up often you'll just memorize what the answers are I already know that 2 to the seventh is 128. I don't even need to do the multiplication but we're going to go through and say 2 times 2 is 4 4 times 2 is 8 8 times 2 is 16. 16 times 2 is 32 32 times 2 is 64. 64 times 2 is 128. 128. what about 3 to the fourth power well again we're going to write this using multiplication and then just do the multiplication so 3 to the fourth power is three times three times three times three again four factors of three the four is the exponent tells me how many factors I have 3 is the base that's the number I'm multiplying by itself so 3 times 3 is 9 9 times 3 is 27 27 times 3 is 81 81. what about 17 squared or 17 to the power of 2. well this is 17 times 17. and I don't know this off the top of my head I would have to either use a calculator or do vertical multiplication so let's just do that on the screen here so 7 times 7 is 49. 7 times 1 is 7 plus 4 is 11. and now the benefits of this is I'm multiplying 1 times 17. so that's easy to do right I just shift this over here and I can just basically write 17 as as my answer and now I can just add bring this down 1 plus 7 is 8 1 plus 1 is 2. so we end up with 289 and so we'll write this as equal to 289 . okay what about 15 squared well again we would write this as 15 times 15 and then we would multiply so I happen to know 15 times 15 I'll stop my head but in case you don't you would do a vertical multiplication this is 225 and this is one that's going to come up quite a bit okay what about 4 to the fifth power so this is 4 times 4 times 4 times 4 times 4. so this one's pretty easy to do in your head four times four is 16 16 has 464 64 times 4 is 256 and then 256 times 4 is 1024. and again this is another one that I have memorized we'll learn later on that 4 to the fifth power is the same as 2 to the 10th power and I know that probably doesn't make any sense to you right now but later on you're going to understand why that's the case so the answer for right now though is again 4 times 4 times 4 times 4 times 4 which gives us 1024. okay lastly let's talk about 10 raised to a whole number larger than zero this right here what I'm going to teach you is going to save you a lot of time because you come across these problems a lot and there's a very simple way to solve them so let's say I have something like 10 to the second power what's that equal to well it's 10 times 10. now I taught you in the lesson where we talked about multiplying with trailing zeros you could do one times one and then attach two zeros to this so this is a hundred notice how 10 squared produced two trailing zeros this is two two trailing zeros look at 10 cubed this equals 10 times 10 times 10. 1 times 1 times 1 is 1 3 trailing zeros one two three so three as the exponent three trailing zeros I think some of you can already see where I'm going with this if I have 10 to the fourth power 10 to the fourth power I get 10 times 10 times 10 times 10 equals 1 times 1 times 1 times 1 which is one and then one two three four trailing zeros one two three four so again the exponent is 4 and I have four trailing zeros so here's the rule if we're working with whole numbers and they're larger than the number one so meaning two and moving out forward okay I didn't include one in that because 10 to the first power is just 10. and we learned that earlier but essentially what you do is you write a one followed by the exponent number of zeros that's it it's very very simple so this is especially helpful when we have some huge number let's say we have something like 10 to the 11th power I could go through and write out okay 10 times 10 times 10. basically 11 factors of 10. you could go through and multiply them I'd be there all day if I didn't use the trick for trailing zeros using this trick with exponents it's very very simple I write a one and I follow it with 11 zeros one two three four five six seven eight nine ten eleven comma comma comma so I end up with 100 billion and go ahead and punch that up on a calculator you see that it's correct and again this works so you could do something like 10 to the 15th that would be a one followed by 15 zeros or ten to the 90th that would be a one followed by 90 zeros so using this technique can help you out a lot because a lot of times on your test you'll get at least one to two of these where they'll say okay what is 10 to the fourth or what is 10 to the seventh and you just write a one and you follow four zeros or with 10 to the seventh you write a one and you follow it with seven zeros and you're done right you don't have to sit there and go through and multiply and do all this extra work hello and welcome to pre-algebra Lesson 16 and in this video we're going to learn about the order of operations so the lesson objective for today would be to learn how to solve problems with multiple operations involved okay so up to this point we haven't really discussed any general rules for solving a problem with multiple operations in it this is what the order of operations is going to do for us the order of operations tells us which operation to perform in which order when we're faced with a problem that contains multiple operations so I'm going to read through the order of operations and then we're going to work a bunch of problems so kind of the first step or you would say your highest priority when you're working a problem would be to work inside of any grouping symbols and then I have a side note here if multiple grouping symbols are present you start with the innermost set and work outward okay then the second step or your second highest priority would be to perform all exponent operations okay so anything that involves an exponent you need to do that second then third are your third highest priority is multiplication and division now a lot of students get confused with this you want to multiply or divide working left to right so multiplication does not come before Division division does not come before multiplication they are on the same level you do them from left to right so if I had something like four times five divided by two I would multiply first so I would do 4 times 5 first and then I would take that result which is 20 and I would divide by 2. so this would be 20 divided by 2 which would give us 10. okay now if I kind of flip the order around let's say we use some different numbers here let's say I had 10 divided by 2 times 3. in this case division occurs to the left of multiplication so we would now divide first 10 divided by 2 is 5. is 5 and then we would multiply by 3 giving us 15. so you can see that multiplication division again it's on the same level it's the same step we do it working working left to right okay then your lowest priority your fourth priority is addition and subtraction so I have here add or subtract again working from left to right so addition and subtraction occur on the same level just as we solve multiplication and division so if you have something like 4 plus 3 minus one you would add four plus three first or plus three is seven and then you would subtract away one if I was to switch this around and put a subtraction sign here an addition sign here you would now subtract first right because subtraction is to the left of addition 4 minus 3 is 1. then 1 plus 1 is 2. okay so we're going to start looking at some problems here and really there's not much to the order of operations other than memorizing what's the priorities one two three four and a lot of students will use kind of some clever tricks to remember it I know that one of them when I was in high school was please excuse my dear Aunt Sally right and you'll hear this all the time you'll hear PEMDAS Pam Das but you have to be careful with stuff like this this is please excuse my dear ant Sally you have to be careful with this because this is basically parentheses exponents multiplication division addition subtraction remember these two occur on the same level and then these two occur on the same level you have to remember for this step it's left to right and I can't stress this enough because I have a lot of students that kind of mess this up because they use things like this and they say well no multiplication comes before division PEMDAS multiplication division left to right left to right addition subtraction again left to right left to right so if you can't remember that please take out a pen and a piece of paper write that down multiplication in division are on the same level we perform them left to right addition and subtraction on the same level we perform them left to right okay now having said that let's start on our first problem and we want to just evaluate each which means we're just going to find the value so when we look at four minus three plus five that's our first problem we have two operations here we have a subtraction operation and we have an addition operation so which one are we going to do first again addition and subtraction occur on the same level so we're going to work left to right so I start at the left part of the problem and I go to the right so 4 minus 3 will be done first and 4 minus 3 is 1. so I'm going to replace this part right here this operation with a 1. and then I'm going to write the rest of the problem so then Plus 5. and then 1 plus 5 I do that that's 6. very very simple next we have 5 times 2 minus 1. so our operations here we have multiplication and we have subtraction so multiplication has a higher priority we need to do that first so I'm going to do 5 times 2 first so what is 5 times 2. let me just highlight this 5 times 2 is 10. so I'm going to replace this part right here this operation with a 10. I'm gonna put a 10 there and then I'm just going to copy what remains so then I have minus 1 minus 1. and now I'm free to perform that operation ten minus 1 is 9. okay for the next one we have 4 squared minus 3. and if we look at what we have here we have an exponent involved and then we have subtraction so the exponent has the highest priority and so we would evaluate 4 squared first 4 squared means that I have two factors of four so four times four that is 16. so I'm going to replace this with 16. and then we copy what's left from the problem it's just minus three minus three so 16 minus 3 would give me 13. okay so now we're going to make them a little bit harder up to now we've only seen things with kind of two operations involved now we're going to kind of get into three four five and more operations involved in one problem so we start out with the quantity two plus one and again I say the quantity because two plus one is inside of a set of parentheses then we multiply by three then we multiply by two now the highest priority and any problem is always what's inside of the parentheses so I would perform this operation before anything else and where students get confused is they say oh no you have addition here well it doesn't matter because when you have parentheses you must work inside of them first so we're going to do this first so what is two plus one two plus one is three so I'm going to write that over here I'm basically just replacing this with a 3. and then I copy the rest of the problem I have times 3 times 2. so I'm just going to work this left to right 3 times 3 is 9. and then 9 times 2 is 18. so this would be a final answer of 18. okay now we have 6 Plus 8 divided by the quantity 5 minus 1. again we have this quantity here 5 minus one because it's inside of parentheses right the 5 minus 1 is inside of parentheses so we perform whatever operation is inside of parentheses first so 5 minus 1 is 4. so I'm just going to replace this with a 4 and copy the rest of the problem so 6 Plus 8 divided by and again the result of this is 4. so now if we look we have addition and we have division so division is a higher priority okay we want to divide before we add so that means I'm going to do this problem first which is 8 divided by 4. so 8 divided by 4 is 2. so again this right here will be replaced with a 2. so I'll have six plus two six plus two and to finish this off six plus two is eight okay for the next problem we have the quantity one plus one divided by and then look inside of parenthesis here we have something that's a little bit longer than what we're used to we have six minus two times two so we have two sets of parentheses here now here's what really matters you need to make sure that you perform the operations inside of the parentheses separately from this division okay so in other words you'll get some result here and you'll get some result here and then you're going to perform your division not before okay that's what those grouping symbols are there for so let's start out with the set of parentheses on the left we're going to do one plus one we all know one plus one is two so we're going to replace this with a 2. and then divide it by and then inside of parentheses we have 6 minus 2 times 2. so we're not doing any division yet we're still working inside of parentheses still working inside of parentheses now once you get inside of a set of parentheses you might have a decision to make you see how you have subtraction here and you have multiplication you have to reapply your order of operations multiplication occurs before subtraction it's a higher priority so we would do 2 times 2 first okay so 2 times 2 we know is 4. so let me just circle this guy this right here is going to be replaced with a 4. okay so equals 2 divided by and inside of parenthesis I'm going to have 6 minus and the result of this again was 4. and we're still continuing inside of the parentheses even though we just have one operation left we want to do 6 minus 4 and that's two so we're going to have 2 divided by 2 right because the result of this is 2. and then we finish this problem up by performing the division two divided by 2 is 1. okay for the next one we have the quantity 14 minus 2 then divided by and here we have a 2 next to a set of parentheses so again that implies multiplication so two times the quantity five plus one so we know we're going to start out by doing the operations inside of parentheses first before we do anything else so we're going to start with 14 minus 2. 14 minus 2 is 12. so I'm going to replace this with a 12. okay and I just copied the rest of the problem next we're going to work inside of this set of parentheses here 5 plus 1 is 6. so I'm going to say this is 12 divided by 2. then remember this implied multiplication so this will now be times 6. okay just because we didn't have a multiplication symbol there again you have to remember that as we move higher and math when you see a number next to a set of parentheses again it implies multiplication so this is our problem now we have 12 divided by 2 times 6. remember if we have Division and multiplication they occur on the same step so we're going to work them left to right so that means I'm going to divide 12 by 2 first and then multiply by 6. so 12 divided by 2 is 6. and then times six so then 6 times 6 is just 36 and that gives us our final answer okay let's take a look at one final problem so we have 6 squared minus and then we have the quantity two plus three minus and then we have another quantity 3 plus 13 and then finally divided by 4. so again the highest priority you're going to face in any problem is always going to be what's inside a parentheses so we have parentheses here we have parentheses here and let's figure out what two plus three is first so I'm going to rewrite this part 6 squared minus 2 plus 3 is 5. so I'm just going to replace this with a 5. and then minus we have this inside of parentheses 3 plus 13. and then divided by 4. now the next step is to work inside of these parentheses we're going to do 3 plus 13. so we're going to have 6 squared minus 5 minus so what is 3 plus 13 that's 16 that's again 16. and then divided by 4. okay so now when we look at this problem I have an exponent I have subtraction more subtraction and Division so the exponent is what we're going to deal with first that's the highest priority six squared is two factors of six or six times six which is 36. so again I just replace this with a 36 and I can copy the rest of the problem so I have minus 5 minus 16 divided by 4. now I have subtraction more subtraction and division division has the highest priority so we would do 16 divided by 4 before we do anything else 16 divided by four is four so I'd have 36 minus 5 minus and again this operation right here resulted in four and now I have just straight subtraction so I'm going to work left to right 36 minus 5 is 31 so that's 31 right here and then we have minus 4. and then lastly 31 minus 4 is 27 and that is our final answer hello and welcome to pre-algebra lesson 17. and in this video we're going to talk about integers okay so for our lesson objectives we want to discover a new set of numbers known as the integers we want to use the number line to determine the relationship between integers we also want to learn how to find the opposite of a number and then lastly we want to learn to simplify with multiple and then you see this symbol here most of you at this point will know it as a subtraction symbol throughout this lesson we're going to see that symbol used for different things so I'm just going to put multiple and then show you this on the screen say symbols and again we're going to change the context of that throughout our lesson all right so let's begin by talking a little bit about the whole numbers now up to this point we've only worked with the whole numbers and typically these are the numbers you work with in elementary school we've reviewed how to add whole numbers and we did that on a number line too how to subtract whole numbers again we did that on a number line we reviewed how to multiply whole numbers and how to divide whole numbers and the reason I spent time on that was that if you had forgotten that when you go to do these operations with integers you're just going to be behind so again the whole numbers start with the number zero an increase in increments of one out indefinitely so again remember you'll have these three dots here after three that just tells you you'd have four then five and six essentially that pattern just continues forever now if you look at this illustration here it's of a number line and remember this is just a visual representation of in this case it'll be the whole numbers but in a second we're going to see that we can draw a visual representation of the integers as well so look how this is drawn we have our first notch or our leftmost Notch occurring where zero is and then each Notch to the right represents the next whole number right an increase of one so you got one then two then three and four so on and so forth at some point we stop and we put an arrow there and the arrow just says that hey numbers are going to continue forever right and since this is a number line that represents the whole numbers you can just basically assume that you'd have you know 9 then 10 then 11 then 12 then 13 and 14. so on and so forth so every number that is larger than zero is known as a positive number we can place a plus symbol okay the plus symbol or you might know it as the addition operator in front of a number to indicate that it is positive we don't typically do this because numbers are by default positive when we work with them right that's what we're used to working with so typically you will not put a plus sign in front of it so for example if you have the number three you would just write it like that you know that it's positive but if you wanted to you could put the plus symbol in front of it to indicate hey this is a positive number this is a positive number positive 3. so when we encounter a number that is less than zero okay less than zero we refer to it as a negative number a negative number so negative numbers always have a and again I talked about this previously in this lesson this symbol right here which we've used for subtraction up to this point so this symbol we put that sign in front so if I wanted to say write down the number negative 3. I would put this symbol directly in front of a 3 and I would have a negative 3. so this is a negative a negative three and again if I wanted to write positive 3 I just write the number 3 or I could put a plus 3 in front of it in this case that plus is optional but when you're writing negative 3 you have to write it it's not optional because nobody will know that that's a negative 3. again the default value of a number is going to be that it's positive right because that's what we normally work with so before we move any further what exactly is a negative number so some of you might say well I've I don't know what a negative number is well in most cases all of you have seen some form of a negative number you might not have worked with them directly in math but you've seen a negative number before one thing or one example that's used in almost every textbook would be a negative temperature so you're watching the weather on TV or maybe on your phone or something like that and you find out that tomorrow it's going to be you know negative five degrees essentially it's colder than zero right it's going to be a lower temperature than zero and they represent that with a negative value negative five another common example would be that you have a negative amount of money in your bank account so let's say you have five dollars in your checking account you go to the store and you buy ten dollars worth of groceries okay let's say your bank covers the charge so you had five dollars you spent ten so you're going to end up with negative five dollars in your account plus probably a fee from your bank saying hey you spent more than you had but that's another example of where we would use a negative value kind of a sports one that you could think about let's say that you're watching a football game and they hand off the ball running back runs and he loses yards let's say there's a loss of four on the play well if you see them represent that with numbers they're going to say hey that's a play that yielded negative four yards so we have negative numbers around us we don't use them as often as we do the positive numbers but they are around us now let's talk a little bit about zero so a lot of people get confused about this they think that zero is a positive number it's not zero is special so it is not positive or negative it is neither later on you're going to hear that zero is an even number but it is not positive and it's not negative okay just think about it as being neutral okay so now let's talk a little bit about the integers now the integers are important because from here on out you're going to work with them a lot so the integers include the whole numbers and their negative counterparts so we already know what the whole numbers are again I just put that on the screen at the beginning of the lesson if you look at the integers here if you just look at this part where it says zero then one then two then three then comma dot dot dot that's the whole number part okay this is the part that we know oh numbers now what we're just introducing is numbers that are to the left of zero so you kind of think about one well it has a negative counterpart negative one you think about two it has a negative counterpart negative two we're just counting like we normally do we're just counting a different way now so the numbers are the same you go one two three four five you just have negative symbols in front of them instead so it's negative one negative two negative three negative four negative five negative six so on and so forth all right so let's take a look at the integers on a number line so you'll see that zero is at the center and then to the right of zero you have your positive numbers okay I'm going to put a plus symbol just out here to say that this is positive so anything to the right of zero is positive anything to the left of zero you see how we use these negative signs okay directly next to the number to say that these numbers are negative these are negative numbers negative numbers so again left of zero is negative right of zero is positive and zero is neutral okay it's neither okay so to understand the relationship between integers we need to do some extra thinking so what if I was to ask you is negative 4 greater than one is that true well in fact it's not and what kind of trips people up as they see four and they see one and they say well four is a bigger number negative 4 is actually a smaller number as these numbers that are negative get kind of bigger meaning you have bigger negatives they're actually getting smaller and to see why let's look at a number line so you remember in the lesson where we introduced the number line I told you that as we move right on the number line the numbers are getting bigger so meaning 1 is bigger than zero two is bigger than one three is bigger than two you know eight is bigger than four so on and so forth if a number is to the right of another on the number line it is bigger the same is true if a number is to the left of another on the number line it's smaller so you know 5 is smaller than 6 because it's to the left of six or two is smaller than four because it's to the left of four well look as we're moving this way we're getting bigger and bigger negative numbers but again we're moving left so the numbers are actually getting smaller and you can kind of think about this using money if you want I think that's the easiest way to think about it if I have negative 4 if I have negative four dollars in my account do I have more money or less money than if I have one dollar in my account what's the relationship between these two numbers well one is a bigger number negative 4 is going to be less than one because in this case I actually owe four dollars not only do I not have any money I actually owe four dollars to somebody so I'm in a much worse situation so this is a smaller value than one so we would say negative four is less than one and if you ever get confused about it you can think about it in two different ways the first way is you can do this you can draw yourself a little sample number line and say okay well you know where is this on a number line where is this you know again the one to the left is the smaller number the second thing you can do is kind of use your common knowledge of numbers that you've worked with in the past so let's say you have two positive numbers you have 50 and you have 10. so everybody knows that 50 is a larger number so 50 is greater than 10. but if I put negatives in front of these numbers this is no longer true we would have to flip this to in fact get a true statement negative 50 is now a smaller number than negative 10 because negative 50 is a bigger negative number all right let's look at a quick exercise here we want to replace each question mark with the inequality symbol less than or greater than so we're going to start out with negative 2 question mark negative 5. so what's the relationship between these two numbers negative two is here on the number line negative 5 is here again negative 5 is to the left of negative two so it's a smaller number I told you that the symbol points to the smaller number so if we look at negative two and negative 5 again my choices would be to use this symbol which is the greater than or this symbol which is the less than point the symbol to the smaller number negative 5 is smaller so we use the greater than Okay negative 2 is greater than negative five and again thinking about it the other way negative 2 is to the right of negative five so it's bigger or again thinking about it the way I just said negative 5 is to the left of negative 2 so what's smaller either way negative two is greater than negative five what about one question mark negative one one question mark negative one so one on the number line is here negative one is here so negative one is a smaller number it lies to the left of one on the number line so we would have one and we would have negative one and I want to point the symbol to the smaller number so again I'm going to use a greater than symbol okay the greater than symbol is going to point to the negative one so one is greater than negative one and again if you get confused on something like that just kind of think about money if I have a dollar do I have more than if I owed a dollar well yes I do so one is a greater amount let's take a look at negative seven question mark negative four so here's negative 7 and here's negative 4. so negative seven is to the left of negative 4 in the number line so it's a smaller number so we would have negative seven is less than again the symbol is going to point to the negative 7 because that's smaller negative 4. okay now let's talk about opposites which are also known as additive inverses so opposites are numbers that have the same distance from zero on the number line but lie on opposite sides of the number line so if you kind of think about this you would have like 1 and negative one two and negative two three and negative three four and negative four basically it's just the number and then the negative of that number so you could make an infinite number of these you can go through and say okay five and negative five six and negative six so on and so forth so with opposites it's a very easy concept so to find the opposite of a number we just change the sign that's all you're doing because if I say hey what's the opposite of one again you're just looking for that same value but just with a different sign so you would just change the sign and say okay well the opposite is negative one so we want to find the opposite of each and again you might see this in your textbook if you're in pre-algebra it might say find the additive the additive inverse okay so just know that those two terms are interchangeable if I say hey find the additive inverse or say hey find the opposite they're asking for the same thing all right so the opposite of 5 the opposite of 5 is negative 5. again all I do is I take the number and I just change the sign so this is positive it becomes negative the opposite of negative 12 would just be positive 12. again just change the sign the opposite of 315 would be negative 315. again just change the song from positive to negative and then the opposite of negative one million seven hundred eighteen thousand three hundred eleven would be positive 1 million seven hundred eighteen thousand three hundred eleven very very easy to find the opposite of a number okay so now we're going to use this concept to actually look at some problems we're going to deal with so I have here to simplify each and we start off with something that looks kind of weird we have what looks like a negative symbol or subtraction symbol and then in parentheses we have negative four so what does this mean well essentially when you have this symbol directly next to a number it's saying hey this is negative this is negative so I would read this as negative 4. when you have this symbol right here and it's outside of a set of parentheses basically what I'm saying is hey change the sign of whatever's inside of parentheses but you could read this as the opposite of the opposite of so this would be read the opposite of negative four and it's just saying hey change the song to negative 4 what would you get so whenever you see this what is the opposite of negative four it would be positive 4. again just as I showed you with these other examples when I said what is the opposite of negative 4 we just change the sign that's all we're doing here you just have to know how to read it the opposite of negative 4 is 4. so that's why we'd write it that way now we have the opposite of negative 3. again this is the opposite the opposite of and then this is negative 3. and again all that's saying is hey whatever's in parentheses which in this case is negative three change the song so whatever's in parentheses again this is negative 3 in parentheses change the sign it would be positive 3. or if you saw a negative symbol outside of a set of parentheses and there was a positive number in there let's say it was 5. you would just change the sign of five hey what is the opposite of five negative five okay that's all you're doing okay so before I kind of get into the next problem I want to address something that I haven't talked about yet but it's the concept of an odd or an even number so I'm going to briefly talk about this so an even number is a number that is divisible by two and when I say divisible by I mean you divide it by 2 and you don't have a remainder okay so in other words 6 divided by 2 is 3 so 6 is an even number or 10 divided by 2 is 5 so 10 is an even number 3 divided by 2 would be one with a remainder of 1. that would be not an even number that's something called an odd number so an odd number is not divisible by two okay not divisible by two five is an odd number 5 divided by 2 would be 2 with a remainder of 1. so that's an odd number and again we'll get more into this when we get to the divisibility rules but you need to have a basic definition of this to apply this trick that I'm going to teach you all right so what about when we see a scenario like this where we have multiple symbols in a row so we have the opposite of the opposite of negative 15. all right kind of the slow way to do this is to think about okay the innermost set of parentheses contains negative 15. if I kind of move out of that I would be asking for what is the opposite of what is the opposite of this negative 15 here well it's 15 right just change the sign so kind of contained in these parentheses here I would just get 15 as a result so I'd have this negative out here or begin asking for the opposite of and then I would have inside of parentheses just a 15. then this negative or again asking for the opposite of is telling me to change the sign of whatever's in the parentheses one more time so what is the opposite of 15 well that's just negative 15. so very very easy but if you had a longer problem it would be ridiculously time consuming the quick way to do this is just to realize that each time you have a pair of negatives it's going to make a positive right so this one and this one are a pair so that makes it positive right you recall that we ended up with a positive 15 inside the parentheses right and then we have the negative out there now there's nothing else to pair up with this so it becomes negative so you think about a pair as being 2 so 2 4 6 8 10 12. these are even numbers so when you have a pair Things become positive so all your problems where you have even amounts of those negative symbols will give you a positive if you don't have an even amount of those symbols meaning you have an odd amount of those symbols you get a negative result we have one two three of those meaning we could only make one pair of negatives and then we couldn't do anything with that one so we had a negative result we got a negative 15. if I had the opposite of the opposite of the opposite of negative 15 like that now I have an even number of these symbols now I'm going to get a positive 15. so working backwards this would make it positive 15 so we'd have the opposite of the opposite of positive 15. and then this would make it negative so we would have the opposite of negative 15 and then this last one would make it positive again right we have another pair so this becomes 15. so just count the number of symbols that you have again pair them up so in other words like this is a pair and this is a pair so we end up with an even number of those symbols so we get a positive result here's one that's pretty long so we have the opposite of the opposite of the opposite of the opposite of the opposite of negative 24. so if you were to kind of work through this the slow way you'd be like okay now this is going to make it positive this is going to make it negative this is going to make it positive this is going to make it negative this is going to make it positive you'd end up with positive 24. but the quicker way to do it again is just to count the number of symbols you have you have one two three four five six of them six is an even number six divided by two is three no remainder so we know that the answer would be positive again you can kind of also look at and see hey can I pair these up so this would pair up and become positive this pairs up and becomes positive this pairs up and becomes positive right each pair of those negatives makes a positive and again if you kind of look at one let's say we only had five so the opposite of the opposite of the opposite of the opposite of negative 24 like that now this would make it positive this would make it negative this would make it positive and then we'd have one more that would make it negative so there's only five one two three four five so this would become negative 24 now again it's an odd number 5 divided by 2 is what it's 2 with a remainder of 1. so it's not an even number it's an odd number so we get a result that is negative okay let's look at one last problem and look how long it is so we have the opposite of the opposite of the opposite of the opposite of the opposite of the opposite of the opposite of the opposite of negative nine so again if I could do this the long way it's going to take some time so in other words this would make it positive this would make it negative positive negative positive negative positive and negative so we'd have negative 9. again the easy way is to count one two three four five six seven eight nine nine is an odd number it is not divisible by two nine divided by two is four with a remainder of 1. so it's an odd number so that's why we get a negative result and again you can also pair them up you can say okay well this would make it positive things would make it positive these would make it positive these would make it positive but then this one is hanging out by itself no other negative pair with it so we're going to end up with a negative result and that's why we have negative 9. hello and welcome to pre-algebra lesson 18. and in this video we're going to be learning about absolute value so for our lesson objectives we want to learn how to find the absolute value of a number and also we want to learn how to simplify when absolute value is present so in our last lesson we were introduced to a new group of numbers known as the integers right and this allowed us to see some negative values or some values to the left of zero so before we start working with these integers adding and subtracting multiplying dividing it's important that we learn the definition of absolute value so if we look on the screen here we see that the absolute value of a number is the distance okay the distance between that number and 0 on the number line so what if I started out by asking you what is the absolute value of the number three and the way we're going to ask for this is we're going to have these vertical bars around the number that we want to find the absolute value for so this right here is basically saying what is the absolute value of 3. okay that's what this means again the vertical bars are surrounding the number we're trying to find the absolute value for now following our definition that we were given the absolute value of a number again is the distance between that number and zero on the number line so if I find 3 on the number line and I find 0 on the number line I just need to count how far away it is and it's just one two three units away so the absolute value of three is just three now one thing I want you to note right away what if I asked you for the absolute value of negative three you're going to find that you're going to have the exact same absolute value because it's 1 2 3 units away from zero as well so the absolute value of negative three is also equal to 3. this goes back to our definition of opposites remember in the last lesson where we talked about opposites we said that opposites were numbers that lied on opposite sides of the number line so in this case you have 3 and negative three they're on opposite sides and they had the same distance from zero so this one has the distance of three right you have to travel three units to the right to get to zero this one has the distance of three you have to travel three units to the left to get to zero so opposites are always going to have the exact same absolute value because they have the same distance from zero on the number line so for example if I said what is the absolute value of seven and what is the absolute value of negative seven in each case the answer is going to be seven right if I look at seven and negative seven in each case I have to travel seven units to get to zero right one two three four five six seven and from here one two three four five six seven so the absolute value of seven is seven and the absolute value of negative seven is seven so another thing I want you to note right away is that the absolute value of a number is never going to be negative it's always going to be zero or some positive value and the reason for that is absolute value represents a distance and a distance is not going to ever be negative right for example in real life you think about waking up and going to work you can either not go to work and drive zero miles or you can drive some amount of miles it could be 10 miles 20 whatever it is doesn't matter and I know some of you will put in the comments that you could back up but no that's not driving a negative amount sorry do we really need to pull out a number line every time we want to find the absolute value of a number of course not the rule is if the number is negative just make it positive so for example if I ask you for the absolute value of negative 37 he would just make it positive it would just be 37 right essentially if it's negative make it positive negative 37 will be 37 units away from zero on the number line if the number is 0 or positive keep it the same okay so if the number zero so if you have the absolute value of zero zero is zero units away from zero in the number line so it's just zero if it's some positive value let's say it's I don't know 42. it's just that number all right it's going to be 42 units away from zero on the number line all right let's look at a few problems so we want to find the absolute value of 19. so again if it's a positive number just keep it the same so the absolute value of 19 is just 19. because if we had a number line there I would count 19 units to the left to get to zero if I started from 19 right it's 19 units away so the absolute value of 19 is just 19. what about the absolute value of zero well again zero is zero units away from itself on the number line so the answer is zero what about the absolute value of negative 24. again if you have a negative number just make it positive so I make this positive and I get 24. okay what about the absolute value of negative 34 422. well again if it's a negative value just make it positive so this would just be thirty four thousand four hundred twenty two okay now we want to replace the question mark with the correct symbol so we have less than greater than or equal to okay for the first one we're going to look at the absolute value of negative 27 question mark the absolute value of 22. all right so first let's think about what these are worth so the absolute value of negative 27 is 27. and the absolute value of 22 is 22. again for this one it was negative so we just made a positive for this one was positive so it just stayed the same so if we think about the relationship here the absolute value of negative 27 is a larger number so it's greater than the absolute value of 22. and again the reason for this is because this ends up being 27 after we apply the absolute value so 27 would be bigger than 22. and again you want to point the symbol towards the smaller number okay now I have the absolute value of zero question mark the absolute value of negative five so the absolute value of zero we know is zero the absolute value of negative five is five again just change the sign from negative to positive so which one is going to be bigger is 5 bigger or zero bigger well five is bigger so the absolute value of zero is going to be less than the absolute value of negative five again the symbol is going to point to the smaller value what about the absolute value of negative seven question mark seven well in this case the absolute value of negative seven is seven so seven and seven are having the same value so we're going to use an equal sign so we're going to say the absolute value of negative seven is equal to 7. so sometimes we're also going to have to simplify when we're working with absolute value and one of the things you have to be careful about in the last lesson I kind of gave you a little trick on how to simplify when you have a bunch of negatives involved so I want you to look here and see that we have what we have the opposite of the absolute value of negative 13. now let me just kind of put this to the side for a second I want to show you this problem so let's say we have the opposite of negative 13. we would count the number of those symbols we have and again if it's even divisible by 2 the result is positive if it's odd not divisible by 2 it's negative so here we have one two of those symbols so that's an even number so the answer is positive right because you'd have the opposite of negative 13 that's just 13. now when you're working with absolute value you can only count the number of symbols you have outside of the absolute value bars and the reason for that is because remember the result of the absolute value operation always makes things non-negative okay always makes it non-negative so the least that's going to be a zero now thinking about this if I count only the negatives outside of this operation here I only have one one's an odd number so I know the result is going to be negative 13. right please don't make the mistake of trying to apply that trick with absolute value and getting it wrong because some students will say okay I have one two of those symbols choose an even number I got positive 13. no because when you evaluate this slow way you would say okay the absolute value of negative 13 is positive 13 right you make the negative positive and then you have the opposite of positive 13 which becomes negative 13. so take close attention when you have the absolute value operation involved okay now we have the opposite of the opposite of the absolute value of 21. so again I would count the number of these symbols that I have outside of my absolute value bars so I have two of them two is an even number so my result would be positive but again if you want to kind of do this the slow way you have the opposite of the opposite of the result of this would be 21. so the absolute value of 21 is just 21 right if it's positive keep it the same so we'll put this in parentheses so this would be 21 here and the opposite of 21 would be negative 21 . so let's write this as the opposite of negative 21 and then one more time we'll change the sign to positive 21. but again the shortcut count the number of those symbols you have outside of the absolute value operation and that's going to tell you if it's even you're going to have a positive number if it's odd you're going to have a negative number okay here we have the opposite of the opposite of the opposite of the absolute value of negative 53. again pay attention you have one two three symbols outside of your absolute value operation please don't count this one because this one right here you got to think about it when we're done with this absolute value operation this is going to be a positive number it's not going to be negative right it's different than if I had the opposite of the opposite of the opposite of negative 53. in this case this ends up being a positive 53 right because I have one two three four negatives four is an even number I'm going to end up with positive 53. this becomes positive then negative then positive here it's going to be different because I start out basically you can think about this right here it's just positive 53. the absolute value of negative 53 will be positive so this is positive it's not really negative so this will end up going to negative then positive then negative right so that's why you'd end up with negative 53. again doing this the slow way let me kind of show you this the opposite of the opposite of the opposite of the absolute value of negative 53 is 53. so I'm going to put that in parentheses now change the sign so this will become the opposite of 53 is negative 53. so the opposite of the opposite of negative 53. change this on again so it'll become positive 53. so the opposite of 53 and then change this on one last time to negative 53. and again I can't repeat this enough if I just look at how many of these signs I have outside of the absolute value operation if it's odd which in this case I have one two three three is an odd number I know I'm going to get a negative result if it's even right divisible by 2 I know I'm going to get a positive result okay last problem we have the opposite of the opposite of the opposite of and then the absolute value of 6 Plus 3. now again I can count the number of symbols I have outside of the absolute value operation I have one two three so I know the result is going to be negative basically I can just kind of eyeball this and see that I'm going to have the absolute value of 6 plus 3 which is 9 and I know I'd have negative 9 right it's it's a very very simple problem but let's kind of go through it the slow way so let's copy this the opposite of the opposite of the opposite of and then we know six plus three is nine so inside of absolute value bars we'd have nine and even at this point we can see we'd have negative nine as a result but again let's keep going so now we just take the absolute value of nine that's nine so we'd have the opposite of the opposite of the opposite of and then we'd have 9. and now we would just go through and take the opposite so the opposite of not would be negative nine so we'd have the opposite of the opposite of negative nine and take the opposite of negative nine you get positive nine so the opposite of positive nine and then the opposite of 9 is just negative 9. so absolute value is a very easy concept overall you just have to remember if you're taking the absolute value of a negative number make it positive if you're taking the absolute value of 0 or a positive number just keep it the same hello and welcome to pre-algebra lesson 19. in this video we're going to learn about adding integers so the lesson objectives for today we want to learn how to add integers using a number line so this will give us a visual representation of what's going on and it can be helpful if you're struggling with this topic and then we're going to move on and learn how to add integers without a number line so this is how we're going to solve our problems moving forward okay so let's start out by talking about the scenario where we're going to use a number line and basically we'll start with the easier scenario where we're adding two or more integers with the same sign so A positive plus a positive which we already know how to do or a negative plus another negative so I have here that we would start at the left number that's on the number line then this Plus here this is basically for a positive number if you have a positive number it indicates a movement right so for example let's say I had 4 plus 5. we know that we could start out at four on the number line and then because this is a positive number I know that I'm going to move to the right and in this case I would move to the right by five units and of course we all know that four plus five is not let's slide down here and we see that this right here is for a negative number and this indicates a movement left so if I'm adding a negative number basically I'm going to move to the left so let's say I had something like 4 plus negative 5 or actually let's do negative 4 plus negative 5 to have the same sign here to make a little bit easier what's going to happen is I'm going to start out at negative 4 on the number line and because I'm adding a negative number here I'm going to move to the left by the absolute value of that number so the absolute value of negative 5 is 5. so I'm going to start at negative 4 and go five units to the left and that's going to put me at negative 9. and we'll see this this in a moment so I have here use the absolute value of the number being added to determine how far to move let's look at some examples we want to add each on the number line and this is not going to be bad at all once you do a few of these it's really really easy okay so the first thing is when we look at the number line remember zero is not positive and it's not negative so it is neither if I look to the right of zero those are the positive numbers to the left of zero those are the negative numbers as we move to the right remember numbers are increasing in value and if we move to the left numbers are decreasing in value so it's really important to remember that so let's start out with four plus five on the number line again all I have to do is start out at 4 and I'm going to add 5. so again this is a positive number that I'm adding so I'm just moving to the right by you could say the absolute value of that number there but the absolute value of 5 is just five so I'm just moving to the right by five units so one two three four five units so I end up at 9. so let me Circle the starting and the ending so we know that's not if I start with four dollars and I make five dollars today well now I have nine dollars so that's pretty simple all right the more challenging scenario occurs when you add two negative numbers so thinking about negative four plus negative five well we saw let me put a little border here that four plus five was equal to nine well here with negative four plus negative five you would start at negative four so let me start here and I'll just circle that and I'm going to add a negative number so that means I'm going to go to the left okay and I'm going to go to the left by the absolute value of that number so the absolute value of negative 5 is 5. so in other words the sign here is telling me the direction if it's positive I'm going to the right if it's negative I'm going to the left and then the absolute value of the number is telling me how far to move so here I'm just moving five units to the left so I'm going one two three four and then finally five units to the left I'm going to end up at negative 9 there okay and we can do this in our head if you just think about it with money let's say that I owe four dollars currently I owe four dollars to maybe the bank or something like that and I go out and I accumulate another debt of five dollars so let's say I go out with my credit card and I spend five dollars well how much do I owe well I can just add those two debts together negative four plus negative five would be negative 9. that's something you do in your head you just say okay four plus five is nine and then I'm attaching the common sign which is negative to show that I owe money right so I owe nine dollars or I could say it's negative nine for the sum let's take a look at another one here we have a negative two plus negative three before I do that let me come over here and just do two plus three so again you can see that there's a pattern emerging so two plus three is what that's five I could start at two and I could just go one and that color doesn't really show up so let me change to Orange here so one two three units to the right and I end up at five okay let me erase this and go back to negative two plus negative three so here I would start at negative two so let me put that there and I'll Circle this I'm adding a negative 3. so again if I'm adding a negative the negative tells me I'm going to the left and then by the absolute value of this number of units so the absolute value of negative 3 is 3. so I'm going to the left by 3 units so one two three units and I end up at negative 5. so this is negative 5. again notice You've Got 5 here and you've got negative 5 here so I can just add the absolute values here and attach the common sign if I owe two dollars and then I owe another three dollars well that's two plus three or five dollars that I owe so that can be represented with negative five so all you're going to end up doing is adding the absolute values of the numbers so just pretending like they're positive and then just attaching the common sign here again you just say okay what if they were positive so two plus three would be five and then attach that common sign which is negative okay let's look at one with more than two integers so now we have three integers all the same sign so it's not really any harder you're going to have negative 1 plus negative 2 plus negative 5. so if I start at negative one and I add negative 2 I'm just going to the left by 2 units so one two units to the left so that's going to put me a negative three and then I add another negative five so I'm going to go five units to the left so one two three four five so I'm going to end up at negative eight there so I started at negative one I ended up at negative eight so let me put negative eight here and again you can think about this by just adding whole numbers let's say I had one plus two plus five one plus two is three three plus five is eight again because all the signs here are negative you would just attach that common sign so I have a debt of a Dollar Plus another debt of two dollars plus another debt of five dollars let's say I owe Steve a dollar I owe Chris two dollars and I owe Mary five dollars well how much do I owe in total I could just add those guys together and I get a total debt of eight dollars which I can represent with negative eight okay let's just do one more with the number line and then we'll move off of it so we have negative four plus negative three plus negative one so let's start at negative four and then we're adding negative 3 so I'm going to the left one two three units so that puts me at negative seven and then I'm adding negative one so I'm going one more unit to the left so I started at negative four and I end up at negative eight so this is negative eight once again you could just add four plus three plus one so four plus three is seven seven plus one is eight and you're just attaching the common sign I owe four dollars I have another debt of three dollars and another debt of a dollar how much do I owe in total well I owe eight dollars which can be represented with negative eight all right so let's look at the rule which I think is pretty easy at this point to understand when adding two or more integers with the same sign you add the absolute values and keep the common sign so very very simple so I have here let's try a few without the number line we have negative fifteen plus negative 11. so again this is negative and this is negative so just put a negative and then just pretend these two numbers are positive what if I added fifteen and eleven a lot of you can do that on your head you know it's 26 but if you need to do a vertical addition we can always stop and do that five plus one is six one plus one is two so you just drag that up there and say this is negative 26. I owe 15 and then I go out and charge up another eleven dollars on my credit card how much do I owe in total well I'm gonna owe twenty six dollars so I can just say that's negative 26. what about this one we have negative 7 plus negative 1 plus negative three again this is negative this is negative and this is negative so the answer is negative just use the common sign then just pretend like the numbers were positive what if I added seven plus one plus three seven plus one is eight eight plus three is eleven so just put that here behind that common sign so this becomes negative 11 as the answer let's do another one so here we have negative 5 plus negative 1 plus negative six plus negative fourteen so every sign here is negative so the answer is negative and then just add the absolute values so five plus one is six six plus six is twelve and then 12 plus 14 I think a lot of you know that's 26 you can do it in your head but let's just do it off to the side just so nobody's lost two plus four or six one plus one is two so you're just going to bring that over here and say that the answer is negative 26. all right so now let's move on and talk about the more complicated scenario so this is where we're going to add two integers with different signs or sometimes you'll add more than two integers so you'll have many integers that you're trying to add and the signs are mixed so what happens when we have mixed signs let's revisit our number line and try a few problems now before we do this this type of problem gives students the most trouble when you look at let's say negative eight plus negative four again this is just an addition problem we just saw this basically you use the common sign the common sign is negative and then you're just adding right you're just saying well what is 8 plus 4 that's 12 and then you're just attaching that sign so this is negative 12. so that's really easy I'm just going to tell you before we start this this procedure is just as easy you're now just doing a subtraction problem okay so think about negative 8 plus 4 first on the number line so here's negative eight and then I'm adding four so that means I'm going 4 units to the right so I'm just going to go from right here one two three four units to the right so that puts me at negative four does that make sense I always tell people to use money in your head if you're trying to think about this let's say I have a bank account and currently I owe them eight dollars I go into the bank and I give them four dollars do I still owe money yes I still owe four dollars so it makes sense that this is negative 4 here again I have not given them enough money to get back to zero or to get back to where I don't owe any money so basically you could think about always that if I have a negative here let's say it's negative eight that I have to add the opposite of that number so let's say in this particular case that's going to be positive 8 to get back to zero and we could think about that problem let me erase this so if I have negative 8 which is eight units away from zero on the number line remember the absolute value of negative eight is eight I would have to go one two three four five six seven eight units to the right to get back to zero so I have to add the opposite of the number to get to zero now if I start adding numbers that are greater than eight what's going to happen is now I'm going to have a positive balance in my account let's say I do negative eight plus nine well I'm going to go one more unit to the right on this number line and I'm going to end up at 1 now so now I have a dollar in my account because I paid more money than I owed so hopefully that does make sense you can pause the video and really start to think about what's going on here but essentially this is going to translate into a nice little procedure for us all we really have to do is first think about which number is bigger in terms of its absolute value so which number has the bigger distance away from zero on the number line let me get rid of this and go back to this as the original problem well in this case the negative 8 does right it's 8 units away from zero on the number line four is four units away from zero on the number line so this guy is going to be the winner in terms of our sign so you use the sign of the number with the larger absolute value then I'm just going to put this in parentheses for right now if you think about these two numbers as if they were positive numbers and you just subtract the larger minus the smaller that's all you have to do so it just becomes eight minus four so eight minus four and so this becomes the negative of 8 minus four is four so this is negative four let's say you change this around and now you have eight plus let's say negative four well this is going to be a positive result you could put a plus out here if you want or you could leave it off it's up to you and then you're just going to subtract the absolute values so what is 8 minus four so eight minus four well that's going to be 4. and that should make sense in this particular case if I had eight let me get rid of this problem right here if I had eight dollars in my account let's start here and I spend four or I have a data for let's say I charge it for my credit card well basically when I settle up I'm gonna have four dollars so I'm gonna go one two three four units to the left and I'm going to end up at four so we'll see later on that eight plus negative four can be directly written as eight minus four you can immediately go to that so if I had something like let's say 10 plus negative 5 I'll immediately just write that as 10 minus 5. so this is a very easy problem to do a conversion with if you have something like let's say negative 10 plus 5 well this one's a little bit more tricky because the negative has the larger absolute value so in this case I just say okay well I'm going to use the negative and now I'm going to convert this into 10 minus 5 in my head 10 minus 5 is 5. so this gives me negative 5. okay let's look at a few more so here we have a 9 plus negative 7. so let's start out at 9 on the number line and I'm adding negative 7. so I've incurred a debt of seven dollars or let's say someone took seven dollars away from me for a product that I bought something like that so I'm just going to move to the left by one two three four five six seven units so I end up at 2. so basically you could have written this as nine minus seven which equals two again the larger absolute value here is nine nine is nine units away from zero on the number line and negative seven is seven units away from zero on the number line so basically I'm not taking away enough to get me to zero or some negative value so the answer is going to be positive right and then you could just subtract the absolute values 9 minus seven is two if I change this around let me put this a little bit tighter in there if I change this around and let's say I do negative 9 plus 7 well now I owe nine dollars right so I'm here at negative nine I go into the bank and I give them seven now I still owe two dollars so I've got to go one two three four five six seven units to the right and now I'm at negative two right so I started at negative nine and I'm at negative two so I have not gotten out of the hole yet so here you would say this is the negative of and then you go nine minus seven so nine minus seven and let me make that seven a little bit better so this becomes negative two okay let's look at one more of these and then we'll just run through a bunch of examples so we have negative seven plus twelve so start out at negative seven and I'm going to add 12. so I go one two three four five six seven eight nine ten 11 12. so I end up at 5. so here's negative seven again that should make sense if I owe seven dollars and I pay Seven dollars off I'm gonna be at zero but now I didn't just pay Seven dollars off I paid twelve dollars into my account so I should have five more dollars left in my account so this really becomes a subtraction problem it becomes twelve minus seven which is five again if this number is larger in terms of absolute value you could just flip it around and turn into a subtraction problem you could really just say okay well this is going to be 12 plus negative seven which becomes twelve minus seven which equals five this is something you're going to do mentally very very shortly after you start working enough problems if you started with something like negative 12 plus 7 well then in this case I owe twelve dollars but I'm only paying off seven so it makes sense that I should still be in the hole right or have a negative account balance so this will be negative and then I'm just going to say okay what is 12 minus seven just subtract the absolute values the larger minus the smaller 12-7 is going to be 5 so this ends up giving me negative 5. all right let's go through the procedure and then we'll just run through a bunch of problems so when adding two numbers with different signs use the sign of the number with the larger absolute value then think about the numbers and terms of their absolute values and subtract the smaller from the larger so you're doing a subtraction problem figure out what your sign is and then just subtract the absolute values larger minus smaller that's all it is okay so let's try a few without the number line so here we have 13 plus negative 8. again you can easily convert that in your head into 13 minus eight and the reason is simple this guy is larger in terms of absolute value I have thirteen dollars you could say I'm incurring a debt of eight dollars whatever it is I'm taking eight away from what I have so 13 minus 8 would just be 5. so the sign of the number with the larger absolute value is positive and then you're subtracting the larger absolute value which is 13 minus the smaller absolute value which is 8 13 minus 8 would give you five if you had negative 13 plus eight well then the 8 that you're adding is not enough to cover the debt right you owe thirteen dollars so this guy will be negative the larger absolute value is negative so this sign would be negative and then you're just doing what is 13 minus 8 which gives you now negative five so that's the difference between these two I started with 13 which is a bigger number in terms of absolute value and I added a debt of 8 or I subtracted away eight however you want to think about that basically I'm going to end up with five dollars in my account here I owe 13 and I only pay off eight so I'm going to have a negative here and then I do 13 minus 8 I end up with a debt of five dollars so that's the difference between these two what about 12 plus negative two again the absolute value of 12 is bigger than the absolute value of negative two so this would be positive and then you could just subtract the larger absolute value minus the smaller so this turns into 12 minus two which is 10. if I had negative 12 plus 2 well now this is going to be negative I owe more than I'm putting back in the absolute value of negative 12 is larger than the absolute value of 2 so this is going to be negative and then you do the same subtraction you're still going to do 12 minus 2. so this part's still 10 but I have that negative there so this would be negative 10. what about 14 plus negative 23. so the absolute value of 14 is 14 the absolute value of negative 23 is 23. so we know this is negative and then you're just going to do a subtraction what is 23 minus 14 let's borrow here and put a 1 there 13 minus 4 is 9 1 minus 1 is 0. so this is negative 9. and again if you change this up and you said well I have a negative 14 plus 23 well now I owe 14 and I'm putting 23 in my account so of course I expect to have a positive balance right so the sine of the number with the larger absolute value is positive so the answer would be positive and now you can just change this into 23 minus 14 which of course we know is 9. what about negative 31 plus 25 well here the sign of the number with the larger absolute value is negative so we're going to use that sign and then you're just going to subtract the absolute values larger minus smaller so 31 minus 25 let's borrow here and put a 2 maybe that 5 could look a little bit cleaner and let's put this as an 11 11 minus 5 is going to be six two minus 2 is 0. so this would be negative 6 here and again if you had changed that up and said you have 31 dollars and you add a negative 25. so basically you spend 25 or you incur a debt of 25 however you want to think about that well what's going to happen is I have enough money to cover that so I know I'm going to have a positive balance in my account because the absolute value of this positive number is larger than the absolute value of this negative number and then I just do my subtraction so larger absolute value minus the smaller what is 31 minus 25 we know that 6. so here that would be 6. okay let's do some problems where it's mixed so what I'm going to tell you is when you do mix addition the way I like to do it is I like to add the numbers with the same sign first it just makes it easier you can reorder this using the commutative property you can always add in any order so I'm just going to say that this is 12 plus 7 done first and then plus negative 15. so we know 12 plus 7 is 19. we've been doing that forever so this is 19 and then plus negative 15. and then of course here 19 has the larger absolute value so the answer will be positive and then you're just doing a subtraction so larger absolute value minus smaller so what is 19 minus 15 well that's going to be 4. what about 5 plus negative 2 plus 1 plus negative 7 plus 11 plus negative 14. so let me put 5 plus 1 plus 11 next to each other and then plus negative 2 plus negative 7 plus negative 14. I'll put those next to each other and again just add the numbers with the same sign first so those guys let me do this in different color and those guys you'll do them separately so what is five plus one that's six six plus eleven is 17. again you can always stop and do a vertical addition one plus six is seven bring down the one so that's 17. so this right here would be plus now you have negative 2 plus negative 7 plus negative 14. so I know that's negative and then just add 2 plus 7 that that's 9 and then 9 plus 14 a lot of you know that that's going to be 23 but 4 plus 9 is 13 so put the three down carry the 1 1 plus 1 is going to be 2. so this would be negative 23 here so now which number is larger in terms of its absolute value well negative 23 is so this would be a negative and then you're just doing a subtraction so what is 23 minus 17 a lot of you again know that 6 already but let's just do this over here so nobody is lost borrow here and that one could be better so this is 13 minus 7 which is six one minus 1 is 0. so this right here you drag that up and you would get negative 6 as the final answer all right let's take a look at one more of these so here we have negative 3 plus negative 1 plus 8 plus negative 5 plus 9. so let's write negative 3 plus negative 1 plus negative 5 plus 8 plus 9. okay so negative plus negative plus negative here would give me a negative and then just add the absolute values so three plus one plus five three plus one is four four plus five is nine so this is negative nine and then Plus we know eight plus nine is seventeen so now if you think about this this number is larger in terms of its absolute value so our answer will be positive and then you're just subtracting the larger absolute value minus the smaller so you're just turning this around into what is 17 minus 9. and a good way to do this I think is to just reorder that and say this is 17 plus negative 9. when I see that immediately I know that's 17 minus 9 and of course that gives me a final answer of 8. hello and welcome to pre-algebra lesson 20. in this video we're going to learn about subtracting integers so the lesson objective for today we're just going to learn how to subtract integers and the procedure for this is pretty straightforward if you know how to add integers then you can automatically subtract integers so I have here the first thing you'd want to do is to leave the first number and that means the leftmost number of the subtraction problem we call that the menu end this is going to be unchanged so don't mess with that at all the second thing is you're going to change the subtraction operation into addition so instead of a minor are going to put a plus then the last thing is to change the second number so that's the rightmost number of the subtraction problem we call this the subtrahend we're going to put this as its opposite so just change the sign of the number so basically you're changing your subtraction problem into addition of the opposite let's take a look at a few problems using the number line I think this is not too bad overall if you can add integers then you can subtract integers it's basically the same thing with just an additional step so here we have 7 minus 9. so before we do anything let's just think about this logically if I have seven dollars in my account then I can only spend seven dollars before I go negative I think we all know that so if I'm spending nine dollars well I'm spending two more dollars than I have so I should expect to have a balance of negative two so if I think about this using my procedure I would keep the first number unchanged so just write a seven then change this subtraction into addition and then change the number being so subtracted away in this case it's 9 you're going to put that into its opposite so just change the sign for positive 9 it would become negative 9. so this is 7 minus 9 is the same thing as 7 plus negative 9. so I have seven and you take nine away from me it's the same thing as if I said hey I have seven and I have a debt of 9 and we're going to settle up right so you're taking nine away from me it's the same thing either way so when I think about this using my procedure for adding integers well I want to use the sign of the number with the larger absolute value the absolute value of negative 9 is 9 the absolute value of seven is seven so negative 9 has a larger absolute value so we get a negative for the result now you're just going to do a subtraction so you're going to do larger absolute value minus the smaller so what is 9 minus seven and this ends up giving me 9 minus seven is two you apply the negative this is negative two but we knew that already because we just thought about it logically if I have seven dollars in my account and I spend seven dollars I'm at zero but I'm spending nine so I'm spending two more than what I have so I should expect to be at negative two for my account balance you can also think about this on the number line let's say again we start out at seven so this is my account balance here and I take away nine or you could say you add negative nine whatever you wanna do there you're basically going to move nine units to the left so I'm going to go one two three four five six seven eight and then finally nine so I started at seven notice when I added negative seven or subtracted away seven I got to zero and then if I subtract two more away or I add another negative two I'm going to be at negative two for the result so 7 minus nine which is the same thing as seven plus negative nine gives us negative two okay let's look at a similar problem so here we have five minus eleven so again if I have five dollars in my account and I go through and I buy eleven dollars worth of stuff I know that I overspent by six dollars right because if I spend five dollars I'm at zero and then basically I'm gonna spend another six from what I have so that's gonna put me at negative six well we can show this by saying that I have five keep that leftmost number the same I'm gonna change the minus into a plus so now this is addition and I'm going to change that 11 into its opposite so that will be negative 11. so this becomes what use the sign of the number with the larger absolute value so negative 11 has a larger absolute value so this would be negative and then inside just subtract the smaller absolute value from the larger or you can say the larger absolute value minus the smaller however you want to think about that it's going to be 11 minus 5 inside so 11 minus 5 is 6. so this becomes negative 6 as the answer again using the number line to see this visually if you need that here's five so I start out with five dollars in my account and I'm gonna spend 11. so first let's say we spend five so one two three four five units to the left at that point I'm at zero so there's no more money in my account well basically I'm going to go another six units to the left because I'm spending six more dollars or I'm adding another negative six however you want to think about that so I'm going to go another one two three four five six units to the left to end up at negative six so basically I started out with five and I added negative eleven or I took eleven away and I went 11 units to the left on the number line so I end up at negative six or I owe six dollars to the bank however you wanna think about that okay we'll do one of these without a number line and then I'm gonna change it up a little bit I'll go back to the number line so for this one we have 15 minus eighteen so just write this as fifteen plus a negative 18. so it starts to become pretty quick for you leave this first number unchanged so 15 is unchanged change your subtraction to addition change the number being subtracted away into its opposite so 15 minus 18 is 15 plus negative eighteen so the negative 18 has the larger absolute value so we'll use that sign and then you're just going to subtract the larger absolute value minus the smaller so 18 minus 15 so this becomes the negative of 18 minus 15 is 3. so this is just negative 3. again if I start with 15 in my account and I spend 18. so I'm taking 18 away from what I have I would expect to basically have negative 3 because I spent three more dollars than what I had okay for this problem we will return to the number line so the leftmost number you'll notice that that's a negative now so this is actually going to make the problem a little bit easier so let's just go with our procedure the leftmost number is negative five I'm not going to mess with that I'm going to change my subtraction operation into addition and then I'm going to change the number that's being subtracted away here that's positive 3. I'm going to put that as its opposite so this will be negative 3. so we can write negative 5 minus 3 as negative 5 plus negative 3. and so we have this common sign that's negative so the answer will be negative and then you just add the absolute value so 5 plus 3 would be eight so the final answer here is negative 8 I think it's also useful to see this on the number line so let's say that I start with negative five so I'm starting with this leftmost number here and then whether I think about this as subtracting away three or adding negative 3 I'm moving three units to the left so I'm going one two three units to the left so I started at negative five I'm going to end up at negative eight all right what about negative 56 minus eighteen so this would be negative 56 plus a negative 18. so basically you know this would be negative because that's the common sign and you could just add 56 plus 18 some of you can do that in your head it's going to end up being 74 but let's just go through this so 6 Plus 8 is 14 4 down carry the one one plus five is six plus one is seven so that's 74. so this becomes a negative 74 for your final answer what about negative 107 minus 29. so let's say this equals negative 107 plus a negative 29. so we know the answer is negative and you're just going to say well what is 107 plus 29. so just add the absolute values 7 plus 9 is 16 6 down carry the one one plus zero plus two is three bring down the one so this is 136. so again drag that up there we'll say the final answer is negative 136. all right now let's talk about something that gives students a lot of trouble in this section and that is subtracting away a negative is the same as adding a positive so you might see something like 4 minus a negative 3 and we say that's equal to 4 plus 3. so minus a negative is plus a positive I'll give you a little bit of an explanation in a moment but right now let's just use the procedure that we talked about earlier so for the leftmost number the menu n is going to stay the same so this is 4 and this is 4. the subtraction operation becomes addition so this is a minus and this is a plus and then the number being subtracted away again called the subtrahend this guy is going to get changed into its opposite so the opposite of negative 3 is positive 3. so just following the procedure we already talked about you would get the right answer so this becomes four plus three which we know is 7. you see that I have 4 minus two equals two four minus one equals three and four minus zero equals four you'll see that each time the number that's being subtracted away or the subtrahend decreases by one the difference or the result from the subtraction operation increases by one that should make sense because I'm taking one less away so the the result or the difference should be one larger so 4 minus 2 is 2. 4 minus one is three four minus zero is four so now when we get into one less than zero well that's going to be negative one so I have four minus a negative one which gives me five so it's the same thing as if I said what is four plus one well that's five then four minus a negative two that's four plus two that's six then four minus a negative three that's four plus three that's seven let's look at some examples so here we have negative seven minus a negative twenty so this becomes negative seven plus twenty so again minus a negative is plus a positive keep this first number unchanged change the subtraction into addition this number right here that's being subtracted away it's negative twenty it becomes its opposite which is positive twenty so this is what the sign of the number with the larger absolute value is positive and so you're just going to change this over into 20 minus seven which becomes thirteen what about negative thirteen minus a negative six so negative 13 plus 6 minus a negative is plus a positive so the sign of the number with the larger absolute value is negative so this will be a negative and then just subtract the larger absolute value minus the smaller so this is 13 minus six so this becomes negative 13 minus 6 is 7. so negative seven what about negative 94 minus a negative 17. so negative 94 plus 17. minus a negative is plus a positive so let me say that the sign of the number with the larger absolute value is negative and then you're just going to go 94 minus 17. so in other words you're saying the larger absolute value minus the smaller so 94 minus 17 so this gives me what let me borrow here put an 8 and this will be 14 14 minus seven is seven eight minus 1 is 7. so this would end up being negative 77. all right let's wrap up the lesson and what we're going to do now is talk about multiple operations so basically subtracting more than two integers so when we perform subtraction with more than two integers the rules are the same change each subtraction to addition to the opposite this is going to help you a lot because remember addition is commutative so you can add in any order that you'd like so here we have 9 minus 3 minus eight minus 12 minus four so let me go equals 9 the leftmost number not gonna mess with that so I'm just going to copy that and then everything else you're going to go plus negative so this will be plus negative 3 and then plus negative 8 and then plus negative 12 and then plus negative 4. so now what I'm going to do is I'm going to add all these numbers over here because they're all negative first right so I'm just going to keep 9 and then I'm going to say negative plus negative plus negative plus negative is going to be a negative and then just add the absolute value so 3 plus 8 is 11 11 plus 12 is 23 and then 23 plus 4 is 27. so this would be negative 27 here and I should probably show that in case you were not using a calculator so let me do that real quick so 3 8 you have 12 and then you have four really quickly two plus three is five five plus eight is thirteen thirteen plus four is seventeen so put a 7 down carry the one one plus one is two so this is 27 that's where I got that from I know some of you are not using a calculator and you're just getting up to speed with addition so now we have 9 plus negative 27 so this right here would be a negative because the sine of the number with the larger absolute value is negative and then you're going to subtract the larger absolute value which is 27 minus the smaller which is 9. so let's do that off to the side so 27 minus 9 borrow here this is a one this is a 17. 17 minus 9 is 8. bring down the one so this is going to be 18 so we'd have negative 18 as the answer okay let's do another one so here we have negative 2 minus is a negative 3 minus the negative 9 minus 14. so negative 2 don't mess with that minus the negative is plus a positive minus a negative is plus a positive and then you have minus 14 that's plus negative 14. okay so I'm going to reorder this and put negative 2 next to negative 14 those have a common sign and then I'm going to put 3 and 9 together and then basically negative plus negative is negative 2 plus 14 is 16. and then plus 3 plus 9 is 12. so now we know the answer would be negative because negative 16 has a larger absolute value and you're just going to do 16 minus 12. I don't think you need a vertical subtraction for that it's just going to be 4. so this would end up being a negative 4 as the answer okay let's just do one more so here we have negative 5 minus a negative 17 minus 8 minus 9 minus the negative four so negative 5 don't mess with that and then minus the negative is plus a positive and then we're subtracting away 8 so that's plus negative eight we're subtracting away nine so that's plus negative nine and then minus the negative is plus a positive okay so I'm going to rearrange this and say that we have 17 plus 4 and then plus negative 5 and then plus negative eight and then plus negative 9. so 17 plus 4 again if you need to you can just do a vertical addition seven plus four is eleven one down carry the one one plus one is two so that's going to be 21. and then for this one negative plus negative plus negative is a negative and then we'll think about 5 plus 8 that's 13 and then plus nine let's just do that also the side three plus nine is going to be 12. two down carry the one one plus one is two so that would be negative 22. I think we can all see that 21 plus negative 22 would be negative one but again to go with the official procedure we know that this would be negative because negative 22 has a larger absolute value and so then you would just do the larger absolute value which is 22 minus the smaller which is 21 and so this is going to be the negative of 22 minus 21 is 1 so basically negative 1 for the final answer all right let's take a look at an alternative way to think about why subtracting away a negative becomes adding a positive we saw this with a pattern I'm going to give you one other approach to use there are many available some people will show this as taking away a debt some people instead of using a horizontal number line like I'm going to use here you'll see a vertical number line and maybe they'll use something like temperatures so a positive and a negative temperature again lots of different ways to think about this so we're not going to actually do a number line subtraction if that's what you're thinking what I'm going to do is think about the distance between two points on a number line so let's start out with point a and that's going to refer to four let me highlight this as I go so point a refers to 4 on the number line and point B refers to 1 on the number line okay so if I ask you for the distance between those two points what am I asking for well if I go from B to a or from A to B how far do I need to travel and remember when we think about distance like we talked about in the lesson on absolute value it's non-negative so it's either zero meaning I get in my car and I don't go anywhere where some positive value so I go somewhere if a is your house and B is the coffee shop well if I go from A to B so from my house to the coffee shop let's say that's three miles on the return trip when I go from B to a and I'm going back well my car's odometer doesn't say okay this is negative three miles because we're on a return trip no I go three miles there and I come three miles back so the distance is three either way so that's why you see this set up like that so we can show this as the distance between a and B as being equal to three now what's another way to do this your book will give you a subtraction formula that involves the absolute value I'm going to keep this simple and I'm just going to say that we're going to subtract the larger number minus the smaller just for the sake of this lesson to keep it simple so I'm going to say this could also be 4 minus 1 which would give us 3. so I can do it with subtraction larger minus smaller or I can go through a manually count either going from B to a one two three or from A to B one two three again don't worry about the direction it's just how far am I traveling so we can see that the distance between A and B on this number line can be found as four minus 1 which is going to give us three so let's go into something we already looked at so what about 4 minus a negative 3 which becomes 4 plus 3 which equals 7. we're going to show this using the distance between these two points now instead of this point B being at one it's going to be at negative three point a will still be a get four so let me highlight this and this and then this and this so let's think about the formula we just had the distance between A and B we saw a moment ago we had four and then we had one so we did four minus one large amount is smaller so now what I'm going to do is 4 minus a negative 3. so again larger minus smaller we know that negative 3 is smaller than 4 because negative 3 lies to the left of 4 on the number line so this right here if I went through and counted starting from 4 well I would go one two three four so at zero I've traveled four units and then to get to negative three I've got to go another one two three units okay so that means that this distance right here is four so from right here to right here and then this distance right here is three so from right here to right here so I basically take that starting number which is four and then I'm going to add on another three so how could I look at that and get that well I need this part right here which is the four and then then I need to add the opposite of whatever this negative number is to get this plus 3 or to tack on an additional three to get the total distance so this statement 4 minus the negative 3 becomes 4 plus 3 which becomes 7 as the answer now looking at another one let's say that we had point a now that's going to be at negative one so a that point is at negative one point B will now be at negative six well I can still use the same strategy so the distance between A and B would be what take the larger number in this case it's negative one negative 1 is to the right of negative six so it's the larger number and then minus a negative 6. so I'm just subtracting the larger minus the smaller so this should give me the distance between these two points so looking at this and just counting I can see that if I go from B to a or a to B it doesn't matter 1 2 3 4 5. that's my expectation so I should get a 5 here but how do I go from negative 1 minus a negative 6 to 5. well I'm going to have to change this minus a negative to plus a positive so we'd have negative 1 plus 6 which would give us 5. hello and welcome in our pre-algebra lesson 21. in this video we're going to learn about multiplying and dividing integers for the lesson objectives for today we're going to learn how to multiply integers and we're also going to learn how to divide integers all right so at this point in the course we should know how to multiply whole numbers together and we know that unless we have a factor of zero if we multiply two whole numbers together we're going to get a positive product so I have here we all know that the product of two positive numbers is positive even if you've never thought about it so we have that a positive times a positive will give us a positive all right so now let's build up to the rules where we're going to multiply with negative numbers so first we have 4 times 3. so from the multiplication tables we know the answer is 12. but going back back to earlier in the course we said that multiplication was a shortcut for repeated addition so really I could write this as 3 plus 3 plus 3 plus 3. so I have one two three four of these guys and so of course I can add three plus three is six six plus three is nine nine plus three is twelve so I get 12 either from this guy or from the multiplication tables and going with this guy so we see that it's just a shortcut for repeated addition now what if I throw something in the mix like four times negative three well what does this mean again if I follow this same thought process well it means that I have a repeated addition here where I have negative three plus negative 3 plus negative 3 plus negative three one more time well we know how to add integers already and basically we have a common sign so we'll just put a negative right there and then you could just add the absolute values so three plus three plus three plus three is going to be be 12. so the answer here becomes negative 12. does this make sense just stop for a moment and think about it let's say that you are losing three dollars a day at your business or whatever you're doing so day one day two day three day four instead of adding those guys together you could do a quick multiplication and say okay well there's four days where I lost three dollars each day I would do four times three in my head and say that's 12 and then I would apply a negative to it because it's a loss so that's basically what we're going to be doing when we multiply integers okay we're going to think about the absolute values being multiplied and then we're going to attach the sign that we need to all right now I want to show you something else because it's a little bit confusing when we go to negative three times four okay the way I like to show this is using my commutative property so let me erase this real quick we already know that four times three is equal to three times four so we can legally switch the order of the factors if I think about 3 times 4 let me write that out so 3 3 times 4 this is what this is now going to be 4 plus 4 plus 4 which is also going to give me 12. all right so 4 plus 4 is 8 8 plus 4 is 12. so if I get something like let's say negative 3 times 4 I can legally switch the order and say that this is 4 times negative three and I'm right back to this definition right here which gives me negative 12. so we can see that a positive times a negative gives me a negative and a negative times a positive gives me a negative let's go through the same thing with this guy right here I'm just going to slide down a little bit so we have 5 times 2. so let's start out with what we know this is two plus two plus two plus two plus two so I have one two three four five of these guys and so we know that 5 times 2 from the multiplication tables would be ten but again you could add two plus two is four plus two is six plus two is eight plus two is ten so we can also do five times negative two in a similar way so this would be negative 2 plus negative two plus negative 2 plus negative two plus negative two and of course this would be negative 10. now if I change the order let's say I was given negative two times five well of course I could just flip it back right so it's going to give me the same answer because of the commutative property of multiplication so I can just legally say that this is the same as 5 times negative 2 which would give me this negative 10. and if you were presented with let's say negative 5 times 2 well you could flip that around into 2 times negative 5 and you could say that this is equal to what this is negative 5 plus negative 5 which would be negative 10 as well okay so let's look at the sine rules here so we saw that a positive times a negative or a negative times a positive is going to give us a negative so that's pretty easy to understand where we start running into problems is where you get negative times negative and we say that's a positive all right let's just quickly think about this pattern to show that a negative times a negative is a positive there's other ways you can show this but I think this is the quickest way and the easiest way to think about it so if you have 3 times negative 3 we say that's negative 9. so we know that already that 3 times negative 3 can be written as negative three plus negative 3 plus another negative three and of course that is negative nine now looking at this pattern as we decrease this Factor on the left by one so in other words now this is going to be a two what's going to happen is the product here is going to increase by three so this 2 times negative 3 is equal to now only negative 3 plus negative 3 and so this would give me a negative 6. so I've increased by three now if I go to 1 times negative 3 I'm increasing by 3 again right so this is negative 3. and the pattern just continues so now this goes to zero and this goes to zero so I'm increasing by three let me actually get rid of this and I'm just going to show this as this going plus three and this going plus three and this going plus three so when you get here to negative 1 times negative three again this is just decreasing by one so from zero If I subtract away one I'm at negative one and this is still increasing by three so plus three this goes down by one this increases by three so plus three and then plus three so negative 1 times negative three again using this pattern would be positive three the negative 2 times negative three using this pattern would be positive six and then negative 3 times negative three using this pattern would be positive nine all right so once you have the sign rules down you're ready to multiply integers and basically if you can multiply whole numbers then you could just blow through these problems the first thing is you want to determine the sign of the product so if you have a positive times a positive or A negative times a negative so if you have the same signs your answer or your product will be positive and then if you have different signs so a negative times a positive or positive times a negative your answer will be negative so you have a negative product in that situation so once you figure out what the sign is you just want to multiply the absolute values so it's really really quick so let's come through here and just blow through some of these problems you have a negative 2 times 7. negative times positive is negative again different signs then just multiply 2 times 7 that's 14 so the answer is negative 14. here we have 4 times negative 4. so you have a positive times a negative that's a negative so different signs negative product then 4 times 4 just multiply the absolute values that's 16 so the answer is negative 16. here we have negative 11 times 5 so you have a negative times a positive that's a negative and then 11 times 5 that's 55. so this is negative 55. here we have negative 12 times negative 9 so when negative times a negative is a positive so when you first start doing this you can write out a plus symbol if you want of course it's not necessary so I'm just going to do it just for the sake of completeness here now I want to do 12 times 9. some of you are still learning how to multiply so I'll do that as a vertical multiplication so 9 times 2 is 18 8 down carry the 1. 9 times 1 is 9 plus 1 is 10. so this is 108. so this is positive 108 of course you can keep the plus sign out there if you want or you can just write 108 but negative times negative is positive and also positive times positive is positive so if you have the same sign negative and negative then basically you're going to have a positive product okay here we have negative 10 times negative 4 so we know this is a positive again you can put a plus sign out there or leave it off it's up to you and then if we're multiplying 4 times 10 right you're multiplying the absolute values there well you just multiply 4 times 1 that's 4 and then just attach one trailing zero so this would be positive 40. then here a negative times a negative again is a positive and then you're doing 15 times 14 let me do that over here so 15 times 14 4 times 5 is 20. zero down carry the two and then four times one is four plus two is six and then let me erase this one times fifteen is fifteen but remember you're starting in the tenths place here so you've got to move this over so you're gonna put a 5 here and a one here and then let me just add this so bring down the zero six plus five is eleven one down carry the one one plus one is two so this would be 210. so this would be two hundred ten all right now let's talk about what happens if we multiply more than two integers so this is just as easy first let me do this the long way and then we'll think about what's going on so let's say I work left to right and I go negative 6 times 1 and that's going to give me negative 6. negative times positive is negative 6 times 1 is 6. so that's negative 6. so let me copy this so times negative 8 and then times negative 4. so now you have a negative six times a negative eight negative times negative is positive and then six times eight would be 48 so this is positive 48 and then times negative 4. so here positive times negative is negative and then 48 times 4 some of you can do that in your head it's going to be 192 but just in case you can't 4 times 8 is 32 2 down carry the 3. 4 times 4 is 16 plus 3 is 19. so bring this over here the final answer is negative 192. now an easier way to do this is to count the total number of negative factors now if the total number of negative factors is going to be an even number and I'll explain what that is in a moment then you're going to have a positive product if the total number of negative factors is an odd number I'll explain what that is in a moment then you're going to have a negative product let me actually erase this real quick I'm just going to put this up here this is negative 192. I want you to see what would happen if I wrote negative 6 times negative 1 times negative 8 and then times negative 4. how would that change the product from this guy going to this guy well what's happening is in this particular case notice that you have these two guys that you can pair up and then you have this one left over so basically you have three negative factors and three is an odd number again I'll explain what that is in a moment so this guy right here gives me a negative product but this guy I have these two that I can pair up and these two that I can pair up so basically each pair creates a positive right each pair of negatives so negative times negative gives me a positive and then negative times negative gives me a positive so because I have one two three four negative factors and four is an even number this is going to be a positive product so this would change this to positive 192. now when I say even and odd what do I mean we haven't gone to what we call the divisibility rules yet but basically if you can divide a number by two and there's no remainder so something like 2 divided by 2 is 1 4 divided by 2 is 2 6 divided by 2 is 3 so on and so forth well that's an even number right you can basically pair things up so if I had let's say 9 negative 6 times negative 1 times negative eight times negative four and let's say I attacked on times negative 1 and then times negative one well here's a pair here's a pair here's a pair I have six negative factors so that means I'm going to be able to pair these negatives up and there's nothing left over and so I'm going to end up with a positive result so this would still be positive 192. now if I took one of these away like this well now the answer is going to change into negative 192 because I can pair these guys up and pair these guys up but this is all left over so that means that an odd number would be something that's not divisible by 2. so in this case you have one two three four five of these guys if I take that 5 and I divide it by 2 what's going to happen is I'm going to get 2 with a remainder of 1. so you have one two and then one left over that one left over or that remainder is always going to give you a negative because I can't pair it with another one if I go back to times negative one well now I can pair it up I have six of these Skies six divided by 2 would be three right one two three of those guys so this would give me a positive 192. let's fly through this now so we have negative eight times three times negative 2 times 11. okay so I have one two negative factors two is an even number two divided by two is one no remainder and so I know I'm gonna have a positive product so now I can just multiply absolute values eight times three is twenty four twenty four times two is forty eight so I need to know what is 48 times 11 now and so 1 times 48 is 48 and then this one again you've got to move down 1 times 48 would be 48 and let's add here so 8 comes down 4 plus 8 is 12 2 down carry the one one plus four is five so let's drag that up there and say the final answer is 528. okay let's take a look at one more so here we have negative 2 times 5 times negative 1 times 3 times negative 20 times negative 30. so this looks a lot worse than it is first thing I would count I have one two three four negative factors four is an even number right 4 divided by 2 is 2 no remainder again you can always pair these guys up so this guy and this guy that's going to make a positive and then this guy and this guy that's gonna make a positive so if you have an even number of negative factors you will get a positive result so here I can just go through and multiply the absolute values so 2 times 5 is 10 and then times 1 would still be 10 and then times 3 would be 30. let me stop and write this out so I have 30 and then forget about the signs here we already know it's positive times 20 and then times 30. okay well what I can do is use my trick for trailing zeros three times two times three three times two is six six times three is eighteen so this is 18 and then I have one two three zeros that I'm going to attach to the end so one two three so the final answer here would be positive eighteen thousand all right now let's move on and talk about dividing integers so this is just as easy as multiplying integers so the first thing is we want to determine the sign of the quotient and then once we've done that we just want to divide the absolute values so the size rules are pretty easy so basically if you're dividing integers and you have the same sign you're going to get a positive quotient if you're dividing integers and you have different signs you're going to have a negative quotient so positive divided by positive or negative divided by negative will give you a positive and then if you have a positive divided by a negative or a negative divided by a positive you will get a negative all right before we get to the problems let me take a few minutes here and think about where the sine rules Come From so let's start with something we know let's say we have 15 divided by 3. so again what does this mean it means I have 15 of something could be boxes could be whatever and I'm going to split it up into equal groups where each group has three so three boxes or again whatever you have so we can use a related multiplication statement or again you can use repeated subtraction so you could go through and say okay this is 15 minus 3 and that would give you 12 and then you can go 12 minus 3 and that would give you 9 and then you would go 9 minus 3 and that would give you 6 and then 6 minus 3 will give you 3 and then 3 minus 3 would give you 0. so I did that one two three four five times so the answer is five and of course we already know that so the answer here would be five or you could have asked the question what times 3 gives me 15 and the answer from the multiplication tables would be five so we know 5 times 3 is 15 so 15 divided by 3 is 5. okay so using that same logic let's go ahead and ask the question what is negative 15 divided by three well this would be what I would put a question mark here and just say well question mark times 3 would give me negative 15. well for multiplying integers we know that if this is negative well that's going to come from a positive times a negative or negative times a positive this right here is positive so this has to be negative so the question mark would be a negative and then of course multiply the absolute values we know that 5 times 3 would give us 15. so this would be negative 5 here so the question mark here would be negative 5. okay so that's one way to think about it I have some things set up over here so now we're doing 15 divided by negative 3 equals what well again what times negative 3 would be 15. well this right here would have to be a negative 5 now because negative times negative would be positive so negative 5 times negative 3 would be positive 15 so that means 15 divided by negative 3 would be negative 5. now coming through here negative 14 divided by negative 7 is what again you could say what times negative 7 is equal to negative 14. so here you would want a positive right because positive times negative is negative so this would be be a two two times negative 7 is negative fourteen so negative fourteen divided by negative 7 is positive 2. so here we have that a positive divided by a positive gives you a positive you have a negative divided by a negative gives you a positive so again same signs you get a positive quotient then you have a positive divided by a negative gives you a negative and then a negative divided by a positive gives you a negative so different signs negative quotient okay let's burn through some examples so you have a negative 50 divided by five so again negative divided by positive is negative and then 50 divided by 5 would be ten so this is negative 10. here you have negative 40 divided by four so negative divided by positive is negative 40 divided by 4 is ten so this is negative 10 as well so then we have negative 200 divided by negative ten so negative divided by negative is positive and then 200 divided by 10 is 20. and then coming down here you have negative 34 divided by negative 17 negative divided by negative is positive 34 divided by 17 is 2. then we have 27 divided by negative 3. so positive divided by negative is negative 27 divided by 3 is 9 so this is negative 9 and then for the last problem we have 16 divided by negative 4 positive divided by negative is negative and then 16 divided by 4 is 4 so this is negative 4. hello and welcome to pre-algebra lesson 22. in this video we're going to learn about exponents with an integer base so for our lesson objective we want to learn how to perform operations with exponents that have an integer base so before we get into the main content let's just briefly review exponents so recall that at this stage we're using exponents to conveniently notate repeated multiplication of the same number so for example something like three times three times three times three times three I have one two three four five factors of three so I can write this as three to the fifth power right the number that's being multiplied by itself gets written as the bigger number this is known as the base okay so that's three in this case the smaller number in the top right hand corner this 5 is the exponent okay it's the exponent and so it's telling me that I have five factors of 3. now if we want to look at another example let's say we had 7 times 7 times 7. the number that's being multiplied by itself is seven so that's my base that is my base and then I have one two three factors of seven so my exponent is a 3. so at this point it's very very easy to work with exponents we've only worked with a base that's positive what we're going to think about today is what happens when we have a negative base well when we're working with exponents we need to be extra cautious when dealing with negative numbers there's some notational issues that are going to come up and we're going to explain that in this video so when working with a negative number that is raised to a power the base does not include the negative part unless we use parentheses okay so let me read this again it's very important that you understand this when working with a negative number that is raised to a power the base does not include the negative part unless we use parentheses so let's think about this problem using an example I want you to pretend that your teacher tells you hey what is negative 2 squared so you have your pencil your paper and you have a calculator and you start out by writing negative 2 squared just like that no parentheses and you go through and from your knowledge of exponents you say okay I have two factors of negative two so I have negative 2 times negative 2. now you say Okay negative 2 times negative 2 is 4. now you pull out your calculator and you verify you say okay I'm going to type in negative 2 I'm going to use the to the key right the to the key and then I'm going to put a 2 in so it's negative 2 to the power of 2 and I'll just erase that real quick and write it like this and your calculator spits out an answer that says negative four and so you're thinking what gives why did my calculator say that this is negative 4 whereas I think the answer is positive 4. well I'm going to tell you it's a notational issue and here's y so let me kind of erase this this is wrong by the way so I'll erase this and show you why all right if I have negative 2 squared written like that really what I'm saying is I want the opposite the opposite of 2 squared so to make this clear let me write this as negative 1 times 2 squared and using our order of operations we know that we would perform any exponent operation before we multiply so essentially I would have negative 1 times the result of this 2 squared is 2 times 2 that's 4. so now I have negative 1 times 4 and that's negative 4. so you can see where we got the wrong answer right we just didn't have the right notation so to get the notation correct for what your teacher asked you for if your teacher said hey what is negative 2 squared well essentially I would need to put parentheses around negative 2. the negative and the 2 need to be enclosed so that they are both part of the base so this now is a negative 2 and I'm squaring that so this would be equal to negative 2 times negative 2. and this is now equal to 4. and you can verify this go ahead and type this and this way with your calculator make sure to put parentheses around the negative 2 and you will see that you get an answer of 4. now you might be saying what if I did this trick up here where I'm multiplying by negative one well it still works out let's say I did negative one times two enclosed in a set of parentheses and I squared it well essentially what's going to happen is I'd have y negative 1 times 2 multiplied by negative 1 times 2. and just to kind of simplify this I just have multiplication involved so I can just remove all those parentheses and just write this as negative 1 times 2 times negative 1 times 2. and commutative property tells me I can reorder this so I'd have negative 1 times negative 1 times 2 times 2. so negative 1 times negative 1 is positive one then positive one times two is two two times two is now 4. so we can see that our mistake was just with the notation right that's all it is so moving forward the only thing we need to realize is that if I have a negative number that I'm working with and I want to raise it to a power I have to enclose that negative number inside of a set of parentheses now there are some scenarios where you'll get the same result although the notation is wrong and let me show you one of those so let's say you had something like negative four to the third power written like this the negative 4 is not inside of a set of parentheses so technically speaking because of the notation this is the opposite the opposite or you could say the negative of 4 cubed I could write this as negative 1 times 4 cubed and of course following my order of operations I would do 4 cubed first that's 4 times 4 which is 16 times 4 again which is 64. so this is negative 1 times 64 which is negative 64. now let's suppose that I enclosed this negative 4 inside of a set of parentheses and the last situation we looked at we went from negative to positive right we kind of corrected our mistake and we got a different answer will we still get a negative or will we get a positive here well what you're going to find is that in this case it will not change our answer and then we're going to explain why that's the case so let's say I enclose this inside of a set of parentheses so now I have negative 4 times negative 4 times negative 4. well I have three negatives involved in this problem three negatives is going to give me a negative three is an odd number in other words I have one pair here of negatives that gives me a positive but then I have one left over right so I'm going to end up with a negative result so I would have a negative 64 in this case also now knowing that I'm going to go through and read some rules that you just want to remember when you're working with exponents when the base is negative an enclosed in parentheses the result is positive if the exponent is even right because I would have an even number of negative factors that gives me a positive result the result is going to be negative if the exponent is odd because I would have an odd number of negative factors that's going to give me a negative result and then the other scenario you need to be aware of is that when the base is negative and not enclosed in parentheses the result is always negative always negative doesn't matter what the exponent is because in that scenario you're just saying Hey I want the opposite of what you're doing over there so for example the opposite of 4 squared because this negative 4 is not inside of a set of parentheses this will always be negative right this will always be negative so in this case it would just be the opposite or the negative of 4 squared 4 squared is 16 so this would be negative 16. if I was to write this with negative 4 inside of a set of parentheses now I would get positive 16. because in this case I have negative 4 times negative 4 2 negative factors gives me a positive result so in this case I'm thinking about okay well my exponent is even I'm going to get a positive result okay let's use these rules and just do some practice so we want to evaluate each all right so we have negative 3 inside of a set of parentheses and it's squared so this tells me that I have what I have negative 3 times negative 3. so this is going to be positive 9. but again what I can do is I can just look at because this is inside of a set of parentheses and my exponent is even I know my answer is going to be positive right I have an even number of negative factors in this case it's 2 and an even number of negative factors gives me a positive result what about negative 3 raised to the fifth power and notice how the negative 3 is enclosed inside of a set of parentheses well I know the result is negative because I have an odd exponent there 5 is an odd number and so I would have an odd number of negative factors and you could just think about 3 to the fifth power now you don't even need to think about the sign anymore 3 to the fifth power would be what it'd be 3 times 3 that's 9 times 3 again that's 27 times 3 again that's 81 times 3 one last time which would be 243. so this would be negative 243. what about negative 1 to the 38th power well 38 is an even number so we know the answer is going to be positive now what's cool about the number one is when you multiply it by something the number is unchanged If I multiply one by itself it doesn't change so 1 times 1 is 1. it doesn't matter how many times I do that if I go 1 times 1 times 1 times 1 times 1 times 1. the answer is one I could do this a million times I still get one so 1 to the 38th power forget about the negative 1 to the 38th power would just be one the fact that we have a negative and it's enclosed inside of a set of parentheses tells me I just need to look at whether this exponent is even which in this case it is or odd since it's even I know I'd have an even number of negative factors and an even number of negative factors gives me a positive so I end up with positive one what about negative 1 to the power of 71. well my negative 1 is enclosed inside of a set of parentheses but my exponent is odd so I would have an odd number of negative factors now that will give me a negative result and I would just think about 1 to the 71st power is just one so I'd end up with negative 1 as my answer okay now we have negative 10 cubed and this negative 10 is inside of a set of parentheses so again I'm looking at this exponent this exponent is a 3 that's an odd number and so that tells me I'm going to have a negative result once I know the song I can just think about okay what would 10 cubed be what would 10 cubed be well I taught you a trick for this basically when you have 10 raised to a whole number that is larger than one you just write down a one and follow it with the exponent number of zeros so in this case the exponent is a 3 so I would write one two three zeros and I would get a thousand you can just drag that up there and attach that negative to it and you would end up with negative 1000 as your answer what about negative 8 squared but again notice how there's no parentheses around negative eight so I'm not looking at the exponent here I don't care that this exponent is even the result is going to be negative right because I'm basically saying hey what's the opposite of 8 squared 8 squared is 64. when I take the opposite of it I get Negative 64. right and let me kind of write this out so you get a little bit more practice seeing this so this is essentially negative 1 times 8 squared so it's negative 1 times 64 which equals negative 64. now what about negative 8 squared where the negative 8 is enclosed inside of a set of parentheses well now I'm looking at my exponent I have a 2 there 2 is an even number so I know my answer is going to be positive and now I can just think about what is 8 squared okay that's 64. and So my answer would just be 64. okay let's take a look at negative 5 cubed where the negative 5 is enclosed inside of a set of parentheses well in this case my exponent is odd so I'm going to get a negative result right an odd number of negative factors gives me a negative and now I just think about what is 5 cubed 5 cubed is 5 times 5 that's 25 then times 5 again that's 125. So my answer is negative 125. but the final problem we look at negative 5 cubed where there's no parentheses around the negative 5. now in this case we also get a negative 125. and you might say well how come I got the same answer again to explain this in this case I'm saying hey what's the opposite of 5 cubed 5 cubed is 125 the opposite of that is negative 125. in this case whether the exponent is odd or even it doesn't matter you always get a negative and looking at this scenario where I have parentheses around negative 5 I look at my exponent if my exponent is odd like it is in this case I'm going to get a negative result I would have had negative 5 times negative 5 times negative 5. all right I have three negative factors three is an odd number so I'm going to get a negative result hello and welcome to pre-algebra lesson 23. in this video we're going to talk about the order of operations with integers so the lesson objective for today would be to learn how to use the order of operations when integers are involved now we previously talked about the order of operations when we wrapped up our section on the whole numbers so now that we're wrapping up our section on the integers we're going to revisit this topic and just do some examples with integers involved so before I go through the order of operations I just want to remind you if you're using something like PEMDAS so this is really really popular in the United States in other countries you might hear bodmas it's basically telling you the same thing you need to be really really careful a lot of people remember this as please excuse my dear Aunt Sally so parentheses exponents multiplication division addition and then subtraction remember when you look at this the m and the D the multiplication of division have the same same priority level and they need to be worked left or right I can't tell you how many times people memorize this and then they're looking at that and they're saying okay I need to multiply before I divide they are on the same priority level they are worked left to right then the other guy here is addition and subtraction so same priority level they're worked left to right so if we go through this we would say the first thing is to work inside of any grouping symbols so that's why we have that P that stands for parentheses but it's also for brackets or absolute value bars or anything that's a group example the second thing would be to perform all exponent operations so that's the E there now included in this would be something like a square root so let's say you had something like the square root of four and we haven't talked about this yet in the course and I'll explain as we get to an example but actually we can convert this over to exponential form so this is actually 4 to the one-half power so if you have a square root involved or a cube root or something like that you would actually do this on the same level so I'll talk more about this as we get into a problem let's just erase it for now so coming down here here you'll see the third and the fourth steps this is where people get lost right so again if I write out the PEMDAS so Pam Das like this this right here is where everybody gets confused so the m and the D I have here multiply or divide working left to right so if multiplication comes before division working left to right you do multiplication first if division comes before multiplication again working left to right you do division first same priority level don't let this ordering of the letters here trip you up then the same thing is going to go for the A and E S so add or subtract again working left or right okay with that being said let's just jump in and look at some problems so we want to evaluate each all right so let's take a look at the first problem we have negative three times then you have negative 2 and notice this negative 2 is wrapped and this is being squared and then plus you have negative 24 and then divided by 4. so what is the highest priority again we start with parentheses the only parentheses we have would be this guy right here which is wrapping this negative two and this guy right here which is wrapping the negative 24. so there's nothing to really do there with parentheses then we move on to exponent operations so do we have anything with an exponent yeah we have this negative 2 that's being squared so I want to stop here and remind you of what we talked about in the last lesson if you have negative 2 and it's wrapped so it's inside of parentheses and I Square it this is equal to negative 2 and then times negative 2 which is positive 4. okay you can do this on your calculator make sure you wrap it so you would wrap it like this negative 2 you would hit that key there and then you would put your 2 in there and then basically you would get a 4 as an answer now this is different versus if I did this let's say I wrote negative 2 like this I did not wrap it and then I squared it like this let's say you key that into your calculator you're going to get an answer of negative 4 and you're probably going to be confused by that because you're saying well that's negative 2 squared it's a notational issue if I ask you for negative 2 being squared you want to use this okay this is basically going to get translated into negative 1 times 2 squared so in other words the base here is actually 2 right it's not negative 2 because this negative is outside of here you could read this as the negative of 2 being squared okay so this would end up being negative 1 times 2 squared would be 2 times 2 or 4 so this is negative 4. so if you key it into your calculator you've got to make sure that you use the right notation so that you get the right answer let me get rid of this and what we're going to do is put equals here and I'm just going to replace this with a positive 4. so we have negative 3 and then times positive 4 and then plus you have negative 24 and then divided by 4. so I've taken care of my exponents the next thing again is multiplication or division working left to right so here I see that I have multiplication and I have division now working left to right this multiplication occur refers to the left of this division so that means this guy right here is the higher priority let me unhighlight that so that means I'm going to do negative 3 times 4. so a negative times a positive is a negative so let me write that as a negative and now I can just say well what is 3 times 4 that's 12. so this would be negative 12 here okay then we have plus negative 24 and then divided by 4. so now I'm still on my multiplication and division step and I have division here so I'm going to divide negative 24 by 4. so negative 24 divided by 4 would be negative 6. negative divided by positive is negative and 24 divided by 4 is 6. so that's negative 6. so we would say this is negative 12 and then basically we would have plus negative 6 or you could write negative 12 minus 6 whatever you're more comfortable with so the last thing here is just to do this addition so negative 12 plus negative 6 you have a common sign both of these guys are negative so the answer would be negative and then you could just add the absolute value so 12 plus 6 is 18. so this gives me a final answer of negative 18 all right now before we move on to the next problem I'm going to do this problem in a different way I'm actually going to change it up so you see we got negative 18 there so what I'm going to do is actually write this problem as negative 3 and then instead of wrapping this I'm just going to put times the negative of 2 being squared like this then plus we're going to have negative 24 and then divided by 4. and I'm going to show you that you get a completely different answer now you're going to get positive 6 and the change is going to be because this guy right here is going to throw everything off because your sign is going to go from a positive 4 to a negative 4. so if you're keying this into your calculator and you're expecting to get the same answer as this I'm going to show you where you're going to run into an issue so let me copy this and let me paste this in I'm just going to run through this really quickly so we know the first thing we would do would be the exponent operations but again because this is the negative of 2 being squared this becomes negative 1 times 2 squared in your calculator 2 squared is done first because of the order of operations so that's 4 so you get negative 1 times 4 which is negative 4. okay it's only positive four if this gets wrapped so I'm going to replace this right here with a negative 4 so you'd have negative 3 times negative 4 then plus negative 24 divided by 4. so now just like we saw before we're going to do this multiplication first but now instead of a negative times a positive we have a negative times a negative so a negative times a negative is a positive so negative 3 times negative 4 would be positive 12. now we have our plus negative 24 and then divided by four so now we would do our division so negative 24 divided by 4 that's going to be negative 6. so this becomes 12 and then plus your negative 6 so you could go 12 minus 6 if you want and obviously the answer there is 6. so you see the other way we did it when we wrap this we got a negative 18. and when we don't wrap this we get a positive 6. so it's very important to understand this notational issue it's going to come up for the rest of your time that you're studying math all right let's take a look at the next problem so here another easy one we have negative 20 divided by negative 5 then minus 12 times 2 plus 3 squared now the first thing we want to do is work inside of parentheses or any grouping symbols so we do have parentheses here but they're just used for clarity right they're just used to wrap that negative 5. so you can ignore that here and just move into working with the exponents so here we have 3 squared so we'd want to do that first so this is the highest priority let me copy all of this so negative 20 divided by you can wrap this or not wrap it I'm just going to wrap it for clarity so divided by negative 5 then minus 12 and then times 2 and then plus so 3 squared is 3 times 3 that's going to be 9. so we've taken care of the exponents and now we want to move into the multiplication or division so let me write this one more time because people do get lost here and it's very important to understand that multiplication division occur on the same level okay they're work left to right so if you're using PEMDAS or please excuse my dear Aunt Sally or you're in a different country than me and you're using bodmas it's the same thing you want to highlight these two and say that they are worked left to right okay you might want to put that down in your notes or something like that it's very important to remember that let me erase this and we're going to think about this I have division right here and then I have multiplication right there which one am I going to do first again they're the same priority level so I want to work left to right so starting on the left this operation occurs before this one okay so that means I'm going to do my division first so I'm going to divide negative 20 by negative 5. so negative divided by negative is a positive and then 20 divided by 5 would be 4. okay so that's done then we have minus 12 times 2 plus 9. now the next step because we're still on the multiplication or division level would be to multiply 12 times 2. now if you want you can convert this over you could say this is plus negative and do it that way or you can keep it like this it's going to give you the same answer in the end it's really up to you so I'm just going to keep it like this so I'm going to say this equals 12 times 2 is 24. so we would have 4 minus 24 and then plus 9. so here we have subtraction and we have addition now when you have integers involved where you're dealing with positive and negative numbers I like to convert all of the subtraction into a related addition problem I think it just makes it easier so I'm just going to go plus negative like this and then because of the commutative property of addition you can add in any order okay so I can just say this is 4 plus negative 24 maybe I could wrap that to make it more clear and then plus 9. so I can add however I want typically what you'll do is you'll add the numbers with the same sign first so I would say 4 plus 9 would be 13. and let me reorder that so nobody's lost let me reorder this to 4 plus 9 plus negative 24. four plus nine would be 13 and now we have plus a negative 24. so now we want to do 13 plus a negative 24. so hopefully you have mastered adding integers at this point I know when you have a different sign involved it can be a little bit tricky for some people but basically you want to look at these two numbers and say which number is larger in terms of absolute value the absolute value of 13 is 13 the absolute value of negative 24 is 24. so negative 24 has a larger absolute value so we're using that sign so the result will be negative so now I can just think about these numbers as if they were positive numbers and I could just do a subtraction subtract the bigger minus the smaller so what is 24 let me do this off to the side what is 24 minus 13. I think a lot of you can do that in your head at this point say it's 11 but just in case you can't 4 minus 3 is 1 and then 2 minus 1 is 1. so this is going to be 11. so this would be negative 11 for the answer all right let's take a look at the next problem so this is a little bit more challenging but still not that bad overall so we have some brackets here and inside the brackets we have a set of parentheses so negative 4 being squared plus 52 that's inside the first set then plus inside the second set you have negative 93 divided by 31 then plus 55. then outside the brackets we have divided by a set of parentheses here which is 2 cubed minus 28 inside all right the first thing I want to do is work inside of any parentheses or grouping symbols well you have brackets here and then you have parentheses here so what I'm going to do is actually start off inside the brackets and when I do that I want to reapply the order of operations so in other words I want to look for parentheses again and let me change my highlighter color here you see that you have that here and then you have that here okay so you want to make sure that you do what's inside the parentheses here and then what's inside of the parentheses here separately and then you can add so it doesn't really matter if you do this one first or this one first I'm just going to work on the one that's here on the left first let me put a little border here and I'm going to say I have a negative 4 being squared plus 52. so let's do that first now like we previously talked about if you're keying this into the calculator make sure that you wrap negative 4 before you square it otherwise you're going to get the wrong answer because if you have negative 4 being squared this is what this is negative 4 times negative 4 which is positive 16. if you have the negative of 4 squared again the negative is not wrapped so this is really negative one times 4 squared which gives you negative 16. two different results in your calculator so if you don't make sure that you're wrapping this you're going to get the wrong answer so very important so when we look at this we see that we have an exponent operation and we have addition so we would do the exponent operation first it has the higher priority we just said that negative 4 squared was 16 and then we would add that to 52. now I can pretty much do that in my head I know that's 68 but just in case you can't and you're not using a calculator again you can always stop and do a vertical addition so 52 plus 16 2 plus 6 is 8 5 plus 1 is 6. so that's how we get the 68. okay so let's erase everything and how much work you want to show is up to you I'm just going to show all the steps so nobody's lost I'm going to put the brackets here the result of this is 68 then plus I'm just going to copy this so you have negative 93 divided by 31 plus 55 so that's inside the parentheses close down the brackets and then we're dividing by so inside these parentheses you have 2 cubed minus 28. okay close that down let me slide down here a little bit all right now the next thing is we're still working inside the brackets so now let's go into these parentheses here here and so basically you have negative 93 divided by 31 and then Plus 55. so we would obviously divide before we add division has a higher priority level negative divided by positive is negative when we think about 93 divided by 31 that one's a little bit tricky because essentially you can't really benefit from doing the long division you really have to say well what is 93 divided by 31 right because 31 goes into 9 zero time so you have to expand this and say well 31 goes into 93 so I'm back to the original division problem if you use multiplication you would find that it's 3 31 times 1 is 31 31 times 2 is 62 and then 31 times 3 would be 93. so you basically have to use multiplication there so 3 times 31 as I just said is 93. subtract here you get zero so there's no remainder so this would end up being negative 3 right so negative 93 divided by 31 is negative 3. okay let's get rid of this and now we're going to add 55. now this one again it's a little bit tricky because you're adding integers with different signs what I like to do is flip the order here and say this is 55 plus a negative 3. and then notice if I had been given 55 minus 3 I can legally go to that we can always convert our subtraction into a related addition so change the subtraction into addition change the number being subtracted away into its opposite okay so this ends up just becoming 55 minus 3 which is going to be 52. so let's get rid of this and let's say this equals inside of brackets 68 plus 52 and we'll close that down and it's divided by again in the parentheses we have 2 cubed minus 28. okay let's wrap things up by doing this addition here I think it's probably something again you could do in your head but there is some carrying there so maybe you want to write it out so 68 plus 52 8 plus 2 is going to be 10 put a zero down carry the 1. 1 plus 6 is 7 plus 5 is 12. so this would be 120. and say this equals 120 so we're done with that divided by again inside the parentheses 2 cubed minus 28. so now I want to work inside of these parentheses here and I want to do 2 cubed first let me make that a little bit more clear I want to do that first because that has a higher priority than the subtraction so 2 cubed is 2 times 2 times 2 which is going to be 8. so let's say this is 120 divided by you would have 8 minus 28. so now I want to do the subtraction again what I like to do is go plus negative here and think about it that way so if I have a negative 28 plus 8 then I'm using the sign of the number let me write 120 here I'm using the sign of the number with the larger absolute value so negative 28 has a larger absolute value so I'm going to put a negative there and then I'm just going to do a subtraction so basically the larger absolute value minus the smaller so basically you would say that 28 Minus 8 would be 20 so this is divided by negative 20. okay so from here it's actually pretty easy we know positive divided by negative is a negative and then you can think about 120 divided by 20. so again we have another one where the long division doesn't really help you that much because 20 doesn't go into 1 or 12 so you have to go all the way into what is 20 into 120 which is basically going to give us 6. right you have to use a related multiplication statement to figure that out so 6 times 20 is 120 subtracting you get 0. so this becomes negative 6 as our final answer all right let's take a look at the next example so this one's a little bit more tedious a little bit more time consuming is what I would really say so we have the square root of 64 times negative 9. let me wrap this here so it's more clear then you have minus a negative 3 then plus remember these vertical bars tell you that you want the absolute value of something and those are grouping symbols so we have the negative of 2 squared so notice this negative here is not wrapped inside of parentheses then times negative 15 and then plus 4 and then divided by 2. I'm going to wrap this also for clarity then down here we have minus and then we have these parentheses here so you have negative 2. notice this one's wrapped being squared plus negative 8 divided by 2 and this is squared the first thing I would do is think about what parentheses or grouping symbols do I have when you have parentheses that are just used like these just to wrap a number don't worry about them okay so you're just looking for grouping symbols like the absolute value bars there or these parentheses here okay so you have some operations inside so where I want to start let me get rid of this highlighting here so nobody's confused where I want to start I'm just going to start inside of the absolute value bars if you wanted to do this guy inside of here first it's really up to you I'm just going to start there so I'm just going to copy this down here so we have the absolute value of again be very careful this is the negative of 2 squared times I'm going to wrap this so negative 15 then plus 4 and then divided by 2. we'll come back up in a moment I'm going to need a lot of room for this let me come down here and let me put equals here so the first thing is you would think about what's my highest priority well inside of here do I have any parentheses well I have these parentheses but they're just used to wrap a number so really I don't I could just move on to exponents and here I have this exponent operation but again if you're keying this into your calculator this right here is going to give you negative 4. this right here is going to give you positive 4. again this is the negative of 2 squared so your calculator is doing negative 1 times 2 squared which is giving it negative 4. this one it would do negative 2 times negative 2 which will give it positive therefore so I've got to make sure that I use the right one if I want the right answer so this right here would give me a negative 4. not a positive 4 because it's not wrapped then times here you have this negative 15 and then plus 4 and then divided by 2. let me close that down and now I'm thinking about okay I have multiplication addition and division multiplication division are on the same priority level they are worked left or right so this one is to the left of this one so I'm going to multiply first so negative 4 times negative 15 negative times negative is positive 4 times 15 is 60. so this right here I'll put the absolute value of 60 then plus 4 and then divided by 2. so now I have addition and division you want to divide first so 4 divided by 2 is 2. so we'll say this is the absolute value of 60 plus 2. and then from here 60 plus 2 is 62 so this would be the absolute value of 62. we can go ahead and do that operation now the absolute value of a positive number is just the number so the absolute value of 62 is 62. and I'm just going to come back up here and put equals I'm going to copy this so the square root of 64 then times negative 9 then minus a negative 3 and the result of this was 62. so now I'm just going to put plus 62 I'm just going to keep working then I have minus this guy right here is not going to fit on this line so let me slide down just a little bit and I'm going to write inside of parentheses you have this negative 2 being squared notice that this is wrapped okay that's very important then plus negative 8 and then divided by 2 and then this guy is squared okay let's put equals here in terms of what to do next you have these parentheses here that you need to work inside so I'm going to think about let me just write this down here I'll just put a little border and I'll say that we have this negative 2 again this is wrapped being squared plus you have negative 8 and then divided by 2 and then we're going to square this I'm just working on this part right now so thinking about my order of operations inside I have parentheses but they're just used to wrap this number and this number so I move on to exponent operations so I have that here right so negative 2 being squared so again negative 2 being squared because this is wrapped you're going to get 4. if you don't wrap it if you keep in your calculator like this you're going to get negative 4. so again we have to make sure that we're paying attention to which form we're using so this right here would be a positive 4 and then plus a negative 8 and then divided by 2 and this whole thing is squared all right so now I have addition and Division I want to divide first so negative 8 divided by 2 will be negative 4. this would end up giving me 4 plus a negative 4. of course you could write that as four minus four if you wanted to and this would be squared so 4 plus a negative 4 those guys are opposites or additive inverses or however you want to think about that the sum is going to be zero so we'll say this is zero squared let me get rid of all this and let me come back up and get rid of this border I'm just going to copy things again so I have the square root of 64 times this negative 9. then minus this negative 3 then plus 62 and then minus this is going to be 0 squared okay so I've dealt with all the parentheses in the grouping symbols so now I want to move on to exponents now this right here is worked on the same priority level as an exponent because the square root of 64 as we're going to learn later on is the same thing as 64 to the one-half power we won't talk about that for now we'll talk about it much later on but you just need to know that if you're seeing some type of square root or cube root or something like that in an order of operations problem it's going to be with the exponents in terms of the priority level okay let me get rid of this and first let's do the one we know we know that 0 squared would be zero so let me write the square root of 64 times negative 9 minus a negative 3 then plus 62 and you can write minus 0 if you want or you can leave it off of course if we subtract away zero it's not going to do anything so let's just leave it off okay let me slide down here and now let's think about the square root of 64. so at this point we have not talked about square roots and basically all you need to think about is this if I have the square root of some number let's just say 4. then basically what I'm asking for is what number when multiplied by itself will give me this number under the square root symbol which is called the radicand in this case for 4 you would immediately think about 2 right 2 times 2 is 4. so the square root of 4 is 2. now some of you are going to say well hey negative 2 times negative 2 is also 4. so we have a different notation for that you have the negative of the square root of 4 which would be negative 2. so if I ask what positive number when multiplied by itself gives me 4 I want this notation here so this is the principle where some people say the positive square root of 4 that is positive 2. then this one I'm asking for what negative number when multiplied by itself gives me the radicand right so this is the negative of the square root of 4 and that's going to be negative 2 because negative 2 times negative 2 would be 4. okay so we understand that General concept and so so with that being said we would say well what is the square root of 64. well what number when you multiply it by itself would give you 64. now you could Factor the number which we haven't talked about yet basically right now I think most of you know that that would be 8 because 8 times 8 is 64. again if you have the negative of the square root of 64 that's where you would get Negative 8. okay we're asked for the principal square root of 64 which is positive 8. so that's the difference between the two so let me get rid of this let me put equals let me put 8 times negative 9 and then minus a negative 3 and then plus 62. so I don't have any exponents anymore I just have multiplication subtraction and addition so I want to multiply before I do any subtraction or addition so this is positive times negative which is negative and then 8 times 9 is 72. so this is negative 72. now you have minus a negative 3 and then plus 62. so we have talked about the fact that minus a negative is plus a positive so I know this is the one that trips up a lot of students when you see minus a negative it is plus a positive so I'm going to rewrite this as negative 72 plus 3 plus 62. again minus a negative plus a positive so then if you want to since this is all addition now you can reorder it I'm just going to say this is 3 plus 62 and then plus negative 72. again that's my commutative property to where I can add in any order let me actually grab this let me paste this in here so we have some room so now I'm just going to add 3 and 62 together 3 plus 62 is 65. so you get 65 plus a negative 72. so at this point you basically have to think about okay I'm adding these integers with different signs which number has the larger absolute value the absolute value of 65 is 65 the absolute value of negative 72 is 72. so negative 72 has a larger absolute value so we're going to use that sign in the answer so this would be negative once you're done with that just work with the numbers as if they were positive and you're just going to do a subtraction of the larger minus the smaller so I'm going to go 72 minus 65. borrow here this is a six this is a 12 12 minus five is seven six minus six is zero so I'm just basically going to drag that 7 up here and say that negative 7 is my final answer now if you didn't get negative seven go back and just be very careful and check your work okay if you're using an online calculator to check or something like that just make sure you're wrapping things correctly a lot of times the smallest error will give you a big change in the result okay so the answer you're looking for here is negative seven all right let's take a look at a tedious problem so for this one before we get started I need to explain something about fractions so at this point in the course we have not talked about fractions but we're going to get to that topic very shortly if I had something like let's say 20 divided by 5 I could actually write this division problem using a fraction so I can take my dividend or the starting amount and put that up here in the top part of the fraction which is called the numerator and then I'm going to have my fraction bar and then this divisor here this five I'm going to put that in the bottom part of the fraction this is called the denominator so I can say this is 20 divided by 5 or 20 over 5 or 20 fifths there's a lot of different ways you could say that but either way this is going to be a result of 4. and we'll cover this in more detail later on in the course I just want you to understand that you can write a division using a fraction bar so when you come across an order of operations problem with a fraction like this involved you want to simplify the numerator and denominator separately and then you will perform the main division so at the end I would perform the final value in the numerator divided by the final value in the denominator just like we did 20 over 5 like that and we got four so we're going to do something similar here okay so what I'm going to do is start off with my numerator and just simplify that so I'm just going to copy it so negative 2 to the fifth power divided by 4 and then times we have this negative 3 and then plus so we have these parentheses like this this I'm just going to use a different color for clarity and then we have this negative 3 being cubed and then plus the absolute value of 4 minus 4 squared let me close that down and let me change back to this color and close that down okay I'm gonna grab this all right let's paste this in and we're going to go real slow just step by step so the first thing is we're thinking about do I have any parentheses or grouping symbols don't worry about the parentheses that are used to just wrap a number so this and this you can ignore that for now you're just thinking about do I have any parentheses with some operations inside that I need to do well yeah I have these parentheses here so I want to start inside of there but once I get inside I need to reapply the order of operations so I need to think about inside of here are there any parentheses or grouping symbols and the answer to that is yes so if I come in here I see that I have the absolute value bars there and so those are grouping symbols so I would want to start with this right here so what is the absolute value of 4 minus 4 squared well inside of there you would think about okay I have subtraction and then I have an exponent so I want to do the exponent first so 4 squared is 4 times 4 that's 16. so this is the absolute value of 4 minus 16. so a lot of you can do that in your head at this point so 4 minus 16 is negative 12. if you can't you can rewrite this as 4 plus a negative 16. okay so that is what you can do if you're struggling with things like that you can always change it into a related addition statement and then you could just use your rules for adding integers with different signs so what I would say is the absolute value of what the absolute value of 4 is 4 and the absolute value of negative 16 is 16. so the larger absolute value comes from this guy right here the negative 16. so you use that sign in the answer so I know this is going to be negative then you would do a subtraction the larger absolute value minus the smaller so 16 minus 4 would be 12. so this is negative 12 and we want the absolute value of that which is going to be positive 12. so let me get rid of all of this just scratch work and I think because this is going to take such a long time that maybe I could just erase things I'm just going to erase this and hopefully you've copied that down and I'm just going to put a positive 12 there so we'll just erase things as we go so now I'm still inside of these parentheses here and now I have an exponent operation and I have addition so the exponent comes first you have negative 3 that is being cubed here so this is negative 3 times negative 3 times negative 3. so it doesn't really matter that this is wrapped here you have an odd exponent so you have an odd number of negative factors that's going to give you a negative right negative times negative times negative will give you a negative and then 3 times 3 is 9 9 times 3 is 27 so this is negative 27. okay let me get rid of this and put in a negative 27 there okay let me get rid of this now again we're adding two integers with difference lines so you want to use the sine of the number with the larger absolute value the absolute value of negative 27 is 27 the absolute value of 12 is 12. so negative 27 has a larger absolute value so if I'm doing negative 27 plus 12 solve I know the answer is negative and then I just do a subtraction larger absolute value minus the smaller let me put this off to the side so you're going to go 27 minus 12. 7 minus 2 is 5 2 minus 1 is 1. so I would drag this over here and then basically you would have negative 15 as the answer so let me erase this and this is now negative 15. okay let me get rid of this now these parentheses you could keep them if you want they're wrapping the negative you could drop them it's really up to you it's just personal preference all right so now that we've gotten rid of all the grouping symbols we need to think about the exponent operations and we have negative 2 to the fifth power so negative 2 to the fifth power now this is an odd exponent so it really doesn't matter if you typed negative 2 like this to the fifth power into your calculator you're going to get negative 32. and if you wrap it you're going to get negative 32 and that's because you have an odd exponent so with an odd exponent then basically you're going to have an odd number of negative factors which gives you a negative result here this is basically always going to be negative it's the negative of 2 2 to the fifth power so your calculator treats this as negative one times two to the fifth power which is negative 32. with this one it's saying well what is negative two times negative 2 times negative two times negative 2 and then times negative two well that's going to be negative 32. so just two different ways to think about things the answer is the same either way but how you go about getting it is different all right let me get rid of this and I'm going to replace this with a negative 32. so negative 32 and so now we have division multiplication and addition and this really looks like an X I'm trying to make this a little bit more clear there that that's a multiplication symbol so you have this division here that's going to occur to the left of the multiplication and again this is where everyone gets tripped up because of the ordering of the letters in PEMDAS everyone thinks that multiplication comes before division they have the same priority level they are worked left to right so because the division occurs to the left of the multiplication you want to divide first so you want to say what is negative 32 divided by 4 negative divided by positive is negative 32 divided by 4 is 8. so this is negative 8. so I'm just going to erase this and put in a negative 8 there okay so now we have multiplication and addition so we have negative 8 times negative 3 that would be done first multiplication is the higher priority negative times negative is positive 8 Times Square is 24. so this will just be 24. and for the last guy here we're basically adding these integers with different signs so again use the sign of the number with the larger absolute value the absolute value of 24 is 24 the absolute value of negative 15 is 15. so we're going to use a positive here because this guy has a larger absolute value so I know this is a positive so then I would just subtract what is 24 minus 15 a lot of you can do that in your head you know it's going to be 9 but again you can always do it off to the side what is 24 minus 15. you'd borrow here this is going to be a one this is 14 14 minus 5 that's going to give you 9. okay let me get rid of this and you don't need that plus symbol there so basically I'm going to come back up and the result of the numerator is going to be 9. then this is over I'm just going to copy the denominator so we have this negative of 4 squared minus the square root of 9. and then plus you have the absolute value of 2 times this negative 3 and then plus 18 divided by three minus 15. let me close down the absolute value bars there then plus you have this negative 1 being squared okay I'm going to grab this paste that in there all right so a lot to do here you have these grouping symbols here so for parentheses and then you also have the absolute value bars there so I'm just going to start with this one and then we'll simplify this completely and then we'll work on this one and then we'll see what else we have so inside the parentheses you have the negative of 4 squared minus the square root of 9. so what is done first well in this particular case you have an exponent operation and then you have a square root here I told you those are the same priority level so it really doesn't matter if I do this one first or this one first I'm going to basically do each one of these operations before I do this subtraction so if I think about again the negative of 4 squared you have to be careful there your computer or your calculator whatever you're using is saying this is negative 1 times 4 squared so this is negative 16. again if you want positive 16 you have to wrap that so you would have to do negative 4 wrapped in parentheses being squared and that's going to give you 16. and I know I keep going over this again and again and again and that's just because it comes up all the time as a common mistake so let me get rid of this and let me replace this with negative 16 here and then the square root of 9 again we are asking for what positive number when multiplied by itself gives you 9 well that's going to be 3 right so I'm going to say that the square root of 9 or at least the principal square root of 9 is going to be 3. so now I have negative 16 minus 3 and so you can go plus negative here to make that a little bit cleaner and let me wrap that so we have the same sign here so I'm just going to put a negative there and then just add the absolute value so 16 plus 3 would be 19 so this is going to be negative 19 as the answer okay let me get rid of this and actually I'm just going to erase things as I go and I'll just say that this is negative 19. all right so now I'm going to mess with this guy right here so inside the absolute value bars you will notice that you have what you have multiplication you have addition you have Division and subtraction so the multiplication occurs to the left of the division so that means it has a higher priority so I'm going to go 2 times negative 3 first a negative times a positive or positive times a negative will give you a negative and then 2 times 3 is 6. so let's get rid of this and just put negative 6 there then I have addition Division and subtraction so you want to divide first 18 divided by 3 is 6. so let's put a 6 there and so now I have addition and then subtraction so the addition and the subtraction have the same priority level we're just going to work them left to right so here we're seeing that the addition occurs to the left of the subtraction you want to add before you subtract so you have negative 6 Plus 6. so I'm adding a number in its opposite or a number in its additive inverse that's always is going to give me zero so this right here is zero so you could write zero if you want or you could leave it off really doesn't matter 0 minus 15 is still negative fifteen so I'm just going to put negative 15 there and if I take the absolute value let me close this down so this is a little bit more clean if I take the absolute value of negative 15 that's going to give me positive 15. okay the last thing here let me move this down the last thing here when we're looking through everything we have an addition we have another addition and then we have an exponent operation so I want to perform the exponent operation first so you have a negative one again this is wrapped and it's squared so this is negative 1 times negative 1 which is one and again I'll show this one more time if you had written the negative of 1 squared this becomes in your calculator negative 1 times 1 squared which becomes negative one right so negative 1 times 1 is negative 1. let me get rid of this and of course that form there because it was wrapped that's positive one okay so now we could just go through and add again I like to add the numbers with like Signs first that's possible because of the commutative property we can add in any order so I could say that this is 15 plus 1 plus a negative 19 like this so 15 plus 1 is 16 and then you have 16 plus a negative 19. now a lot of you can eyeball this and see that it's negative three but again you can always go through the official procedure so I would use the sign of the number with the larger or absolute value the absolute value of 16 is 16 the absolute value of negative 19 is 19 and so negative 19 has a larger absolute value so we're using its sign so the result is negative and then you just subtract the larger absolute value minus the smaller so 19 minus 16 is 3 so this is going to be negative 3 as the answer so let's go back up and let's put a negative 3 there so at this point again you can convert this over remember we said that 20 divided by 5 could be written as 20 over 5 which was 4. well if I have 9 over negative 3 I can convert that over if it's easier for you to 9 divided by negative 3. so either way I'm just going to keep it like this as a fraction but if you need to do that for right now because you're not used to working with fractions that's fine basically 9 divided by negative 3 a positive divided by a negative or a positive over a negative that's going to be a negative and then 9 divided by 3 is 3. so the final answer here is going to be negative 3. now this problem has so many different operations that if you didn't get negative three you really have to go back a lot of times problems like this you just rip the paper up because there's just so many places you can go wrong but you really want to go back and just check every step and you can follow along with the video to see what answer dot I get for each step and then as I said before the smallest mistake the smallest mistake on a problem like this can give you you a really really different answer okay so you're looking for negative 3 as the final answer hello and welcome to pre-algebra lesson 24. in this video we're going to look at some divisibility rules so our lesson objective for today is just to learn how to determine if a number is divisible by then we have the numbers 2 3 4 5 6 7 8 9 10 11 or 12. so essentially it's the numbers 2 through 12. now we're going to hold off on this number seven and this number 11 until the very end because the divisibility rules for 7 and 11 are a bit more time consuming I'm not going to say they're complex but they're more time consuming than the other ones so what does it mean when we say a number is divisible by another so a number is divisible by another if the result has no remainder so for example 25 is divisible by 5. so 25 is divisible by 5 because 25 divided by 5 is 5 and it does not have a remainder 30 is divisible by 3 because 30 divided by 3 is 10 with no remainder if I look at something like 19 divided by 4 is 19 divisible by 4 well no it's not if I take 19 and divide it by 4 I'm going to get 4 with a remainder of three so because there's a remainder involved 19 is not divisible by four so when we have a number that is divisible by another the result has no remainder so what we're going to do is we're just going to look at the divisibility rules I'm going to do 2 through 12 and we're going to exclude 7 and 11 until the very very end the ones 2 through 12 excluding 7 and 11 are very very easy you just need to write them down as you go you will memorize them as you practice and then eventually you won't need your paper anymore so let's begin by looking at divisibility by two so a number is divisible by 2 if it is even so we've briefly kind of talked about even numbers and I kind of gave you the definition of even and said that a number is even if it's divisible by two well yeah that's true but kind of the shortcut is the final digit is a zero two four six or eight that's the way you can tell if something's going to be even and therefore divisible by two now if you looked at let's say the first 10 numbers you can see the pattern if you look at zero two four six eight and ten those are the even numbers here the other numbers and let me change the color here one three five seven and nine those are called odd numbers so if a number is not even it's odd so they alternate you go even then odd and even then odd and even then odd and that's just how it goes forever and ever and ever so again for divisibility by two you're looking at the final digit of the number if it's a zero two four six or eight then it is automatically divisible by two it is an even number so for divisibility by three the sum of its digits is divisible by 3. and if you don't get it at first the first time you add the digits you say well I don't know if this is divisible by three you can keep adding digits you can keep going until you find out hey is this number divisible by three for divisibility by four the final two digits of the number forms a number that is divisible by four for divisibility by five the final digit is a zero or a five for divisibility by six the number is divisible by both two and three so for divisibility by eight it's similar to the rule that we had for divisibility by four so the final three digits of the number forms a number that is divisible by eight then for nine it's similar to the rule for three so the sum of its digits is divisible by nine and just like we saw with three when we're checking for divisibility by nine if we form the sum of the digits and we don't recognize whether that sum is divisible by nine we can take that sum and we can sum the digits again for divisibility by 10 the number ends with a zero and then finally for divisibility by 12 the number is divisible by both three and four all right so let's get some practice going so we want to determine whether each number is divisible by two three four five or six so I'm starting with the number 232. so is it divisible by 2. is it divisible by two well a number is divisible by two if it's even and again I told you that a number is even if the final digit is a 0 2 4 6 or 8. the final digit here is a two so this number is even so yes it's going to be divisible by 2. now let's check the visibility by three so a number is divisible by 3 if the sum of its digits is divisible by three so what I would do is I would go through and say okay what is 2 plus 3 plus 2. 2 plus 3 is 5 5 plus 2 is 7. so you would then ask yourself if I divide 7 by 3 would I have a remainder and yes you would 7 divided by 3 is 2 with a remainder of 1. 2 with a remainder of 1. so 7 is not divisible by three and therefore 232 is not divisible by three what about four so a number is divisible by four if the final two digits of the number forms a number that is divisible by four so just look at the final two digits of the number just think about only those two digits so the number 32 is that divisible by four well yes it is 32 divided by 4 is 8 with no remainder so this would be a yes and what about five a number is divisible by five if the final digit is a zero or a five so if I look at the final digit here it's a two that's not a zero and it's not a five so the answer would be no and then lastly we want to ask is it divisible by six well a number is divisible by 6 if it's divisible by both 2 and 3. so 232 we found earlier is divisible by 2 but it's not the visible Y3 so if it's not divisible by both 2 and 3 it will not be divisible by 6. so this is a no okay for the next one we're looking at the number 93 and again we're doing 2 3 4 5 and 6. so for the number two remember we're looking at is it an even number meaning does it end in 0 2 4 6 or 8. this ends in a three 93 is an odd number it's not even it doesn't end again in a zero a two a four or six or an eight so it's not going to be divisible by two this is a no what about three well to be divisible by three the sum of the digits for the number would be divisible by three so you do 9 plus 3 that's 12. 12 divided by 3 is 4 with no remainder so therefore because the sum of the digits which is 12 is divisible by 3 the number 93 is divisible by 3. so this is going to be a yes this is a yes now next is this number divisible by four now generally you're going to ask okay the final two digits of the number forms a number that is divisible by four but we only have a two digit number here so we have to think about another trick now one of the things I'm going to teach you and you'll understand why in the next lesson 4 is built from 2 times 2. so for that reason if a number is not divisible by 2. if a number is not even it will not be divisible by 4 automatically so we don't need to go through and say okay 93 divided by 4 is there going to be a remainder 93 is an odd number there is no odd number that's going to be divisible by 4. and so this is going to be no next we look at 5. so for a number to be divisible by 5 the final digit is a 0 or 5. here our final digit is a 3 so this is going to be a no and then the last one is this number divisible by 6 a number is divisible by six if it's divisible by both two and three so we got a yes on three but a no on two so it's not going to be divisible by six so no all right let's look at one final problem and then we'll look at some practice problems that deal with 8 9 10 and 12. so what about 300 so again we're working with two three four five and six where's the divisible by two well yes it is when your final digit is a zero it tells you it's an even number and for divisibility by two we need the number to be even what about three well you can see that that's yes right away because if you sum the digits here three plus zero plus zero is three three divided by three is one no remainder so that's a yes as well what about four if you look at the final two digits of this number now you might find it odd you see Zero Zero you say well that's what number does that form it's just form zero so basically you're saying is zero divisible by four well yes because zero divided by anything other than 0 is 0 with no remainder so this is zero no remainder so because it doesn't have a remainder you can say that yes 300 is divisible by four so this is a yes then what about five again for five we're looking at the final digit of the number and in this case it is a zero so yes if a number is divisible by five it's going to have a final digit of a zero or a five so this is a yes as well and then lastly for six we know that that's a yes because we just look at what we've already done for two and three those are both a yes so this is a yes right for a number to be divisible by six it needs to be divisible by both two and three and in this case it is okay now we're going to look at eight nine ten and twelve so to determine whether each number is divisible by 8 9 10 or 12. all right so we have 918 again we're working with 8 9 10 and the number 12. so the rule for eight is that the final three digits of the number forms a number that's divisible by eight now that's not really going to help us here because we have a three digit number one of the things you can kind of think about is 8 is built out of two times two times two so if it's not an even number which in this case it is so if you had something like 917 you could just say no right that's an odd number it's not even it's not divisible by two so you could just say no right off the bat but in this case we do have an even number so we really just have to check it by doing a long division so we'd say 918 divided by eight eight goes into 9 once one times eight is eight subtract you get one bring this down 8 goes into 11 once one times eight is eight subtracting you get three bring down eight eight goes into 38 how many times now here's where you could stop because 8 times 5 is 40 That's too big eight times four is thirty-two so you know you're going to have a remainder all right 4 times 8 is 32 I get a remainder of 6. so it didn't divide evenly meaning we had a remainder so we know that the answer here is just going to be no so this is new what about divisibility by nine well that's a similar rule to three the sum of the numbers digits will be divisible by not so we just sum this nine plus one is ten ten plus eight is eighteen think about 18 divided by nine that's two with no remainder so this one's going to be yes this will be yes then what about 10 . so the rule for 10 is you look at the final digit and if it's a zero it's going to be divisible by 10. if it's anything else it's not so because this is an 8 for the final digit so the answer is no so what about 12 well for 12 we look at the fact that it needs to be divisible by both three and four okay three and four so we don't know if it's divisible by 4 yet because we haven't checked that but for three if something's divisible by 9 it's automatically divisible by 3 because 9 is 3 times 3. so we got a yes on the divisibility by three is it divisible by four look at the last two digits of this number if I had 18 divided by 4 would I get a remainder well yes I would 18 divided by 4 is 4 with a remainder of 2. all right if I did 18 divided by 4 or goes into 18 four times four times four is sixteen subtracting you get two so four with a remainder of 2. so this one is going to be no and again although it's divisible by 3 it's not divisible by 4 and so because it fails that one it's not divisible by 12. okay what about sixteen thousand two hundred ninety nine so again let's look at 8 9 10 and 12. so the number is not even the final digit is a nine so it's not an even number it needs to end in a zero a two a four a six or an eight to be even so because it's not an even number it's not divisible by two so there's no way it's going to be divisible by eight so this is a no what about the number nine Well for now we sum the digits so we have one plus six that's seven seven plus two is nine nine plus nine is eighteen eighteen plus nine is 27. so 27 divided by 9 is 3 with no remainder this is a yes what about ten a number is divisible by 10 if the final digit is a zero in this case our final digit is a nine so we're going to put a no there a number is divisible by 12 if the number is divisible by four and three although it would be divisible by three again if a number is divisible by 9 is automatically divisible by 3 it's not going to be divisible by 4. and the reason for that is it's an odd number right if it's not divisible by 2 it can't be divisible by 4. so this is an automatic node all right we'll look at one last one and then we're going to talk about the rules for 7 and 11. so here we have ten thousand eight hundred let me kind of change the colors up so we have 8 9 10 and 12. so for eight again we look to see first and foremost is it even and yes it is right it ends in a zero and then we can look at the final three digits of the number that's 800 is 800 divisible by eight well yes it is and that should be easy for you because 800 divided by eight you can pretty much do that in your head that's a hundred right because 100 times 8 would be 800. so we know that this would be a yes next we look at nine so we would add here one plus zero is one plus eight is nine plus zero plus zero is still nine now 9 divided by 9 is 1 with no remainder so this would be a yes as well then what about 10. look at the final digit of the number and it's a zero so this is yes and then what about 12. well we know it's divisible by 3. if a number is divisible by 9 it's automatically divisible by 3. so that part checks out but what about four the final two digits of the number would form the number zero zero divided by four would be zero with no remainder so it's divisible by four so the answer here would be yes because it's divisible by both three and four it is going to be divisible by 12. okay so what about the rules for 7 and 11. so I'm going to tell you the rules for 7 and 11 a lot of teachers just skip them over because they can be a little confusing but in all reality if you practice them just like anything else it's not that hard so for seven there's some steps you need to follow the first thing you're going to do is remove the last digit from the number then you're going to take the number you removed and double it subtract this away from the shortened original number and I'll put here that we may need to repeat the process a few times sometimes you have a large number and you'll do it once or twice and you you just don't know if it's divisible by 7 yet so you just keep going until you have a single number left and you can say okay is it seven or you could have a multi-digit number like 14 and say okay well 14 I know that's divisible by 7. all right so let's practice determine if each number is divisible by 7. all right so we have twenty six thousand four hundred eleven now the first thing I'm going to do is I'm going to remove the last digit from the number so basically just think about it as I'm cutting this number off and I'm turning the number into 2641. that's my new number now the number that I cut off what I'm going to do is I'm going to double it meaning I'm just going to multiply it by 2. so 1 times 2 is 2 and then I'm going to subtract it away from this shortened number so I'm going to subtract the weight 2. now what am I going to get well I need to borrow here 11 minus 2 is 9. bring down the 3 bring down the six bring down the two does anybody know if 2639 is divisible by seven I sure don't so you would continue right you would just do the same thing so now I'm going to go okay 2639 cut off the last digit bring that guy over here I have 263 now as my shortened number and I'm going to double nine so I multiply by 2. 9 times 2 is 18. so 9 times 2 equals eighteen and I'm going to subtract that away from 263. so I'll borrow here this becomes 5 this becomes thirteen thirteen minus eight is five five minus one is four bring down the two I have 245. I still don't know if this is divisible by 7 so I'm going to continue I'm going to bring this over here I have 245 cut off the last digit that's a five and so that leaves me with a short number of 24 double five five times two is ten subtract that away from 24 and I get 14. so now I do have a number and I know if it's divisible by 7. 14 divided by 7 is 2 with no remainder so after going through this process I can see that my original number which is twenty six thousand four hundred eleven is in fact divisible by seven okay let's try another one we have six thousand four hundred eighty and essentially what we want to do here again is just chop off the final digit so I'm going to write this number 648 and then I take that final digit and I double it so what happens if I take 0 and I multiply it by 2. I get 0. so I can say 648 minus 0 if I wanted to but we all know that 648. so I continue I chop off the final digit of the number again and I'll have 64. and I'll take this over here and I'll say what is 8 times 2 right you're going to double 8 times 2 is 16. and we'll subtract that away 14 minus 6 is 8 5 minus 1 is 4. so I have 48 now and I can already tell that 48 is not divisible by 7. and I know that through my multiplication tables if I think about 48 divided by 7 equals what work backwards what times 7 equals 48. well I know that 6 times 7 is 42 7 times 7 is 49 that's too big so I don't have a whole number here that's going to work and so I'm going to have a remainder right so I know that 48 is not divisible by 7 and therefore this number 6480 is not going to be divisible by 7 as well okay so let's look at the rule for 11 now 11 is less complicated but it's still more complicated than the rules that we looked at earlier so the first thing you do is you find the sum of the digits in the odd places and subtract away the sum of the digits in the even places and I'll show you how to do that in the example then the next thing is that if the result is a zero or a number that is divisible by 11 then the original number is divisible by 11. so let's look at some examples determine if each number is divisible by 11. so what you'll do if you get this on a test just go through starting at the leftmost number and say okay this is position one then two then three then four then five so remember the difference between odd and even an odd number would be something like 1 3 5 7 9 so on and so forth an even number is a number that ends with zero two four six or eight so two four and then if we had a a six digit an eighth digit something like that we'd have even so I want to start by just forming the sum of the digits in the odd places so nine plus seven plus zero so nine plus seven plus zero then I'm going to subtract away the sum of the digits that are in the even places so we'd have 0 and then two so zero plus two is two zero plus two so nine plus seven plus zero is sixteen zero plus two is two sixteen minus 2 is 14. 14 is not divisible by 11. 14 divided by 11 is one with a remainder of three so this number ninety thousand seven hundred twenty is not divisible by 11. all right let's look at 361 152 and again we're going to start by labeling it so the leftmost digit of the number we're going to say that's in position one then two then three then four then five then six and what we're doing is we're going to sum the digits in the odd places so the digits that's in the position of one three and five so three plus one plus five three plus one plus five it's just a coincidence that the digits that's in the fifth position happens to be a five I know that can be confusing but that's a coincidence now we're going to subtract away the sum of the digits that are in the even places so in this position of two four and six so six is the number in the second position plus 1 which is the number in the fourth position plus two which is the number in the sixth position so again six plus one plus two so three plus one is four four plus five is nine subtracting away six plus one is seven seven plus two is nine as well so when we do this subtraction we get zero now remember we said that it would be divisible by 11 if the result was a zero or divisible by 11. here we got a result of zero so that tells me that this number 361 152 is in fact divisible by 11. hello and welcome to pre-algebra lesson 25. in this video we're going to learn about factoring whole numbers so the lesson objectives for today we want to learn how to determine if a whole number is prime composite or neither and additionally we want to learn to write a whole number as the product of prime factors all right before we get into factoring whole numbers let's think about the concept of a prime number so we have that a prime number is a whole number larger than one that is only divisible by itself and one so first let's think about the whole numbers again so we've talked about these a lot in this course so let me write out the whole numbers here and then basically I'm going to start with the smallest whole number which is 0 and then we increase in increments of one so next comes one then two then three then four and of course at some point we have to stop we can't list all the whole numbers so I'm going to put a final comma and then the three dots here to show that the pattern continues for up when we think about this definition a prime number is a whole number larger than one so automatically I've got to go into these numbers here and I've got to basically say okay I'm only looking at the whole numbers starting with 2 and going to the right or starting with two and anything that's going to be larger so it has to be divisible by only itself and one we know that every non-zero number is divisible by itself so a non-zero number divided by itself will always give you one so two divided by two is one no remainder we also know that every number is divisible by one if you divide a number by one you just get the number so for example 13 divided by by 1 is 13. so when I think about let's say the number two that's going to be a prime number because it's only divisible by itself 2 divided by 2 is 1 and 1. 2 divided by 1 is 2. in each case you get no remainder but there's nothing else you can really do with the number two so for that reason 2 is a prime number and we're going to find that 2 is the only even prime number so that's really important any even number larger than 2 is going to be divisible by 2 and therefore not a prime number because it would be divisible by something other than the number itself and one all right it might be extremely helpful to memorize the first few prime numbers so I have here the first 10 prime numbers we have two that's the smallest prime number and then we go to 3 5 7 11 13 17 19 23 and then 29 so that's the first ten there's an infinite number of prime numbers and how many you want to memorize is up to you by just doing the first 10 will greatly speed up up your work in many cases all right so how can we check to see if a number is prime so we're going to use the divisibility rules up to the square root of the number all right let's briefly touch on the concept of a square root again so something like the square root of 64 is equal to 8 because 8 times 8 is 64. when I ask for the square root of a number remember there's two different notations the first one is the principal or positive square root of in this case 64. I am asking for what positive number when multiplied by itself gives me this what we call radicand or number under the square root symbol since 8 times 8 is equal to 64. the square root of 64 is equal to 8. and again this is the principal or positive square root if you want the negative square root you need this notation here so this is the negative of the square root of 64 and that equals negative 8. so this is asking for what negative number when multiplied by itself gives me the radicand or number under the square root symbol so since negative 8 times negative eight is going to give us positive 64. the negative square root of 64 is going to be negative 8. so in this lesson I'm only going to ask for the principal square root so don't worry about this negative answer here so if I asked you for something like the square root of 4 you're just going to answer 2. if I said what is the negative square root of 4 which I'm not going to do in this lesson that would be negative 2. now the reason it's important to understand about square roots when we look at the factors of a number and of course this is just going to be with positive factors we would have something like for 64 1 times 64. then 2 times 32 then 4 times 16 then 8 times 8. now 8 is the square root and what's going to happen is after this you're going to have factors but they're going to repeat so in other words the next one will be 16 times 4. then you would have 32 times 2 and then you'd be back to 64 times 1. so this one right here and this one right here those are just reversed in terms of the order same thing for this one and this one and then this one and this one so you basically have duplicates after you hit the square root of the number now in most cases you're not going to get an integer answer when you take the square root of a number so we're going to talk about that here all right so what if I asked you is 103 a prime number so to figure this out we would need to use our divisibility rules up to the square root of 103 but you might stop for a minute and say well what is the square root of 103 I know that something like the square root of 100 is equal to 10 because 10 times 10 gives me 100 or here I've written that 10 squared is 100. then the square root of let's say 121 I know that's 11 because 11 times 11 is 121. when we think about a number like 100 or 121 it's called a perfect square so a perfect square is a number that can be formed as let's say an integer times itself to give you that number so here 100 we could form it as 10 times 10 or you could also do negative 10 times negative 10 but let's not worry about that for right now and for 121 we could do 11 times 11. so that's why it's a perfect square later on you'll hear about something called rational numbers and you will extend that definition to say that a perfect square is any number where you could basically take a rational number and multiply it by itself to get the number but in this case we're just dealing with integers so we're just going to go with a simpler definition now 103 is not a perfect square there's no integer that I can multiply by itself to get 103. so if I say what is the square root of 103 I'm kind of stuck but really what you would want to think about is 10 squared is 100 and 11 squared is 121 well okay this number right here is between these two perfect squares in other words writing this as an inequality I could say 103 is greater than 100 and it's less than 121. so if I take the square root of 103 it should be greater than the square root of 100 and a less than the square root of 121. let me actually get rid of this one right now so if I think about this I'm just going to drag this up here this would be that 10 which is the square root of 100 is going to be less than the square root of 130 whatever that ends up being Which is less than the square root of 121 which is 11. so just thinking about this I can say okay well this would be 10 and I know we haven't gotten to decimals yet 10 point something okay so it's going to be between 10 and 11. I don't really care what comes after that decimal point I only care that it's going to be between 10 and 11 and I'm just going to go with that lower integer value so I'm just going to go with 10. so I would check up to that if you're lucky and you can use a calculator and you punch this in you would get about and I'm going to round this to the nearest hundredth and we haven't talked about that yet but I'm just going to put 10.15 so we haven't talked about decimals we haven't talked about rounding decimals just trust me for right now that this is the answer so it's about about 10.15 you can see that that would be greater than 10 and less than 11 but you really don't need any knowledge of decimals you're just going to if you're using a calculator cut this part off and just take that whole number just take that 10 and you're going to check up to whatever that is okay so now that we've established that we're going to check up to 10 we can just run through the divisibility rules for this one it's going to be actually pretty quick so we know that this is an odd number so immediately you can Mark out 2 4 6 8 and 10. then when you think about divisibility by 3 you sum the digits so 1 plus 0 plus 3 1 plus 0 is 1 plus 3 is 4. so you think about is four a number that's divisible by three no four divided by 3 would give you one with a remainder of 1. so because there's a remainder there you can mark this up and then if a number is not divisible by 3 it can't be divisible by 9 so mark that out so now we're left with 5 which is pretty easy and then seven which is a little bit tricky so five you just look at the last last digit of the number is it a zero or is it a five in this case it's a three so five is out so cross that out and then we think about seven which is the most challenging we would think about cutting off the final digit of the number so in this case that's three so you just cut it off and then basically over here I'm going to double it so three times two and you're going to subtract that away so this would be what three times two is done first so that would be six so you get ten minus six which gives you four so if this number is zero or some number that's divisible by seven then basically the number you were working with is divisible by seven here this is a four so that's obviously not zero and it's not going to be divisible by seven so this will not be divisible by seven so let's go ahead and say that this number 103 is a prime number now you can see how much work it is to actually go through and check and see if something is prime so in most cases your teacher is going to give you something from a list that you've already memorized but in case you get something that's not you can go through and use this strategy to determine if something is prime all right now let's think about a composite number so a composite number is a whole number that is greater than one and is divisible by a whole number other than itself and one so when we say it's a whole number that is greater than one against the same numbers that we're working with we write out the whole numbers and then basically we start with two and then we move to the right on that list or you say anything that's two or larger that's what you're thinking about so it's got to be divisible by a whole number other than itself and one so here are the first 10 composite numbers so when you look at four six eight nine ten twelve fourteen fifteen sixteen and eighteen notice that 4 6 8 10 12 14 16 and 18 those are even numbers again they are automatically going to be divisible by 2 because they're even numbers so they can't be prime numbers they're automatically going to be composite numbers because they are divisible by some whole number other than itself and one so now looking at the other ones we have this 9 here 9 is divisible by three you have 15 15 is divisible by 3 and also five and that's pretty much it for the first 10 composite numbers obviously there's an infinite number of composite numbers just like there was an infinite number of prime numbers you can't memorize all of them so you just have to go through the strategies to determine if you have something that's prime or composite all right now on your test you might be asked about zero and one remember in the definition for a prime number and also in the definition for a composite number we specifically said it was a whole number larger than one so basically we're starting with two and we're looking for anything larger so what about zero and one the numbers 0 and 1 are not time so they're not Prime and they're also not composite if you get that question on a test they are neither okay you want to put that down they are neither they're not Prime and they're not composite all right let's run through a few of these so we want to determine if each number is prime composite or neither we're going to begin with 123 and coming down here you see that I have some numbers listed here so starting with 2 and then going up to 11. so why did I stop with 11. remember we only need to check up to the square root of the number now in this particular case 123 is not what we call a perfect square there's no rational number or some books will say integer that multiplies by itself to give you that number so when I have this situation I'm going to think about the fact that 11 squared is going to be 121 and then 12 squared is going to be 144. so these are things that I know if you don't you can do a multiplication so if you're not using a calculator you're just going to think about the fact that okay well then the square root of 121 is 11 and then the square root of 144 is going to be 12. well this number 123 is going to be between those two numbers in other words 123 is greater than 121 and a less than 144 so then the square root is going to be somewhere between 11 and 12. I don't actually care where it is because I'm just going to go down to 11 right so I only have to check up to 11. if your teacher is nice then it's a lot quicker you can just say what is the square root of 123 in your calculator and we're going to approximate this and I get if I round to the nearest hundredth about 11.09 okay so about 11.09 and again just cut this part off just get rid of this and just take the whole number part and basically I'm going to check up to 11. that's all you need to do so looking at this I know that this is an odd number so immediately it's not going to be divisible by 2 or 4 or 6 or 8 or 10. okay get rid of that right away then when I think about 3 well this number you would sum the digits one plus two is three three plus three is six well yes 6 divided by 3 is 2 no remainder so yeah immediately I know this is a composite number so this is a composite number what about the number 51. again if I'm thinking about the square root of 51 if I'm lucky and I have a calculator well I would say that's about 7.14 if I round to the nearest hundredth so I only need to check up to seven again if you don't have a calculator then just think about the fact that 7 squared is 49 and then 8 squared is 64. so if you think about this get rid of this for a moment we know that 51 is greater than 49 and less than 64. so that means when I take the square root of 51 I'm going to get something that's between 7 and 8. I don't actually care what it is I know that I'm just going to go down to seven so checking this I need to look at two three four five six and then 7. so just going up to seven this number is not even so you can immediately get rid of two four and six and then just think about is it divisible by 3 5 plus 1 is 6 6 divided by 3 is 2 no remainder so yeah this is going to be a composite number all right what about 59 so again I can take the square root of 59 in my calculator if I'm lucky and that would give me about 7.68 if I round to the nearest hundredth or if I don't have a calculator I can think about the fact that 7 squared again is 49 and then 8 squared is 64. so the square root of 59 because again 59 is greater than 49 and less than 64. it's going to be between 7 and 8. again I don't care about this 0.68 part I only care about the whole number so I only care about that 7 just going down so I want that 7 there so I only want to check up to 7. this is not an even number so I can get rid of 2 4 and 6. think about 3. 5 plus 9 that's going to be 14 so you can get rid of 3. 14 is not divisible by 3. now for five again the final digit has to end with a 0 or a five this does not end with a zero if it did it would be an even number and it does not end with a 5. so let me get rid of that so now we have to check seven for that one we know the rule is a bit complicated it might actually be quicker to just to a division but what you do is actually cut off the final digit of the number and you double it so 9 times 2 and you subtract that away so basically 9 times 2 is 18. so this will give me 5 minus eighteen and this will give me negative 13. and forget about the fact that's negative just say it's 13 divisible by 7 the answer is no so you can cross this out and then we can conclude that 59 is going to be a prime number so it's only divisible by itself so 59 divided by 59 is 1 and then also 159 divided by 1 is 59. all right now let's actually move into factoring whole numbers so I'm going to start off with a basic definition that of course we already know but I want you to recall that two or more numbers that are multiplied together are known as factors so in this multiplication problem 3 times 5 equals 15 the 3 and the five are known as factors they're being multiplied together the 15 the result of the multiplication operation is the product then 8 times 7 equals 56 the 8 and the 7 are the factors the 56 is the product so when we have factors that are prime numbers they are known as prime factors now a lot of times you're going to be asked to find the prime factorization of a number and that just means you're going to break the number down into the product of prime factors so I have here that it is often useful to write a number as the product of prime factors we're going to have a lot of operations moving forward that are going to depend on this knowledge so I have here to write a number as the product of prime factors we continue to factor each factor until it's Prime so I'm going to start off with this 80 here and then I'm going to show you how to do this with a factor tree which is a great way to visually organize your work so here I have let's try to write 80 as the product of prime factors so I'm just going to start off with 80 is equal to just pick any two numbers other than itself in one so I'm not going to pick 80 and 1. I'm going to pick something like let's say 8 and 10 so 8 times 10 and then just keep breaking these numbers down so eight I know that's four times two two is a prime number and then times for 10 I'm going to put 5 times 2. so 5 is Prime and 2 is prime but I can break 4 down and say that's 2 times 2 so then times two and then times let me reverse the order here so put 2 there and then times 5. so what do I have I have one two three four factors of two so let's say this is 2 to the fourth power and then times one factor of 5. so 2 to the fourth power is 16 and 16 times 5 is 80. right so we've basically broken this down into the product of prime factors and again you're going to need to do this a lot moving forward so this is the process you could use using a factor tree is generally a little bit better because it helps you to organize your work this right here is kind of a mess and not organized so in order to use a factor tree you want to write the number to be factored and extend two lines below so these lines that you're extending below they're going to basically be called branches then you want to list any two factors of the number so then other than the number itself in one and you're going to place each under one of the lines or what we call the branches then we're going to circle any prime factors and for any factors that are not prime we just repeat the process so you're going to extend two branches list any two factors other than the number itself in one and then you're going to circle the prime factor so I have here this continues until all factors are prime okay let's look at some examples we will look at 80 again using a factor tree so let me write 80 here and then these are going to be my branches so you can do anything other than the number itself in one so before we did 8 and 10 let's change that up and do 16 and 5. so now once you've listed two factors check to see if one of them is prime 5 is prime so you're going to circle it you're going to stop for 16 it's not prime I could say this is 4 times 4 and then you're going to check neither of those is Prime and so this is 2 times 2 and then this is 2 times 2. and so you're going to circle these so this this this and this those are all prime numbers so this gives me that 80 is equal to 1 2 3 4 so 2 to the fourth power times five now let me do this a different way just to show you that you get the same result let's say I didn't start with 16 and 5. I could have done 8 and 10. so let's say I do eight and ten and so this neither of those is prime so I'm going to go five and two five and two are both Prime so I'm going to circle and stop and then for eight let's go with four and two and then 2 is prime so Circle that and stop and then for four it's two times two so Circle those and stop so basically you get 2 times 2 times 2 times 2 which is 2 to the fourth power and then times five so same thing either way let me just do it one other way just to show you that it doesn't really matter so I'm going to go 4 and 20 okay so 4 is 2 times 2 and we're going to stop right so this is Prime and this is prime 20 is 4 times 5. 5 is prime so we're going to stop then 4 is 2 times 2 Circle those and stop so no matter how you do this you end up with 2 times 2 times 2 times 2 which is 2 to the fourth power times five okay let's look at 95 so 95. so for this one you might think about okay how do I get started 95 is not something I really work with that often if you notice the number ends with a 5 you know it's divisible by 5. so I would come over here if you don't have a calculator and just do a quick long division 5 goes into 9 once 1 times 5 is 5. subtract 9 minus five is four bring down this five here five goes into 45 9 times 9 times 5 is 45 subtract and you get zero so I could do 19 and then 5. so let me get rid of this and we know 5 is a prime number what about 19 well I did list this as one of the first 10 prime numbers if you didn't know that again you can go through the process we talked about earlier in the lesson and you could check and confirm that this is a prime number so this right here 95 would be equal to 19 times 5. what about 240 240. so if a number ends with a zero it's divisible by 10 usually that's the easiest to start with so I'll just say this would be 24 times 10. so then from here I know that neither of these is going to be prime so this right here would be 5 times 2 Circle these five and two are both Prime for 24 a lot of different things you could do you could do four and six you could do eight and three you could do twelve and two whatever you feel like let's just go with four and six so neither of those are going to be prime so 6 I'm going to do three and two and this is going to be 2 and 2. let me Circle this and this and this and this all of those are prime so 240 would be equal to one two three four factors of two so two to the fourth power and then times you've got one factor of three and then times you've got one factor of five now let's say just for the sake of completeness I get rid of this and I want to start with let's say 20 times let's say 12. I'm just showing you it doesn't matter how you start so 20 and 12 neither one is prime this one you could do five times four five is prime four is two times two two is Prime and two is prime 12 you could do four times three so three is Prime 4 is 2 times 2 2 and again you get one two three four factors of two one factor of three one factor of five so it doesn't matter what you start out with just pick any two factors other than the number itself in one and you're going to be good to go just keep going down and breaking each number down this way all right what about 126 so this number it's an even number so I know it's divisible by two and then one plus two is three plus six is nine so that tells me it's divisible by three and also nine but because it's divisible by both two and three it's going to be divisible by six let me do that off to the side so 126 divided by six six does not go into one but it will go into 12 twice two times six is twelve subtract you get zero bring down the six six goes into six once one times six is six subtract then we get zero so basically you could do 21 times 6. so this would be 21 and 6. now 21 and 6 neither of those is going to be prime 6 is going to be 3 times 2. let's Circle these 3 and 2 are both prime numbers 21 is 7 times 3. let's Circle those 7 and 3 are both prime numbers so let's say 126 would be equal to let's go one factor of 2 and then times we have two factors of 3. so let's say 3 squared and then times one factor of 7. all right what about 828 so it's kind of a big number I know it's an even number but I would look at the last two digits here this is a 28 so this number is going to be divisible by 4. and I would also check and see if it's divisible by three eight plus two is ten ten plus eight is eighteen so it will be divisible by 3 and also 9. now if a number is divisible by both 3 and 4 it's going to be divisible by 12. so I'm just going to start with that so 828 divided by 12 and you could do something else it's really up to you so 12 goes into 82 how many times well I know that 6 times 12 is 72 so let's put a 6 here and 6 times 12 again is 72. let's subtract 2 minus two is zero eight minus seven is one so this is ten bring down the eight so 12 goes into 108 how many times well from the multiplication tables that would be nine because nine times twelve is a hundred eight so we'll subtract here and get zero so I could go 69 times 12. 69 times 12. and again you could have started with something else it's really up to you so for this one I'm gonna go four and three three is prime so let's Circle that and stop 4 is 2 times 2 those are prime over here for 69 let's think about that for a moment so it's not an even number and 6 Plus 9 is 15 so this is divisible by 3. so let me start with that so 69 divided by 3 3 goes into 6 twice 2 times 3 is 6. subtracting you get zero bring down the nine three goes into nine three times three times three is going to be 9 subtracting you get 0. so this would be 23 times three let me get rid of this so hopefully you'll remember that 23 is a prime number that was in the list of the first 10 prime numbers and three of course is a prime number again if you don't know that 23 is a prime number you can quickly find that out using your divisibility rules going up to the square root of the number so let me write out that 828 is going to be equal to we have one two factors of two so two squared let me slide this down just a little bit here so it fits and then times and then you have two factors of three so three squared and then times you're going to have this 23. let's do one more and this is a really big number usually you get numbers that are at most three digits but I'll do one with four digits just so you get some practice so 4320. again this ends with a zero so it's not that bad I would just start by saying okay well this is going to be 432 times 10. and then for this one this is 5 times 2 Circle these you're done over there so now you got to think about 432. so I know immediately looking at the final two digits that this guy right here since 32 is divisible by 4 432 will be divisible by 4. so you could start with that but then again if you look at four plus three that's going to be seven seven plus two is nine so this number is also divisible by 3 and 9. okay so if it's divisible by four and three you can go with 12. there's other things you can do but I'm just going to go with 12. so let's say 432 divided by 12. so 12 will not go into four it will go into 43. we're going to do that three times three times twelve is 36 let's subtract I'm going to borrow here and say this is a 3 and this is a 1. 13 minus 6 is 7 and then 3 minus 3 is 0. so we're going to bring down this 2 here so 12 goes into 72 exactly 6 times 6 times 12 is 72 subtracting you get zero so this right here is going to be 36 times 12. all right for 36 we could do let's say six times six you could also do four and nine whatever you want to do and then for six we could go three times two and then three times two so we'll Circle these these are all prime numbers then for 12 let's go four and three I'll Circle three and then for four four is gonna be hard to fit so let me Branch down here like this this is going to be 2 times 2 so let me Circle those okay this one's pretty big let me actually slide this whole thing over to the left so I can fit it all right let me write this four thousand three hundred twenty I'm just going to highlight my twos so one two three four five factors of two so this is equal to two to the fifth power then times so now let me go through with a different highlighter color and I'm going to look for my threes so I got one two three of those guys just so I don't miss anything so this is three cubed and then times you've got that one factor of five let me highlight that four consistency here so I'm just going to put times five and you can verify that if you have a calculator but basically two to the fifth power is 32 so this is 32 and then times three cubed would be 27 and then times five so if you want to point that into a calculator real quick you'll see that you do get 4 320 or of course you could do a vertical multiplication with these two right quick and then do another one with this one and verify it the Long Way hello and welcome to pre-algebra lesson 26. in this video we're gonna have an introduction to fractions so for our lesson objectives for today we want to gain a basic understanding of fractions we also want to learn the difference between a proper fraction improper fraction and mixed number so up to this point in our pre-algebra course we've really only thought about the whole numbers and the integers but what happens when we have an amount that is larger than zero but less than one something that's a part of a whole amount well when we come across this scenario we can turn to fractions for help and I know some of you watching this video already have extensive experience with fractions While others have no experience whatsoever no matter what all of us have worked with fractions in our everyday life even if we haven't realized it so a typical scenario would involve you splitting up a pizza with your friends so as an example let's take a look at this pepperoni and cheese pizza that we have on the screen so the first thing I want you to notice is that the pizza is cut up into four equal parts right we have one two three four equal parts now when we work with fractions we're going to have some whole amount that's cut up into equal parts so our whole amount would just be the pizza so one pizza is our whole amount it's cut up into four equal parts so that means if I have four parts out of four parts I have one pizza now let's suppose that three of your buddies come over to the house and there's a total of four of you so let's put that there's four people now in the interest of fairness we're going to make sure that each person gets the same amount of pizza so everybody's going to get one slice out of a total of four equal slices so each person would get one fourth of the pizza now you might say where did that one-fourth come from well the one is called the numerator this is the numerator it's the top part of a fraction and it tells us the number of Parts being discussed or used since we're talking about how much each person is going to get the numerator is a one because each person is getting one part or in terms of the pizza one slice now the bottom number here is the denominator this is the denominator this tells me how many equal parts my whole amount is split up into my one pizza my whole amount has been split up into four equal parts so my denominator is a four now this middle part is called a fraction bar called a fraction bar and later on we're going to learn that the fraction bar represents division we actually have the numerator divided by the denominator but we don't need to think about that for now just think of the fraction bar as a way to separate your numerator or your top part from your denominator so in this scenario where there's a total of four people and we have four equal slices for our pizza each person gets one slice out of four or one-fourth of that whole pizza so as we just saw fractions are generally used to describe how many equal pieces okay how many equal pieces we have of a whole amount now in the previous example we talked about how many equal pieces each person got right each person got 1 4 of a whole pizza so let's take a look at this cookie here so again the cookie represents our whole amount so one cookie is what we have and again you'll notice that it's split up into four equal parts four equal parts three of the parts you'll notice have chocolate chips that's what these black circles are and then one part has macadamia nuts okay that's what the white circles are what if I was to ask you how much of this cookie has chocolate chips well again three parts out of the four have chocolate chips so our numerator is going to be a three okay our numerator is going to be a three again the numerator is the number of Parts being used or discussed in this case we're discussing how much of the cookie has chocolate chips that answers one two three so we have three there so that's our numerator so let me just label this this is the numerator we have our fraction bar to separate the numerator from the denominator and the bottom number of the denominator is a 4 because we have four equal pieces in our whole amount and this is the denominator what if I asked you how much of the cookie has macadamia nuts well that would be one-fourth right we have one piece or one part out of a total of four equal parts so the numerator is going to change now we're discussing macadamia nuts so that's going to be a one the denominator will not because I still have four equal parts in this cookie so now we're talking about the fraction one-fourth as another example let's take a look at this apple that's cut into two equal pieces so we have one piece and then a second piece so let's suppose it's me and you and we're in a room and we both want the Apple so we decide in the interest of fairness that we'll cut it down the middle just like we have each person has an equal amount how much will each person get well each person gets one piece out of a total of two equal pieces right the numerator is a one the denominator is a two all right the numerator again tells me the number of Parts being used or discussed and again we're discussing how much each person is going to get each person is going to get one part so that's why the numerator is one the denominator is 2 because there's two equal parts in the whole amount right in the Apple we've cut it up into two equal parts so the denominator is two each person is going to get one half of the Apple now let's suppose as we're about to bite into this apple six more people show up and they say hey we each want an equal amount so now we've got to cut up the Apple into eight equal pieces eight equal pieces and each person asks you how much am I going to get well they're going to get one piece out of a total of eight equal pieces or the fraction 1 8 right so one again is the numerator and eight is the denominator so again each person would get 1 8 or one piece out of a total of eight equal pieces for this apple let's say that you're watching a video on the web and this video has three equal parts so let's say this is the first part this is part one and then this is part two and this is part three okay and I know this might not be perfectly equal how I drew it but let's just pretend that it's three equal parts now the part in red shows you what you've already watched so what if I ask you the question how much of this video have you watched so far well if we look at the red progress bar we see that that takes up one part out of a total of three equal parts for the video so you've watched one part again out of a total of three equal parts that's the fraction one-third again where one is your numerator this is the numerator and 3 is the denominator now what if I ask you the question how much of the video was left well I have this part here and then this part here so unwatched on watched we would have two parts out of a total of three equal parts so that's two-thirds so two is the numerator in this case and 3 is the denominator again in each case the video was split up into three equal parts so that's where our denominator came from in the watched and the unwatched for the watched we had seen one part out of the three so that's how we got a numerator of one for the unwatched we have not watched two parts out of the three so that's how we got a numerator of two all right so I think we have the basics down on how to come up with the numerator and denominator let's just do a quick practice exercise to see if you can identify the numerator and denominator in each and we're going to start out with one-fifth so the top number is the numerator and I'm going to write it out for the first one and then I'm going to abbreviate after that and the bottom number of the 5 is the denominator so for the fraction one-fifth you have one as the numerator 5 is the denominator so this means I have one part out of a total of five equal parts next we'll look at the fraction two-ninths the 2 is the numerator so I'm just going to put a capital N the 9 is the denominator and put a capital D so I have two parts out of a total of nine equal parts next we have the fraction one-fourth one is the numerator that's the top number four is the denominator that's the bottom number so for this scenario I have one part out of a total of four equal parts then for 3 8 3 is going to be our numerator and 8 is going to be our denominator again if I have 3 8 of something I have three parts out of a total of eight equal parts okay we'll look at two more 5 7 5 is the numerator 7 is the denominator again the numerator is in the top the denominator is in the bottom and if I have 5 7 of something I have five parts out of a total of seven equal parts then for the last one I have fifteen seventeenths so 15 the number on top is the numerator 17 the number on the bottom is the denominator if I have 15 17 of something then I have 15 Parts out of a total of 17 equal parts so there's kind of more diffractions than just what we covered so far once you start getting higher in math you're going to replace your division symbol with a fraction bar and I kind of talked about this in the beginning when I said the fraction bar represents division so I want to just quickly cover some important division rules that we need to remember when we're working with fractions so the first one is going to be that any non-zero number that is divided by itself is going to result in one so if I have something like 4 divided by 4 I can now write this using a fraction as four over four or four fourths and this is going to be equal to one and think about the scenario we had with a pizza let me just go all the way back up real quick so kind of looking at this I see that I've split up my pizza into four equal slices so if I consume four fourths of the pizza or I eat four pieces out of a total of four equal pieces then I've consumed one whole pizza the next rule to remember is that zero divided by any non-zero number is always zero so 0 divided by let's say 3 equals 0. and now we can write this using a fraction as 0 over 3 equals zero and just think about having something that's cut up into equal parts of three and you get zero parts so how much of it did you get you got zero right if I get zero out of a total of three equal parts I'm still getting zero so fractions represent division of the numerator by the denominator we can write a division problem using fractions so again if we see something like let's say eight divided by four this is traditionally how we write it in elementary school now we can write it using fractions as eight divided by 4 like that the numerator divided by the denominator here we have eight divided by 4 8 the leftmost number the dividend goes up on top as the numerator for your divisor goes on the bottom as your denominator so 8 divided by 4 we know that's two or something like 20 I don't know divided by five we could write this as twenty over five right it's just the numerator divided by the denominator so the leftmost number your dividend goes on top as the numerator your rightmost number your divisor goes on the bottom as your denominator so 20 divided by 5 is 4. and we can do one more let's say we had I don't know 100 divided by 25. so we could write this as a fraction we take 100 that's going to be our numerator and then 25 is going to be our denominator right numerator divided by denominator 100 divided by 25 is the same as 100 over 25 and this is going to be equal to 4. okay so let's kind of wrap up our lesson by just talking about the differences between a proper fraction an improper fraction and a mixed number so a proper fraction is used to describe a quantity whose value is less than one so the easy way to tell if you have a proper fraction is just to look at the numerator and see if it's less than the denominator if it is you have a proper fraction so something like two-fifths or 5 7 or 2 11 or 1 9 in each case the numerator is smaller than the denominator if I got something like 15 7 that's not a proper fraction the numerator is larger than the denominator this is an improper fraction now before we go any further let me just try to warn you that your teacher might try to trip you up with some negative fractions let's say you get something like negative two-thirds negative two-thirds if you get a negative fraction ignore the negative sign and just pretend the fraction is positive so I'm just going to think about this as positive two-thirds and just look at the two and the three the numerator is smaller than the denominator so it's a proper fraction I can just put my negative sign back and say Okay negative two-thirds is a proper fraction if I had something like negative 11 9 ignore the negative sign pretend it's 11 9. and look at the 11 it's bigger than 9 so it's not a proper fraction okay it's not a proper fraction so that means negative 11 9 is not a proper fraction either so kind of next up an improper fraction is used to describe a quantity equal to or larger than one so kind of the easy way to think about this is that if your numerator is the same as or larger than your denominator you have an improper fraction so like eight over eight the numerator is the same as the denominator so this is improper right so equal to or larger than 1. if you had something like 13 over 11 equal 2 or larger than one again if you get negative numbers involved just think about them as if they were positive right take the absolute value before you think about the definition so if I had something like negative 15 over 14 . let's say I try to go by this definition and improper fraction is used to describe a quantity equal to or larger than one well any negative number is less than one so if you try to apply this definition using this as a negative you get the wrong answer so that's what I'm saying just ignore the negative sign just look and see if the numerator is bigger than the denominator 15 is larger than 14 and so you would know this is an improper fraction right negative 15 14 is improper okay the last thing we're going to learn about is called a mixed number so a mixed number is kind of a fancy way to write an improper fraction right you can go back and forth and we'll see how to do that in the next lesson so basically what it is is the sum of a whole number and a proper fraction so something like 3 and 1 4. and what this is is it's three plus one-fourth we just write it like this for convenience right and a lot of students get confused and think that's multiplication because they're sitting next to each other well no it's three plus one-fourth or something like 6 and 1 8 right this is 6 plus 1 8. but it's very important that you have a whole number and then a proper fraction it can't be a whole number an improper fraction okay so let's look at one final quick exercise we just want to determine whether each is a proper fraction improper fraction or mixed number so we're going to take a look at seven fifths so the numerator is larger than the denominator this is improper again that's all you need to look at 3 11 the numerator is smaller than the denominator so this is proper for four-fifths the numerator's smaller than the denominator so this is proper for 9 and 1 8 it's easy to see that this is a mixed number right it's a mixed number and I don't think anybody gets confused between mixed numbers and proper fractions and improper fractions they mostly get confused on the definition between a proper fraction and an improper fraction for a mixed number you have a whole number just hanging out to the left so it's pretty obvious that it's a mixed number for 10 7 the 10 the numerator is larger than the seven so this is an improper fraction for six sevenths the sixth the numerator is smaller than the denominator seven so this is proper hello and welcome to pre-algebra lesson 27. in this video we're going to talk more about mixed numbers and improper fractions so for our lesson objective we want to learn how to change a mixed number into an improper fraction and vice versa meaning we also want to learn how to take an improper fraction and change that into a mixed number all right so we're going to begin by learning how to take an improper fraction and change that into a mixed number so this is very very simple just write down the steps use them as your practice and just like anything else over time you can just kind of throw your paper away because you will memorize this so the first step is just to divide the numerator by the denominator so if I have something like let's say 10 thirds okay let's say 10 thirds I would take 10 and I would just divide it by three just like that so 3 goes into ten three times three times three is nine subtract and get one so this division gives me 3 with a remainder of 1. so let's kind of copy this we're going to bring it down to the next step okay so I have here that the quotient is the whole number for the mixed number so this 3 here which remember that's our quotient that's going to be the whole number for the mixed number so the 3 will be your whole number part then the remainder which in this case that's that one there is the numerator of the fraction part so I'm going to put a 1 there now where do I get my denominator for the fraction part well it's the original denominator from the improper fraction so that doesn't change that's so your denominator here for the fraction parts and remember that was a three three and one-third is the same thing as ten thirds okay three and one-third is the same thing as ten thirds all right let's take a look at some examples we're gonna start out with eighteen over seven or eighteen sevenths so again my first step is just to do a basic division so I'm going to take 18 and I'm going to divide by 7. so 7 goes into 18 twice 2 times 7 is 14. subtract and you get four so the answer here is 2 with a remainder of 4. so again to convert this into a mixed number the quotient part which is 2 is going to be the whole number part the remainder which is 4 is going to be the numerator for the fraction part and then your denominator for the fraction part is the denominator from the improper fraction so that's going to be 7 or you could just say okay it's my divisor from when I did my division right 18 divided by 7. however you want to remember that so 18 7 is the same as or is equal to 2 and 4 7. all right let's take a look at another one we have 19 thirds so again we just set up a basic division 19 divided by 3. so 19 divided by 3 and 3 is going to go into 19 six times six times three is eighteen subtract and we'll get 1. so that's our remainder so we get 6 with a remainder of 1. so again the 6 the quotient part is your whole number the one which is your remainder is going to be the numerator for your fraction parts and then the denominator for the mixed number is going to be three right it's the same denominator as from the improper fraction so 19 thirds is the same as organ is equal to six and one third what about eleven sevenths so again we're going to divide we have 11 and we're dividing by seven seven goes into 11 once one times seven is seven subtract then we get four so the remainder here is four so one with a remainder of four and let me kind of move this down so I can write my answer move that over there so this is equal to again the whole number part will be one right that's your quotient the numerator for the fraction part will be 4 that comes from my remainder and then the denominator for the fraction part is the denominator from the improper fraction so that's a 7. so 11 7 is equal to one and four sevenths all right what about 31 over 15. so again we set up a long division we have 31 divided by 15. so 15 goes into 31 twice 2 times 15 is 30. subtract then we get one so this is 2 with a remainder of 1. so again the 2 the quotient part is your whole number one the remainder is your numerator for the fraction part and then 15 which is the denominator from the improper fraction is going to be the denominator for the fraction part so 31 15 is equal to 2 and 1 15. all right let's take a look at one more so we have 59 8 or 59 over 8. so again divide 59 by 8 eight goes into 59 seven times seven times eight is fifty-six subtract and get three so this would be 7 with a remainder of 3. so this is going to be equal to again the whole number part is going to be 7. you get that from your quotient the numerator for the fraction part is going to be 3. again that comes from your remainder and then the denominator for the fraction part is eight that comes from your improper fraction so 59 8 is equal to 7 and 3 8. so really that was pretty easy overall your general strategy for those type of problems again just write down your steps use the steps as you practice and then after you get enough practice you won't need to look at those steps anymore you will have memorized them right so these things will become automatic for you now we're going to talk about changing from a mixed number to an improper fraction and this isn't going to be any more difficult so the first step is to multiply the denominator of the fraction part by the whole number part so if we had something like let's say three and one eighth I would take the denominator of the fraction part which in this case that's this 8 here and I would multiply it by the whole number part so I would have 3 times 8 to start so 3 times 8. and let me copy this we're going to bring this down to the next step then the next thing we want to do is add the result to the numerator of the fraction part so I have this 3 times 8 here that's to be done first when that's done I'm going to add the numerator of the fraction part so that's one so I can put plus 1 here if I want and because of the order of operations I would do 3 times 8 first and then I would add 1 to that result so I really don't need to do anything to clarify that if I just follow the order of operations but if you want to to make it completely crystal clear you can put parentheses around 3 times 8 just to say hey I absolutely must do three times eight first before I add 1 to that the result of this is the numerator for the improper fraction so 3 times 8 is 24 and if I add 1 to that I get 25. all I'm missing now is my denominator and the denominator is going to be the same as the denominator from the fraction part for the mixed number so that's going to be an 8. so we have 25 8 here now what I want to show you is we know how to go from 25 Ace to 3 and 1 8. again we set up a division so 25 divided by 8 8 goes into 25 three times three times eight is 24. subtract and get one so the remainder is 1. so if I convert between this and a mixed number I would get a 3 which is what we have here for my whole number part my remainder is a one so that's the numerator for the fraction and my divisor is an 8 or my denominator for the fraction is an 8 so that's going to be my denominator for the fraction part of the mixed number so 25 8 is the same as or is equal to 3 and 1 8. all right so we're going to begin with eight and one-third and really very simple overall again if you wrote down the steps step one is to multiply the denominator of the fraction part by the whole number part so I'm going to take 3 which again is the denominator of the fraction part multiply that by 8 which is the whole number part so I would start out with eight times three eight times three now once I've done that it tells me to add the result to the numerator of the fraction part and recall that I put parentheses around 8 times 3 to make it crystal clear that we multiply that before we do any addition but again you don't need to do that because of the award of operations if I come over here and I put plus one okay because I'm adding the numerator from the fraction part next whether I have parentheses around 8 times 3 or not it doesn't matter right the order of operations tells me hey multiply before I add but for this lesson I'm going to put parentheses around it just so we're clear we're doing 8 times 3 first before we add that result to 1. so 8 times 3 is 24 and 24 plus 1 is 25. and now where do I get my denominator from again the denominator is the same denominator from the fraction part of that mixed number so this is going to be over 3 over 3. so 8 and 1 3 is equal to 25 thirds okay let's take a look at nine and four sevenths so again the first step is to multiply the denominator of the fraction part by the whole number part so this right here this 7 is getting multiplied by 9. so what is nine times seven I'm going to put that in parentheses and then when we're done with that we add the result to the numerator of the fraction part so the numerator for the fraction part is four so I'm going to say plus 4 when I'm done and my denominator is going to be the same as the denominator from the fraction part of the mixed number so that's going to be a 7. so 9 times 7 is 63 and then if I add 4 to that I'm going to get 67. so this would be 67 over 7 or I can say 67 7. what about 3 and 5 8 well again we want to start out by multiplying the denominator of the fraction part by the whole number part so I'm going to take a to multiply it by 3. so 8 times 3 then when I'm done I'm going to add to that the numerator of the fraction part so in this case that's this 5 here and this is going to be over my denominator from the fraction part of the mixed number is an 8. so that's going to stay the same right that's going to be my denominator for the improper fraction as well so now I'll just crank this out 8 times 3 is 24 24 plus 5 is 29. so this is going to be 29 over 8 or again 29 8 and we could just say that 3 and 5 8 is equal to 29 8. what about 13 and 5 11. well again I'm going to take this part right here this 11 the denominator from the fraction part and I'm going to multiply by the whole number part so we would have 11 times 13. now once we've performed this operation we're going to add the result to the numerator of the fraction part so I'm going to add that to 5. so Plus 5. and this is going to be over again we use the denominator from the fraction part of the mixed number so this is going to be over 11. so what is 11 times 13 11 times 13. a lot of you already know that it's 143. let's go ahead and crank This Out 3 times 11 is 33. and then 1 times 11 is 11. so we're going to add bring down the three three plus one is four bring down the one so again 143 if I add 5 to that I'd have 148. so this would be equal to 148. again that denominator is going to be 11. so 13 and 5 11 is equal to 148 11. what about 107 9. well again we're going to start by taking the denominator from the fraction part multiplying it by the whole number part so you'd have 9 times 100 9 times 100 and then when we're done we're going to add to it the numerator from the fraction part so that's a 7. I'm going to add a 7 on there my denominator stays the same as the denominator from the fraction part of the mixed number so that's going to be a 9. now let's just crank this out what's 9 times a hundred well we have two trailing zeros here so I just need to do nine times one that's nine then attach two trailing zeros to the end so that would give me nine hundred then I would add seven to that so that would be nine hundred seven and then over nine so one hundred and seven ninths is the same as nine hundred seven ninths hello and welcome to pre-algebra lesson 28. in this video we're going to learn how to find the greatest common divisor which I've abbreviated as gcd so the lesson objective for today is just to learn how to find the greatest common divisor again abbreviated gcd for a group of numbers so before we kind of get into the lesson I want to clear something up that causes a lot of confusion for students as someone who makes a lot of video tutorials one thing I've noticed is that if I post a video on let's say finding the greatest common divisor like we have here students will watch that and they will post in the comments hey do you have anything on finding the greatest common factor and my reply will always be yeah you just watched it if someone's asking you to find the greatest common divisor for let's say two numbers it's the same thing as if they asked you to find the greatest common factor for those two numbers so if I said something like what is the greatest common divisor of let's say 12 and 30. it's the same thing as if I asked you what is the greatest common factor of 12 and 30. same exact thing and to kind of think about this for a second we all know that if we have 4 times 3 equals 12 that 4 and 3 are called factors they're called factors of 12. but really they're also divisors right so if I have 12 and I divide by 4 I get three there's no remainder there if I have 12 and I divide by 3 I get 4. there's no remainder there if a number is a factor of another it's also a divisor meaning if I take 12 and I divide it by any of its factors I'll get a result that doesn't have a remainder so when we ask for the greatest common divisor and we ask for the greatest common factor we're asking for the same thing so when working with two or more numbers the greatest common divisor again I abbreviate this gcd is the largest whole number that is a divisor of all numbers and I could have just as easily said when working with two or more numbers the GCF the greatest common factor is the largest whole number that is a factor of all numbers again we can interchange those now actually finding the greatest common divisor or the greatest common factor for a group of numbers is very simple go ahead and write down the steps and then use them as we practice and then again as you do more practice you won't need to look at the paper you will have committed the steps to your memory so for finding the greatest common divisor the first thing you want to do is find the prime factorization of each number we all know how to do that at this point we learned a few lessons back you can use a factor tree or some other method that you've learned if that's more comfortable for you the next thing we're going to do is create a list of prime factors that are common to all numbers and this part's important when you're creating that list make sure that the prime factors are common to all numbers if you're trying to find the greatest common divisor for let's say five numbers and you have a prime factor that's common to four of them but not common to the fifth number you can't use it it has to be common to all numbers then the last thing you're just going to take that list that you made and you're going to form the product of the numbers on the list that's going to give you your greatest common divisor right so the greatest common divisor is the product of the numbers on the list very very easy so we're going to jump in and look at some practice problems we want to find the greatest common divisor for each group of numbers we're going to start out with this gcd of 12 and 20. so what is the greatest common divisor or I could have just as easily said what is the greatest common factor of 12 and 20. the first thing we're going to do is find the prime factorization of each number I'm going to do that using a factor tree but again if you have another method you're more comfortable with use that so for 12 I'm going to start out with 4 times 3. again you can just start out with any two factors for 12 it doesn't matter what it is and then anytime I have a prime factor I just circle it and I stop so 3 is prime I Circle and I stop 4 is 2 times 2 2 is prime so I'm going to circle both and then now let's do 20. I'm going to start out with 5 times 4. again any two factors for 20 would work 5 is prime Circle that and stop 4 is 2 times 2 2 is prime so Circle both now that we have the prime factorization for each number and let's just write it out we have 12 is 2 times 2 times 3 and 20 is 2 times 2 times 5. we're looking to see what is common to both and I know a lot of people will make a table and we'll do that kind of when we get to some bigger numbers for right now we can kind of eyeball and see that we have one two that's in common and then a second two that's in common and nothing else that's in common so if we were to form a list that list would contain two twos right tutus so for my greatest common divisor remember I just formed the product of the numbers on the list so the greatest common divisor is going to be 2 times 2 which is going to be equal to 4. right and I could have just as easily written greatest common factor of 12 and 20. is 2 times 2 which is 4. again these are the exact same thing I want to embed that in your head so that you understand it all right let's take a look at the greatest common divisor of 36 and 54. so again start out by finding the prime factorization for each number for 36 it's 6 times 6 we could do so 6 is not prime so we're going to continue 6 is 2 times 3. and 2 times 3 over here and 2 and 3 are both Prime so we're going to circle all of these and we're done now for 54 I could think about 9 times 6 and neither number is prime so we're going to continue 9 is 3 times 3 6 is 3 times 2 and all of those are prime so we're going to circle all of these and we have our prime factorizations so 36 is 2 times 2 times 3 times 3. 2 times 2 times 3 times 3. 54 is 3 times 3 times 3 times 2. so I'm going to write this as 2 times I'm just going to put a space here because I don't have another two and I'm going to put times 3 times 3 times 3. and I'm basically just doing that so that we can glance at this and see what's common right so I can look right now and see that I have a 2 that's common of both I have a 3 that's common of both and another three that's common of both so if I was to make a little list it would contain one two one three and then a second three so the greatest common divisor of 36 and 54 is going to be equal to 2. times three times 3. so 2 times 3 is 6 6 times 3 is 18. so your greatest common divisor for 36 and 54 is going to be 18. all right let's take a look at one with three numbers involved so not any more difficult just a little bit more tedious the more numbers you put in there the more tedious it gets so we have the greatest common divisor of 25 80 and 90. so we're going to find the prime factorization for each number 25 is 5 times 5. and 5 is prime so we're going to circle both 80 I'm going to start out with 8 times 10 8 is 4 times 2. 4 is 2 times 2. now 2 is prime so I'm going to circle all of these guys and then 10 is 5 times 2. 5 and 2 are prime so I'm going to circle those guys and then what about 90 well I'm going to do 90 as 9 times 10 neither is prime so I continue 9 is 3 times 3. 10 is 5 times 2 and all of these are prime so I'm going to circle them so now I have my prime factorizations so 25 is 5 times 5. so I'm just going to write this like this 5 times 5 80. is 2 times 2 times 2 times 2. then times 5. and then 90 I got 2 times 3 times 3 times 5. so one thing we can do is we can write the numbers that are common on top of each other so it makes it completely clear and I kind of did that in the last example but as you get better at this you really don't need to do it particularly when you look at 25 you see that it's 5 times 5. so right away you know that when you go to the other numbers you're only looking for a five right because that's all that's involved in here and it has to be common to everything so I'm not looking at any of these twos and I'm not looking at any of these twos or threes here so really I would just highlight this five this five and this five I just have one five that's common in everything now there's a second factor of five and the prime factorization of 25 but it's not matched in the prime factorization of 80 or 90 right there's just one five in the prime factorization for 80 and 90. so we can only put one five on our list so it makes it really easy the greatest common divisor of 25 80 and 90 is going to be 5. I don't need to multiply anything here there's only one number that's in common to everything and again that's five all right for the last problem let's look at one that's a little challenging so we have the greatest common divisor of 198 165 and 132. so let's start by just factoring 198. so I know it's an even number so I know it's at minimum divisible by 2. now it's not going to be divisible by 4 because if I look at the final two digits here 98 that's not divisible by 4. so let's go ahead and just start out with dividing this guy by 2 and just seeing what we get 2 will go into 19 9 times 9 times 2 is 18. subtract you get one bring down the eight there two goes into eighteen nine times nine times two is eighteen subtract should get zero so 99 times 2 gives you 198. so 2 is a prime number so we can Circle this and stop 99 is not right away when I look at 99 I'm thinking is it divisible by 9. 9 plus 9 is 18 so yeah it's divisible by 9. so this is 9 times and you might know by now this is 9 times 11. but again in case you didn't you could just do a long division 99 divided by 9. now we should know at this point that 11 is a prime number so we'll Circle that and stop and 9 is 3 times 3. so 3 is prime we're going to circle these we have our prime factorization for 198 it's 2 times 3 times 3 times 11. let's write this on the next page so 198 is 2 times 3 times 3 times 11. okay let's do 165 now so just like with 198 I don't know two factors for this number off the top of my head but again using my divisibility rules I see it ends in a five so I know I can do 165 divided by 5 . 5 goes into 16 three times three times five is fifteen subtract and get one bring down the five five goes into fifteen three times three times five is fifteen subtract and get zero so 33 times 5 gives me 165. now 5 is a prime number so I can Circle this and stop 33 is not I know that 33 is divisible by 11. 33 divided by 11 is 3. so 33 would be 11 times 3 and 11 and 3 are both Prime so Circle both of those and so for 165 we get 3 times 5 times 11. okay for the last one we have 132 and I can see that this number is even right away so I know it's divisible by two the last two digits form the number 32 so that's divisible by four and one plus three is four four plus two is six six divided by three is two no remainder so it's divisible by four and three so it has to be divisible by 12. so 132 divided by 12 12 goes into 13 once 1 times 12 is 12. subtract and get 1 bring down the 2 12 goes into 12 once 1 times 12 is 12. subtracting gets 0. so 12 times 11 okay 12 times 11 would give me 132. again we know 11 is prime 12 we could write as 4 times 3 3 is prime 4 is 2 times 2. 2 is prime so we're going to circle both so 132 is 2 times 2 times 3 times 11. now we're going to form a little list and look for what's common so I have a 2 in the prime factorization of 198. I don't have it in 165 so it's not going to be common to everything although I have two factors of 2 here I can't use two in my list because again it's not common to everything what about three I have that in 198 I have it in 165 and I have it in 132. so I can put one factor of 3 in that list now I have a second factor of 3 here but I don't have a second factor of 3 and 165 so it can't go into the list now the last thing we look at in the prime factorization of 198 is 11. and that's going to be common to everything so there's an 11 here and here so really the only thing that's common to all three would be one factor of three and one factor of 11. so when I build my gcd my greatest common divisor for 198 165 and 132 it's going to be equal to again the product of the numbers on the list so 3 times 11 or 33. hello and welcome to pre-alterville lesson 29. in this video we're going to learn about simplifying fractions otherwise known as reducing a fraction to its lowest terms so the lesson objective for today is just to learn how to simplify fractions and again as I just said this is also referred to as reducing a fraction to its lowest terms so I just want to make you guys aware of a little notational change that's going to start in this video now for a lot of you who haven't taken algebra yet you've only used this symbol to imply multiplication right this is for multiplication so if I wanted something like 6 times 7 I write 6 then that symbol then seven and we know that equals 42. but kind of As you move higher in math this symbol is going to go away you're going to use some other method right and there's multiple methods you can use I know earlier in the course we talked about A number being next to parentheses implies multiplication you're going to have that but primarily what we're going to use is a DOT okay we're going to use a DOT so if I wanted 6 times 7 I could write it like this this symbol also means multiplication so 6 times 7 equals 42. or we could do 2 times 8 that equals 16. all right I'm just replacing this familiar symbol with this new DOT and that's all that we're going to change now you might ask yourself why do we go back and forth between these symbols well as you get higher in math you start working with variables and this this symbol here will get confused with the variable X which is very very common in algebra so we have to switch to something else so that it's Crystal Clear what we want you to do so before we kind of get into the math of simplifying fractions I want you to take a look at these two rectangles on the screen now we have part of each rectangle that's shaded green and part that's just White if I was to look at this top rectangle here it's cut up into two equal parts here's the first part here's the second part so if I was to ask you to represent the amount that's shaded green with a fraction you would come up with a fraction one-half all right so shaded you have one part that's going to be your numerator that's shaded out of two equal parts so that's the fraction one-half now when we look at the rectangle below we see that it split up into four equal parts so one two three four but I want you to observe that these two parts here take up the same space as this one part here these two parts here the parts that are shaded green take up the same space as this one part here so if I wanted to represent the Shaded amount for this bottom rectangle okay so what's shaded in green I now have two parts okay two parts so that's my numerator out of a total of four equal parts so four is my denominator so now we can visually see that one half is the same as or is equal to two-fourths but you might say how is that the case how is that mathematically possible to have the same value from two different numbers well let's just erase this two-fourths for a second and just start out with one half all of you know at this point that if I multiply a number by one the number remains unchanged so for example if I multiply 5 times 1 I get five If I multiply 277 times 1 I get 277. if I multiply one-half times one I get one half now all of us also know that any non-zero number divided by itself is one so if I had if I had 2 divided by 2 2 divided by 2. this is the same thing as the number one so let's say I multiply one-half times two over two I'm basically multiplying one half by one now I know we haven't multiplied fractions yet officially but essentially all we're going to do is multiply the numerator times the numerator and put that over the denominator times the denominator so I would do 1 times 2 that would give me 2 and I would multiply 2 times 2 and that would give me 4. so I took one half and I multiplied it basically by one it's a complicated form of one but two over two is one and I ended up with two fourths so that's how we can have the same value out of two different fractions and I could keep going I could have multiplied one-half by let's say five over five five over five is one one times five one times five is five two times five is ten so 5 10 is the same as one-half so equivalent fractions are fractions that have the same value and we just saw that with one half two fourths and five tenths those are all equivalent fractions because at the end of the day they have the same value so now that we understand how two or more fractions can look different but have the same value let's talk about reducing a fraction to its lowest terms so a fraction is considered simplified okay simplified or your teacher or your textbook might say reduced to its lowest terms when the numerator and denominator have no common factors other than one okay other than one so to reduce a fraction to its lowest terms or again to simplify a fraction you start out by factoring the numerator and denominator completely then you're going to cancel all common factors between the numerator and denominator all right so we're going to jump in and look at some practice problems now we want to reduce each fraction to its lowest terms okay we're going to begin with 15 over 27. so the first thing we want to do is Factor the numerator and denominator completely now you've probably gotten pretty good at factoring whole numbers at this point so you can use a factor tree if you need to but for some of the smaller numbers you probably have memorized the prime factorizations already like 15 you should know is 5 times 3. you shouldn't need a factor tree for that 27 you know it's three times three times three three times three times three if you need a factor tree pause the video and make one you know you have 9 times 3 3 is prime Circle that minus three times three Circle both of those so you see that you get 3 times 3 times 3. now after you've done that you're looking to cancel common factors between the numerator and denominator so in other words if I have a 3 in the numerator I can cancel that with one of the threes in the denominator essentially all I'm doing is I'm saying hey I have 3 over 3 and that's the same thing as 1. so when I go through I don't have this 3 anymore I just have a 5 in the numerator and when I go through here I don't have this 3 anymore so I just have a 3 and another 3 in the denominator and 3 times 3 is 9. so I'm basically saying that 15 27 or 15 over 27 is the same thing as 5 9. and again reverse this process so I kind of get a little insight as to what happened so you have 5 9 if you multiply it by three over three again three over three is the same thing as one so I'm not changing the value of the fraction five times three is fifteen nine times three is twenty-seven okay let's look at 18 over 60 . so again we want to factor the numerator and denominator completely so for 18 let's just do a factor tree let's start out with six times three so 3 is prime let's Circle that six is two times three two and three are both Prime let's Circle those so 18 is two times three times three two times three times three for sixty let's do ten times six is two times three those are both Prime so Circle two and circle three ten is five times two and those are both Prime as well so Circle five and circle two so we're going to have two times two two times two times three times five so what's common between numerator and denominator well I have one factor of 2 in the numerator that I can cancel with one factor of 2 in the denominator 2 over 2 is equal to one so we can just remove that now I have one factor of 3 in the numerator that I can cancel with one factor of 3 in the denominator and then that's all I have this 3 up here but I have no other three down here to cancel it with then I have this 2 down here but no other 2 up here to cancel it with right I've already canceled out this 2 I can't use it again and then finally I have this 5 here no 5 up here to cancel it with so our simplified version of this fraction I have a 3 in the numerator and then in the denominator I have 2 times 5 or 10. so 18 over 60 is equal to 3 over 10 or 3 10. let's take a look at 95 over 20. again we want to factor each number so 95 I know is 19 times 5. and those are both prime numbers so we're going to circle those so 19 times 5 20 I know is 4 times 5 and 4 is 2 times 2. so 2 times 2 times 5. now what's common between numerator and denominator I have a 19 in the numerator No 19 in the denominator I have a 5 in the numerator and I have a 5 in the denominator so I can cancel those common factors between numerator and denominator and then I have two twos in the denominator but I don't have anything in the numerator that matches that so my answer I'm going to have a 19 in the numerator and I'm going to have 2 times 2 or 4 in the denominator so I end up with 19 4 as my answer all right what about negative 26 over 156 . so we've never worked with a negative yet and basically when you see a negative when you factor the number just write a negative 1 out to the side and then just write the prime factorization for the number 26. so for 26 is 13 times 2 so I'm going to write negative 1 times 13 times 2 all right negative 1 times 13 is negative 13 negative 13 times 2 is negative 26. all right then we have 156. so let's make a factor tree for that one so 156 I know that it's divisible by two and it's also divisible by 4 because 56 is divisible by 4. so let's do that off to the side what is 156 divided by 4 4 goes into 15 three times three times four is twelve subtracting get three bring down the six four goes into 36 9 times 9 times 4 is 36 subtracting gets zero so this would be 4 times 39 4 is 2 times 2. and 39 is 13 times 3. so let's Circle all of these because they're all prime so we would have 2 times 2 2 times 2 times 13 times 3. now what can we cancel between numerator and denominator we can cancel this 13 in the numerator with this 13 in the denominator we can cancel this 2 in the numerator with one of the twos in the denominator doesn't matter which one just one of them now all I have left in my numerator is a negative 1. so that's going to be my numerator and in the denominator I have 2 times 3 or 6 so our simplified version is negative 1 6. all right let's look at negative 33 over 77 so again I have a negative value here just write negative 1 times and then do the prime factorization for 33 and that's just going to be 3 times 11. what about 77 well 77 is 7 times 11. so 7 times 11 and as time goes on you're going to recognize right away that the greatest common divisor of 33 and 77 is 11. so you're just going to kind of mentally divide each number by 11. so in other words if I had negative 33 over 77 once I'm good at this I'm not going to go through these steps I'm going to say okay 33 divided by 11 is 3 so this would be negative 3 over 77 divided by 11 is 7. so I end up with negative 3 7. so you're going to be able to mentally do this after some practice but for right now it's best not to skip any steps so we see that we have a common factor of 11 between the numerator and denominator and that's all we can cancel so we end up with negative 1 times 3 or negative 3 over 7. so you end up with negative 3 7 as your answer what about 64 over 288 so 64 I know is 2 to the sixth power so 2 times 2 times 2 times 2 times 2 times 2. again if you didn't know that you could do a factor tree right 64 is 8 times 8. 8 is 4 times 2. 2 is prime so we're going to circle these and then 4 is 2 times 2. just after a while you're going to start to recognize what the prime factorizations are for some of these common numbers so I got one two three four five six factors of two which we have up there and then what about 288 well I know it's divisible by two and 88 is definitely divisible by four so I can start with that let's divide 288 by four and see what we get 4 goes into 28 7 times 7 times 4 is 28 subtract and get zero bring down this eight here four goes in eight twice two times four is eight subtracting gets zero so 72 times 4 is 288. now without making a factor tree I know that 4 is 2 times 2. I know that 72 is 9 times 8 okay 9 is 3 times 3. 8 is 3 factors of two again if you're not comfortable with this yet go ahead and make the factor tree it's not a big deal so 288 again 4 times 72 or is two times two 72 is eight times nine eight is four times two minus three times three 3 is prime 2 is prime and then 4 is 2 times 2. and then that's Prime so we have one two three four five factors of two so two to the fifth power times two factors of three or three squared so two to the fifth power times three squared and that's exactly what we got up here we have one two three four five factors of two and two factors of three okay so now we're looking to cancel common factors between numer and denominator I can cancel this 2 with this two this two with this 2. this two with this two this two with this two and then this two with this two now I have a last two up here because I have one two three four five six factors up here but I only have five factors of two in the denominator so I'm going to have one factor of 2 up here that I can't cancel with anything because I had one more up here than I had down here these threes I don't have anything to cancel with up in the numerator so my simplified version of this fraction I'm going to have a 2 in the numerator and 3 times 3 or 9 in the denominator so we end up with 2 9 as our answer all right so now we have 25 over negative 135 25 over negative 135 25 is just 5 times 5. negative 135 again if you have a negative involved just write negative 1 and then just do the prime factorization for 135. so 135 it ends in a 5. so I know it'll be divisible by five and what is that 135 divided by 5 would go into 13 twice 2 times 5 is 10 subtract and get three bring down the 5 there 5 goes into 35 7 times 7 times 5 is 35 subtracting get zero so 27 times 5 is 135 and we know 5 is a prime number we can Circle that and 27 we know is three factors of 3. all right it's 9 times 3 3 is prime 9 again is three times three and three is prime so we're going to circle both of these so 3 times 3 times 3 times 5. so what can we cancel between numerator and denominator well all I can do is cancel this 5 with this 5 and that's it all right I have one more five here but nothing to cancel with down here and I have some threes down here nothing to cancel with it up there so we'd just end up with 5 in the numerator and one two three threes in the denominator 3 times 3 is 9 9 times 3 is 27 then multiply by negative 1 you get Negative 27. now I can keep it as 5 over negative 27. I could also write this as negative 5 over 27 or I could write this as negative and then 5 over 27. all of these are the same thing remember that when we divide integers a positive divided by a negative is a negative or a negative divided by a positive is a negative or you can just have the negative out here just saying hey the fraction is negative and then 5 27. all of these mean the same thing you have a negative 5 27 as your answer okay let's take a look at 4 19. so 4 we know is two times two 19 is prime if I looked at the greatest common divisor or again the greatest common factor of 4 and 19 your answer would be 1. so when you have this situation you're not going to be able to simplify the fraction any further and you just end up with 4 19. as your answer hello and welcome now pre-algebra lesson 30. in this video we're going to learn about multiplying fractions all right so the lesson objective for today is just to learn how to multiply two or more fractions together I'm going to give you an initial procedure and then I'm going to modify it slightly we're going to start doing something called cross canceling and that's going to allow us to simplify while we're doing the multiplication problem so that we don't have to simplify after but let me give it to you this way first and then we'll do the modification so to multiply two or more fractions you're going to multiply the numerators together and then you're going to place the result over the product of the denominators and I have here we want to simplify but we're going to change this up a little bit as we work through the problems now before we get started let me give you a brief explanation in terms of why we do this as the numerator times the numerator over the denominator times the denominator and then we simplify so let's suppose that I start off with a pizza and I'm going to split this up into four equal slices so each slice will be 1 4 of the pizza so let's say this is one two three and then four slices suppose that I eat a slice of this pizza for breakfast and then another one for lunch and then a final one for dinner so how much of the pizza did I eat three-fourths how much of the pizza is left one fourth so we know that at this point but how could I have gotten that three-fourths I would have said that well I ate one two three so I could just count pieces of pizza out of a total of four equal pieces of pizza that were available in this whole amount so I could have said that this is one-fourth that's me eating at breakfast plus one-fourth that's me eating at lunch plus one-fourth that's me eating at dinner so this should give me three-fourths now we're not gonna really talk too much about adding fractions here this is something we'll cover later on it's a little bit complicated but basically if you have the same denominator so here you have a four a four and a four what you do is you keep that denominator the same and you just add the numerators so you see that this is one plus one plus one so my one slice from breakfast plus my one slice from lunch plus my one slice from dinner that one plus one plus one that gives me three slices that I ate out of a total of four slices in that whole pizza so now that we understand that we should get three fourths from one-fourth plus one-fourth plus one-fourth what we can do is transform this into a related multiplication statement remember if you have something like let's say three times five this is going to be five plus five plus five so either way whether you do the multiplication from the multiplication tables or from the repeated addition here you're going to get 15. so the same thing should be true here if I write this as three because I have one two three these guys times one fourth I should get three fourths and how would we do that well you could write three as three over one three divided by one is going to be three so I haven't changed anything and then you could just multiply the numerators together 3 times 1 would give me three and then you could put that over the product of the denominators so 1 times 4 would give me four so that's how you would get 3 4. so you'll see me say we want to multiply the numerators together put this over the product of the denominators all right let me give you another example here because it's not going to be that straightforward when you see something like let's say one-half times 1 4. well what in the world does this mean if you try to translate it into repeated addition we saw that when we had three times one fourth this became 1 4 plus 1 4 plus 1 4 if we write it as a repeated addition but if I write one-half times one-fourth well I don't have three of these what I want is one half of this so basically it's like saying I want one half of one-fourth so I have this first pizza which is cut into four equal slices what we're gonna actually have to do is take this one fourth let me just mark this off right here so this would be one-fourth of the pizza and let me do this over here now this guy is cut into eight equal slices so let me Mark this off what you're gonna have to do if you say that you want one half of something well you need to cut it into two equal parts so I'm gonna slice this guy let me use a different color here I'm going to slice this guy right here and I want one out of those two equal parts so I want this slice right here let me count these up so this is one two three four five six seven eight so I want one slice out of a total of eight equal slices so this would give me one eighth because basically I'm starting with one-fourth of this whole amount so one piece out of a total of four equal pieces and I'm saying hey give me half of that so it's kind of like I got this slice and I said okay well we need to split it so we split it like this or I cut it in half and now how much do I have well I have one piece out of eight pieces now instead of having one piece out of four pieces so this becomes the fraction 1 8. now when you look at it again it's the same procedure multiply the numerators one times one would give me one and then put this over you're going to multiply the denominators 2 times 4 would give me eight so one half times one fourth would give me one eighth all right let's take a look at some examples so you'll find this process to be very very easy we have two fifths times one-seventh so just following the procedure we're going to multiply the numerators together so two times one would give me two we're going to place that over we're going to multiply the denominators together so 5 times 7 would be 35. now 2 is a prime number and 35 we just saw that came from 5 times 7 5 is a prime number and 7 is a prime number so really there's nothing here that I can cancel right the GCF is going to be one so we would say the simplified answer here is 2 over 35. all right so here's one that's going to be a little bit tricky so first let's use our procedure so we have three tenths times five twelfths I'm going to multiply numerator times numerator so 3 times 5 is 15 this is over denominator times denominator so 10 times 12 that's 120. okay well here is the problem you see that this guy right here is the correct answer 15 over 120 but it's not simplified so if you report an answer and it's a fraction and it's not simplified your teacher will probably at minimum take points off some of them will mark it wrong it just depends on your teacher so what you'd actually have to do is Factor this and you're going and undoing what you just did right so 15 you're going back to 3 times 5. and let me make that three a little bit better and then for 120 if I break that down completely 10 is going to be 5 times 2 and then 12 is 4 which is 2 times 2 and then times 3. let me extend that a little bit now we know how to simplify fractions already basically I'm looking at the numerators and the denominators and I'm seeing what's common so I can cancel this with this 3 divided by 3 is 1. now I would advise you at this point to put a 1 there a lot of people will leave it off it just depends on your preference some teachers will make you do it some teachers will tell you to put a 1 there and put a 1 there and some teachers will say it doesn't matter but the thing is when you look at this one I'm going to cancel this 5 with this 5 so I'm going to put a 1 here and a 1 here now if your scratch work looks like this let's say you get rid of all of this stuff well the problem is the Temptation for most students they're going to say okay there's nothing in the numerator I've canceled everything away so they write a zero really it's a one right because 3 divided by 3 was 1 and 5 divided by 5 was 1. so that's why if you put a little one there as a reminder you can say okay well I need to multiply one by one that's going to give me one and this is over 2 times 2 times 2 that's going to be 8. so remember it's just like I did the problem 3 over 3 times 5 over 5 times 1 over let's say 2 times 2 times 2 that's going to be 8. this is exactly going to give me the same answer as this what happens is 3 over 3 is 1 right so I can just say this is one five over five that's one so it's 1 times 1 times 1 8 which is 1 8. so this is the correct answer you don't want to put a zero up there okay let me get rid of this when we look at what we did we double worked right first we multiplied then we had to factor and break the multiplication back up and then we canceled what we actually want to do is cancel before we multiply and this process is called cross canceling so to do this you first want to look at your fractions and see can I simplify them so three tenths can't do anything there 5 12 can't do anything there so once you're out of options there you want to look at the numerator of this fraction let's just say the left fraction and the denominator of the right fraction so this numerator and this denominator and then you're going to do the same thing let me change my highlighter color here so then this numerator and this denominator and you're just going to ask the question can I cancel any common factors between this numerator and this denominator or this numerator in this denominator basically are there any common factors between any numerator and any denominator that's what you're asking so to make this easier to see I'm just going to break it down like we had it so I'm going to put 3 over we know 10 is 5 times 2 and then times I'm going to put 5 over 12 is 2 times 2 times 3. this 3 can cancel with this 3. so This numerator and this denominator there's a common factor of three again I would put a 1 here you can even put a 1 here if you want for clarity you don't need it but you can put it then here you look at this numerator and this denominator you can cancel this 5 with this 5. again you could put a 1 here and a 1 here and then when you go through and multiply well 1 times 1 would give you 1 and this this would be over let me make that a little bit better there you would have again you can think about this as one times two or just two because one times anything is just itself so really it's just 2 times 2 times 2 which is going to be eight again this one would not change anything if you multiplied by it so you end up with 1 8 which is the simplified answer and notice that we didn't have to double work we went ahead and canceled before we did the multiplication so once we get our product it's already simplified all right now that we understand the procedure let me go a little bit quicker through the process so we have 9 over 20 times 5 over 33. so I can't simplify this and I can't simplify this now you could continue to do this and break it down all the way so you could say this is 3 times 3 over 20 is 5 times 2 times 2 and then times you have 5 over 33 is 3 times 11. you can do it this way and then again I can just look at this numerator in this denominator so let's cancel this 3 with this three again you could put a 1 here if you want and one here if you want just as I'm mental reminder and then you can cancel between this numerator and this denominator this five with this 5. so let's put a 1 here and a one here all right so this would give me one so you have 3 times 1 which is three over you're going to have 2 times 2 which is 4 times 11 which is 44. so you end up with 3 over 44 as your simplified answer now let me show you a faster way to get this let me get rid of this and we're going to go even faster so here's how I do it I would look at this guy and say okay well 33 and 9 in my head I'm immediately thinking okay well 33 is divisible by 3. 33 divided by 3 is 11. so I'm going to cross this out and just put 11. then 9 divided by 3 is 3 so I'm going to cross this out and put a 3. so basically I'm dividing this guy by 3 and this guy by 3 and I'm just putting the result there 33 divided by 3 is 11. 9 divided by 3 is 3. it's the same thing as if I factored and canceled so this guy would look at 20 and I would look at 5. well 20 divided by 5 is 4. so cross this out you can put a 1 there air and Crosses out you could put a 4 there so 5 divided by 5 is 1 20 divided by 5 is 4 is a little bit faster than factoring so now you just multiply three times one that's three and this is over 4 times 11 that's 44. okay let's look at one with three fractions involved so we have 7 over 15 times 21 over 25 times 3 over 14. this is simplified so is this so is this so what I would think about is 7 and 14 have a common factor of seven so seven divided by seven is one fourteen divided by seven is two and then coming through here now I have 21. well 21 is 7 times 3. well there's no 7 in any denominator now because I cancel this one away but there is a 3. 21 divided by 3 would be 7 and then 15 divided by 3 would be five so I'm just canceling a common factor of 3 there and then looking at this last one here I have a three but this is a five this is five times five and this is a two so basically I'm stuck I can't do anything else so this would end up being one one times seven which is 7 times 3 which is 21. over you'd have 5 times 25 you can stop and do a vertical multiplication for that I think a lot of you do know that 5 times 5 times 5 or 5 cubed is 125 so basically it would be 125 times 2 which is going to be 250. so 21 over 250 would be your simplified answer again if that's a little too fast for you you can always stop and break things up so you can say that this would have been let me do this down here so 7 over 5 times 3 and then times 7 times 3 over 5 times 5 and then times three over seven times two if you need to do this it's fine so I'm going to cancel this 7 with this 7 I'm going to cancel this 3 with this 3 and basically that's it right so I'm just left with I'm going to put a 1 here again if I left that off then I would have to go back and say okay well this cancel this is a one so a lot of times I leave that off because I know that but for your scratch work I think you should put it at this point so you're not confused so this would be 1 times 7 which is 7 times 3 which is 21. over you have 5 times 5 times 5 again it's 125 times 2 which is 250. so if you want to do it like this because it's a little bit easier for you right now then it's fine to break up the numbers completely and Factor everything into the product of primes in a lot of cases As you move forward you won't need to do that because it's just a little bit too time consuming okay let's talk about the next topic here so we're going to multiply whole numbers by fractions so we have six times one-third so this would be what you just take 6 and put it over one and you multiply it by a third and you can cross cancel here 6 divided by 3 would be 2 right so basically this is 2 and it's 2 times 1 which is 2 over again you could write a 1 there 1 times 1 is 1. so this is two over one or two and if you think about this six times a third six times a third would be what well this is one-third plus one-third plus one-third let me make that three a little bit better plus one third plus one-third and then plus another one-third so if I had one third of a Pizza Plus another third plus another third at this point right here I'd have a full pizza so this is one pizza and then I'm doing the same thing over here so this right here this would be another pizza so one plus one is two so this tells me I should have two pizzas again if you do this using the addition that we're gonna learn about later you would basically keep this denominator the same because it's called a common denominator and you would just add the numerators so one two three four five six of those guys you basically have a six up there so you'd have six divided by three basically which is going to be two okay let's take a look at seven times three over forty nine so this would be seven over one times three over forty nine and I'm going to say that 49 divided by seven is seven so this is seven so now basically this is a one right here so you can just forget about it you could say it's 1 times 3 which is three over 1 times 7 which is 7 but you could also say one times anything is itself so you could just say okay this is going to be 3 7. all right here we have 19 over 15 times 5 over 38 times 36. so basically what I would do is treat this as 19 over 15. times 5 over 38 times 36 over 1. let's go through and see if we can cancel so I know that 38 is 19 times 2. so for this one this would cancel with this this would be a one and this would be a two again 38 divided by 19 is 2 19 divided by 19 is 1. then here 15 divided by 5 is going to be 3 and 5 divided by 5 is going to be 1. now for this one as you get faster you're not going to say okay well 36 divided by 3 is 12 and then you'll say well 12 divided by 2 is going to be 6. you would do that in one shot so in other words what you could have done here let me get rid of these cancelings here you could say that 36 divided by 6 is going to be 6 and then basically cross those out because 3 times 2 is 6. now again you can write this as a one or as you move forward you're just going to realize that that's canceled so it's one so basically this is a one this is a one one times one is one and one times anything is itself six over one is six so this gives me a final answer that's just going to be six all right let's look at another one so we have 4 over 15 times 5 over 24 times 7 over 12 times negative four okay so 4 over 15 times 5 over 24 times 7 over 12 times negative four I'm going to write that over one all right let's think about what we can do here to cancel so starting with the four four divided by 4 is 1 24 divided by 4 is 6. and then 5 divided by 5 is 1 15 divided by 5 is 3. and then if we keep going I have a 7 here 12 is going to be 2 times 2 times 3 this is 6 this is 3. so I can't do anything with that all right then I have this negative 4. let me write that as negative 1 times 4 and what you could do is divide four by four you get 1 and 12 divided by 4 would be 3. so looking at what I have left in the numerators one times one you can forget about those it's not going to change anything then times seven so let's just start with 7. so basically 7 times negative 1 that is negative seven and this is over down here you have three times six times three so three times three is nine nine times six is fifty-four and then times one that's still 54. so this would give me negative 7 over 54 as my answer all right let's take a look at two quick word problems so we have Fred spent two-fifths of his fifty dollar allowance on Comics how much did he spend so this is typical when you see two-fifths of this fifty dollars so what you're going to do is take two fifths and multiply it by fifty and so I'm going to cancel this 50 with this 5 50 divided by 5 is 10. again you could put a 1 down there if you want and then just multiply 2 times 10 is going to be 20. again you can go over 1 if you want but it's just going to be 20 at the end of the day so when it asks how much money did he spend the answer is going to be twenty dollars right that's two-fifths of his 50 dollars all right now we have a recipe requires one half cup of sugar if we only want to make three-fourths of the recipe how much sugar is needed so again just take three fourths and multiply it by one half and what is this going to give me nothing's going to be able to cancel here this is simplified this is simplified 3 and 2 1 and 4. nothing I can do so just multiply 3 times 1 that's 3 over 4 times 2 that's 8. so basically we could say that you would need 3 8 cup of sugar hello and welcome to pre-algebra lesson 31. in this video we're going to learn about dividing fractions all right so the lesson objectives for today we want to learn how to find the reciprocal of a number and then additionally we want to learn how to divide fractions all right so now we're going to talk about how to find the reciprocal of a number this is something we need when we start dividing fractions in a little while so to find the reciprocal of a number we interchange the numerator and denominator so what do I mean by interchange well let's start with a simple example let's say we have the fraction four-fifths and I want to Interchange the numerator and denominator well basically I'm going to take this denominator from the fraction that I'm starting with and that's going to become the numerator for the reciprocal so this would be a 5 up here and then let me put my fraction bar I'm just going to take this numerator here from the fraction that I'm working with and drag that down into the denominator for the reciprocal so the numerator becomes the denominator and the denominator becomes the numerator that's what I mean by interchange the numerator and denominator so the reciprocal of four-fifths is going to be five-fourths and you could reverse that and say the reciprocal of 5 4 would be four-fifths now when we multiply a number by its reciprocal the result is always going to be one so let's say we go back to four-fifths and the reciprocal is five fourths so why is the result going to be one well think about it this denominator becomes this numerator so it's basically five over five which is one right you can cross cancel here let's just put a one here and a one here and then this numerator becomes this denominator so it's four over four that's one so cancel this with this and I'll put a one here and a one here well if we multiply these together you could just think about this as one times one that's going to give you one all right so now let's look at a few problems we are asked to find each reciprocal so we are given two-thirds if you want to find the reciprocal again I'm just going to take this denominator and drag it into the numerator and then I'm going to take this numerator and drag it down into the denominator so the reciprocal of two-thirds will be three halves again you can go two-thirds times three halves if you want to check your work so two over two is one and then three over three is one and so you have one times one which is one again the product here should always be one okay what about eight ninths so again let's take this denominator here bring it up into the numerator let's take this numerator here and drag it down into the denominator so the reciprocal of eight ninths is going to be nine-eighths all right what about one with a negative involved so let me take some time with this one so if you have this set up here the first thing you could do is just put a negative out in front and then basically you could interchange the numerator and denominator so I could bring the five up here and then I could bring the one down here so the denominator becomes the numerator the numerator becomes the denominator and I just basically copied the negative sign so the reciprocal here is negative five over one which is just negative five but there's other ways that you could have thought about this you could say that this is equal to negative one over five you could also say this is equal to one over negative five remember a negative divided by a positive or A positive divided by a negative that's going to give you a negative in each case so three different ways to write the same thing so you could choose to do it this way you could say this is negative one over five and then you can bring this denominator up into the numerator and you can bring this numerator down here into the denominator so either way it's going to be negative five right five over negative 1 is just negative five okay let's look at one more problem so now we're given an integer so we have negative four so what are we going to do well basically you wanna change this into a fraction by writing it over one so I can write negative 4 over 1 like this dividing by 1 leaves the number unchanged so this is still negative 4. so now I'm just going to take this denominator and bring it up into the numerator and take this numerator bring it down into the denominator so the reciprocal of negative four is basically negative 1 4. you can write this again as negative 1 over 4 or you could write it as 1 over negative 4 as we have here or we can say it's negative one fourth whatever you want to do all right now let's move on and talk about dividing fractions so I have here to divide fractions we multiply the first fraction so that's going to be your fraction on the left we call that the dividend in your division problem by the reciprocal of the second fraction so that's the fraction on the right that's going to be your divisor in the division problem so for example let's just say I had something like 3 8 divided by one half this first fraction or the leftmost fraction will be unchanged so you would just copy that down then you would multiply by the reciprocal of the second fraction or the fraction on the right so this is times two over one so basically at this point it's just a multiplication problem so between 8 and 2 there's a common factor of two eight divided by two is four two divided by two is one and then you would multiply three times one that's three over four times one that's four so the answer here would be three fourths all right so the process of dividing fractions is pretty easy overall but some people have a really difficult time understanding where the process comes from so on the next two examples I'm really going to break this down I'm going to start off with a whole number divided by a fraction I think that's the easier one to understand so I'm just going to start off with some things we already know so right now I have four boxes on the screen and I'm just going to say this is one two three and then four just to count them out now let's suppose I asked you what is 4 divided by four what does that mean again let's go back to basic division well I'm asking how many equal groups of four can be made from four so here I have one two three four so this whole thing would be a group right so this would be a group of four so four divided by four would be one I can make one of those groups I think we all know that at this point let me get rid of this and let me change the problem up so let's say we said four divided by 2 now so we know this equals two but again I'm asking how many equal groups of two can be made from four well here's a group here's a group right so one two and then another one two so four divided by 2 would be two now what if I change this up a little bit and I said what is 4 divided by one half well using the procedure I just gave you you could write 4 as 4 over 1 and then multiply by the reciprocal of this which is two over one so that should give me eight well let's think about this down here what does it mean to have one half of something well it means I have a whole amount let's say it was a pizza or in this case a box and I split it into two equal parts and then each part is one half so this right here is one half and this is one-half then this is one half and this is one half this is one half and this is one half and then this is one half and this is one half so if I go through and just count them how many one-halves do I have or how many one half of a boxes do I have well I have one two three four five six seven eight okay well that starts to make a little bit of sense in terms of why I got eight as the answer when I say four divided by one half what I'm actually asking is how many equal groups of one-half can be made from four well what you need to do is start out with the fact that you have four of these guys so four boxes so I'm just going to write that and then in each box there's going to be two half boxes right so two half boxes so I need to actually multiply by two because here I'm getting two here I'm getting two here I'm getting two and here I'm getting two so that's why this turns into the problem four times two which gives us eight if we go a little bit further and we think about something like four divided by one fourth well it's the same principle here let me go through and write these in so this is one-fourth this is one-fourth this is one-fourth and this is one-fourth okay so thinking about this if you went through and counted what would you have so this is one two three four and then you have four of those so that's 16. so basically this turns into four of these boxes one two three four and then in each box there's four one fourths so I'm just gonna do four times four so this becomes sixteen again four stays the same multiply by the reciprocal of of 1 4 the reciprocal of 1 4 is 4 over 1 or just four so this becomes 16. okay let me give you an alternative way to look at this a lot of people will teach this with what we call a complex fraction so let's say we had four divided by one half so I'm going to write this as four over one divided by one-half now we haven't talked about complex fractions yet we're going to get to them later on in the course for right now a complex fraction is a fraction that contains another fraction in its numerator its denominator or both so basically all I'm going to do is take this part right here which is the dividend and I'm going to bring that up here into the numerator of the complex fraction so this is four over one so this right here is the numerator and then I'm going to have my fraction bar so this is the fraction bar for the complex fraction you could think about it as the main so the main Division and then down here I'm going to grab my divisor and that's going to become my denominator here for the complex fraction so this is one half and let's just say that this right here would be your denominator okay so the idea with this method is you're just going to transform the denominator into one and in order to do that you would multiply this one half by its reciprocal which is two over one a number of times this reciprocal is always one like we talked about earlier but to make that legal you'd have to also do that to the numerator so the problem is four over one over and then you have one half and all I'm going to do is multiply the denominator by two over one and to make that legal I'm going to multiply the numerator by two over one so I'm basically just multiplying the numerator and denominator of the complex fraction by two and this would be equal to what well these guys down here these are reciprocals so 2 over 2 is 1. so let me put a 1 here 1 times 1 would just be one so the denominator of the complex fraction is one now then up here I just do four over one times two over one so 4 times 2 would be 8 you could put over 1 I guess if you wanted to but 8 over 1 is just eight so at the end of the day this is eight over one which is eight over one which becomes eight so that's another way to show where it comes from again you can use either method to prove it to yourself and think about how things work I like using the visual example with the boxes but some people prefer this method all right let's look at one that's a little bit more complicated to explain so suppose you had one half divided by 1 8. again let's use the procedure to start so one half times the reciprocal of 1 8 is eight over one and so essentially you can think about eight and two have a common factor of two eight divided by two is four and then two divided by two is one so this is canceled you could just write this as a one so it's basically one times four which is just going to be four how can we show this and prove that we should take one half and multiply by the reciprocal of 1 8. well again you could have written this as a complex fraction so you could do that again so you could say this is one half over 1 8 and again if I want to make this denominator for the complex fraction into one I'm going to multiply by the reciprocal of this fraction here so this is times 8 over 1 and then I'm going to do that to the numerator to make that legal so you see that this right here is going to cancel so this is a 1 and you can put a 1 down here so this is a 1 now and so this would be equal to well basically it's this one half which we're starting with over here on the left times this 8 over 1 which is the reciprocal of this rightmost guy here this one over eight so we end up with what again like we just did 8 divided by 2 is 4 and 2 divided by 2 is 1. so this is just 1 times 4 which is going to be 4. I mean you can go through and say this is 4 over 1 and then over 1 and then this becomes four over one like this and this is 4 but I think we can all see that this is four all right let me give you a visual explanation in terms of Y one half divided by 1 8 ends up giving you one-half times take the reciprocal of this is eight so this ends up giving you 8 over 2 is 4. Okay so we've seen this already using the complex fraction method but if you want to look at this visually what you would need to do because you're dividing by 1 8 just start with some whole amount you could take a pizza a box whatever you want to work with and you're going to divide it up into eight equal parts if you're working with a pizza you're going to have eight equal slices so let's say this is one two three four five six seven eight if I was asking myself the question how many groups of one-eighth would be in ones one divided by 1 8 this would be one times take the reciprocal of this it would be eight so this is eight and you can go through and count that I have one two three four five six seven eight one eighths in this whole pizza now I don't have a whole pizza I only have half of a pizza so let's mark this out like this so maybe this is going to be my half of the pizza and I'm not really good at drawing so I'm gonna go over but basically if you ask yourself the question well how many 1 8 are there in one half well you have one two three four so it makes sense that we're getting four we can visually see that but again why do we end up multiplying by eight well eight is the number of slices or equal parts in the whole amount and then I don't have a whole amount I only have half of that so you're asking yourself the question what is one half of eight and that's how we end up getting four so that's how you wanna think about this so hopefully this makes sense for you visually you can always pause the video and think about it or make your own example with a pizza or boxes or whatever you want to use all right so now that we understand the procedure let's just blow through some problems I think you're going to find this to be really easy once you get started so we have two thirds divided by four ninths so leftmost fraction you're going to leave that unchanged and then we're going to multiply by the reciprocal of the rightmost fraction so basically I'm going to take 4 9 to find the reciprocal that's going to be 9 4. the numerator becomes the denominator the denominator becomes the numerator so now we can just go through and multiply Let's cross cancel so basically between 2 and 4 you have a common factor of 2 so 4 divided by 2 is going to be 2 2 divided by 2 is going to be 1. and then between 9 and 3 have a common factor of 3 so 9 divided by 3 is 3 and then 3 divided by 3 is 1. so this right here is canceled right this is a one so basically you're just going to have 1 times 3 halves which is three halves so that's going to be your answer what about 6 7 divided by 15 over 28. so the leftmost fraction the first fraction stays unchanged and we're going to multiply by the reciprocal of the second fraction or the rightmost fraction so this 28 the denominator becomes the numerator then this 15 the numerator becomes the denominator so the reciprocal of 15 over 28 is going to be 28 over 15. okay now we're ready to do some multiplication so basically looking at 28 and 7 well 28 divided by 7 is 4. so 28 divided by 7 is 4 and 7 divided by 7 is 1. so then between 6 and 15 each is divisible by 3 so 15 divided by 3 is 5. 6 divided by 3 is 2. so what I'm left with is 2 times 4 which is 8 so that's my numerator over 1 times 5 that's 5 that's my denominator so our answer here is going to be eight fifths what about one-third divided by 7. so the leftmost fraction one-third we're going to keep that unchanged and we're going to multiply by the reciprocal of what's on the right now again if this is not a fraction if it's an integer or a whole number or something like that then basically you're going to write it over 1. so let me actually change this up a little bit let me put divided by 7 over 1 and so this equals you'd have one third times the reciprocal of seven over one is one seventh so again this 7 which is the numerator becomes the denominator this one which is the denominator becomes the numerator now we just multiply so nothing to cancel here so 1 times 1 would be 1 over 3 times 7 that's going to be 21 so you get 1 over 21 as your answer but about 2 9 divided by negative 4. so again we can go 2 9 divided by negative four over one so let's say this equals you have two ninths so leftmost fraction that's unchanged and then times the reciprocal of the rightmost fraction so again the reciprocal here let's just write negative 1 over 4. again you could write the negative down here if you want or you can put it out in front it really doesn't matter because when you're multiplying a positive times a negative will give you a negative so you can just put the negative sign in there and then forget about the signs and just work with the absolute values so looking at it now you can see that you have nine and one nothing you can really do with that and then you have two and four so there's a common factor of two so four divided by two is two two divided by 2 is 1. so we already know that that this is negative so let's just multiply 1 times 1 that's going to be 1 over 9 times 2 that's going to be 18. so the final answer here is negative 1 over 18. what about 12 divided by 1 3. so let's write this as 12 over 1 and then basically instead of divided by 1 3 I'm going to go times three over one so this guy stays unchanged I just wrote 12 is 12 over 1 and then I'm multiplying by I'm just going to take the reciprocal of 1 3. so 3 comes up into the numerator one goes down into the denominator so now you just multiply so 12 times 3 is 36 and this is over 1 times 1 that's 1 and of course anything over one is just itself so this just is equal to 36. all right let's take a look at the next one so we have negative 7 over 24 divided by 9 over negative 8. okay so what I'm going to do here is say the first fraction negative 7 over 24 is unchanged and then times the reciprocal of this fraction I'm just going to put the negative out in front and then I'll write eight ninths like this and make these negatives a little bit better here so this is negative 7 over 24 times negative 8 9. okay so we know that negative times negative is positive you can put that there as a little reminder and then erase it or scratch it out after or you could just leave it it doesn't matter and then I'm just going to work with the absolute values so forget about the negative Signs Now I'm going to think about okay well 24 and 8 there's a common factor of 8 24 divided by eight is three eight divided by eight is one and then seven and nine nothing really in common there 7 is a prime number and 9 is 3 times 3. so what I'm going to do is multiply we're going to say 7 times 1 is 7. remember I already know this is positive so forget about the negatives here and then down here I have 3 times 9 that's going to be 27 so again you can erase this at this point you know it's positive so you would say the answer is 7 over 27. all right let's take a look at a problem with two divisions involved so here we have three fifths divided by twenty one tenths and then divided by nine elevenths so the way I would do this problem I would convert these divisions into multiplication by the reciprocal it's just going to make it a little bit faster the other way you could do it is you could do three-fifths divided by 21 tenths so you could find that result and then divide that by 9 11. so that's up to you we'll give you the exact same answer I'm going to do this as three-fifths I'm going to convert this division to multiplication by the reciprocal so times 10 over 21 and then I'm going to convert this division into multiplication by the reciprocal so times 11 over 9. so once you have it written this way now it's a little bit easier because I can go through and just cancel things in one shot so I can say are there any common factors that I can cancel away between any numerator and any denominator so looking through here I have a 3 here and I have a 21 here so 3 divided by 3 is 1. 1 21 divided by 3 is 7. so now looking at 10 and 5 well yeah 10 divided by 5 would be 2 5 divided by 5 would be one now looking at everything else it looks like I'm basically done so I would just multiply the numerators 1 times 2 times 11 that's going to be 22 and put that over you'd multiply the denominators 1 times 7 times 9 that's going to be 63. so the answer here would be 22 over 63 and again if you want to pause the video and try it a different way you can do this division first and then get that result and then divide by this you will still get 22 over 63. hello and welcome to pre-algebra lesson 32. in this video we're going to learn about finding the LCM otherwise known as the least common multiple so our lesson objective for today is just to learn how to find the least common multiple again that's abbreviated LCM for a group of numbers so before we start talking about finding the least common multiple we have to have a general understanding of what a multiple is and there's a lot lot of different definitions that kind of float around but for the purposes of pre-algebra we're going to use this one so to obtain the multiples of a number we multiply the number by each non-zero whole number so in other words if we think about the whole numbers remember they start with zero and increase in increments of one out indefinitely so all I'm going to do is exclude Zero from this list so in this case I'm going to start with 1 and increase in increments of 1 out indefinitely so I'm going to take a number multiply it by one then two then three then four then five each result is a multiple of that original number so let me give you an example so let's take a look at finding the multiples for the number three so again we want to multiply 3 by each non-zero whole number to obtain the multiples so I'm going to start out with three times one so three times one is three and then I'd move on to three times two three times two is six then I'd go to three times three three times three is nine then three times four three times four is twelve and three times five three times five is fifteen and you can kind of see where I'm going with this right I'm starting with one and then I'm increasing in increments of one so I'm multiplying by one then two then three then four then five and if I continue to be six then seven then eight so on and so forth but really you can just list the multiples you can kind of see that we're just increasing in increments of three every time right so if I think about kind of the multiples of three we have three then six then nine then 12 then 15. and again we're increasing in increments of three so then after 15 comes 18 then 21 and I'm going to put three dots here to say this pattern is going to continue forever right increasing in increments of three and I'm just going to put here that these are the multiples of three let's take a look at another one what if I ask you to find some multiples of 11. so I take that number 11 and again I'm going to multiply it by each non-zero whole number I'm just going to start with 1 and just increase so 11 times 1 is 11. then 11 times 2 is 22 then 11 times 3 is 33 and 11 times 4 is 44 then 11 times 5 is 55 and obviously you can continue this forever right there's no largest whole number after 5 would be six then seven and eight so on and so forth but I just want to generate a few of them so you can get an idea of what a multiple is so let's write the first few multiples for 11. so it's 11 22 33 44 55 then it would be what 66 right we're increasing in increments of 11 then 77 and of course at some point I can stop and put three dots to say hey this pattern where we're increasing in increments of 11 is going to continue forever so these are some multiples some multiples of 11. so now that we have a basic understanding of how to find the multiples of a number let's introduce this next concept called finding common multiples so common multiples are just multiples that are common to all numbers of a group so let's say that we have two numbers and these two numbers let's say they're four and five so let's say I want to find the multiples find the multiples of four so I'm going to write them just right here so I'd start with four right because that's 4 times 1. then I'd increase in increments of four so I'd go to 8 then 12 then 16 then 20 then 24 then 28 then 32 then 36 and I'm going to continue this down here then 40 and 44 then 48 then 52 then 56 then 60 and I'm going to put comma dot dot dot so the three dots just tell me that hey we're going to keep increasing increments of 4 out indefinitely and again the way I figured this out is I start by multiplying 4. the number that I want to find the multiples for by each non-zero whole number just starting with one so 4 times 1 would be 4 and I'd increase to the next largest whole number so now I'd multiply by 2. 4 times 2 is 8. between here and here I just increased by four so that's how I knew to go from four to eight now when I multiply by three I'm just increasing by 4 again right this is plus four and I'm going to do that again this is plus four so now I'm going to go to 12. and you just keep doing that twelve plus four sixteen sixteen plus four is twenty twenty plus four is twenty four again so on and so forth now if I did the same thing for the number five let's say I want to find the multiples of five so I would start out with five times one and that's going to give me 5. then 5 times 2 is 10. then 5 times 3 is 15. each time I'm just increasing in increments of 5. so then I'd have 20. then 25 then 30 then 35 then 40 then 45 then 50 then 55 then 60. and we're going to stop there okay again the three dots are telling me that this pattern where we're increasing in increments of five is going to continue forever now what's important here is to look through these two lists that we've generated the list that we have for the multiples of four and the list that we have for the multiples of five we want to see what's common to both meaning what number do we have in this list that's also in this list so looking through here I see that 20 appears here and also here I see that 40 appears here and also here and I see that 60 appears here and also here so what it looks like is that every 20 numbers we're going to have a multiple that's common to four and five right the next one would occur at 80. right if you went through here you'd have 60 then 64 then 68 then 72 then 76 and then 80 and here you'd have 60 65 70 75 80. so it would next occur at 80 then 100 then 120 so on and so forth just every 20 numbers so we can see that we'd have an unlimited number or an infinite number of common multiples between four and five because these numbers continue forever right they're not going to stop but what's important for this lesson is to be able to see what the least common multiple is meaning what's the lowest in value and that occurs here at 20. right at 20. that's where it first happens and that's the lowest in value right 20 is less than 40 it's less than 60 it's less than 80. it's less than all the other common multiples that will come in the future if we were to list out all the multiples let's try another one let's say we want to define the least common multiple between 15 and 20. so let's say we had 15 and we want to find the multiples for that number we'd start out with 15 times 1 that's 15. then you go to 15 times 2 that's 30. then 15 times 3 is 45 and again you can see that you're just increasing in increments of this number right I'm going up by 15 each time so after 45 is 60 then 75 then 90 then 105 then 120 so on and so forth I'm just going to put the three dots to say hey we're going to keep increasing increments of 15 out indefinitely now what about the multiples for 12. so for 12 we'd start at 12 and just increase in increments of 12 out indefinitely so 24 36 48 60 72 84 96 108 and let's put 120 and comma dot dot dot again the three dots are telling me that that pattern where we're increasing in increments of 12 is going to continue forever now when I look through the two lists here I'm looking for multiples that they share otherwise known as common multiples so the first one is going to actually occur at 60. all right we have a 60 in this list and a 60 in this list and then you're going to see that they have common multiples at 120 right so 60 more started at 60 then it goes 60 more and it happens again and then it'll happen again 60 more after that so it happened at 180 and then what happened at 240 then it would happen at 300 so on and so forth but again if we want the least common multiple we want the lowest value so we want this one that occurs at 60. the one that started it all the first one that kind of appears between the two lists and so the LCM the LCM of 15 and 12 we can kind of notate that this way would be equal to 60. all right that's the least common multiple again 120 is a common multiple but it's not the least common multiple very important that you understand that so so far we've found the least common multiple by just listing the multiples of each number and just looking for the smallest that's in common now this method is slow it's tedious and it's impractical so when you start taking tests on finding the least common multiple you're going to use a different procedure and basically what you're going to do is you're going to use the prime factorization of each number to build your LCM so we're going to cover this right now so finding the LCM from each numbers prime factorization so the first thing you're going to do is you're going to factor each number completely then the next thing we want to do is list the prime factors of each number in a table and this isn't completely necessary it's just a way to kind of organize your numbers and see what you're going to need the next thing you're going to do is generate a list that contains each prime factor but I'm going to tell you that this is where people get tripped up at although you need each prime factor when a prime factor is repeated between two or more numbers include only the largest number of repeats from any of the prime factorizations now this step right here again is going to cause a lot of confusion especially if you've never found the LCM before so just write this down and as we practice I'm going to explain what this means now the last step is to just generate your LCM and the LCM is just going to be the product of the numbers on the list that we have created all right so let's look at some practice problems we want to find the LCM for each group of numbers we're going to start with a pretty easy one we want to find the LCM for 9 and 12. so the first thing we want to do is Factor each number completely at this point you probably don't need a factor tree to factor 9 or 12. so 9 we know is 3 times 3 and 12 we know is 4 times 3 and 4 is 2 times 2. so now what we're going to do is make a little table to organize our factors just going to make it nice and clear for us to see what we have in common so I'm going to make a nice little table I'm going to put 9 here and 12 here and when you make your table you're looking for the smallest prime factor between any of the numbers so the smallest prime factor is a 2 that occurs in 12. I have two factors of two so let's start out by doing that and then if I look in 9 I don't have any twos so I'm just going to put like a little space here to say hey there's no there's no twos in nine I'm just going to kind of x that out to say they do not share that now the next thing I'm going to do is look for my next smallest factor which is a 3. in 12 I have one of those and nine I have two of those so I'm going to put a little space here to say they don't share a second copy of three although they do share one copy of three so let me just highlight that now here's where it gets kind of tricky for the LCM of 9 and 12. each and every prime factor that occurs between these numbers has to go in that list so in other words we have one two and then a second two that occurs in the prime factorization of 12 so that has to go in we have 1 3 and then a second three that occurs in the prime factorization of nine so that has to go in the tricky part is not including this three here that's in the prime factorization of 12. you might say why not if I put a third three in here I would get a common multiple for 9 and 12 but I wouldn't get the least common multiple when there's a prime factor that is shared you only use the largest number of repeats so in other words when we look at this we see that we have three that's shared between 9 and 12. once I see that I only put the largest number of repeats that occurs between the two prime factorizations and the prime factorization of nine I have two factors and the prime factorization of 12 I have one factor so the largest number of repeats would be two factors so when I build the list for the LCM it only gets two factors of three not three factors of three okay that's very important that you understand that once you have that down you simply multiply the numbers on the list two times two is four four times three is twelve twelve times three is thirty-six and we have our LCM right the LCM of 9 and 12 is simply going to be 36. and if you wanted to verify that you can go through and kind of list out the multiples of nine so it would be nine and 18 then 27 then 36 then 45 then 54. you know so on and so forth and then for 12 you'd have 12 24 36 48 60 again so on and so forth you can immediately see that 36 is the least common multiple all right 36 is the least common multiple all right let's take a look at another easy one so we want the least common multiple for 8 and then 42. so 8 we know at this point is three factors of two 42 let's just use a factor tree for that I know it's 21 times 2. 2 is prime 21 is 7 times 3 7 and 3 are prime so 42 is going to be 2 times 3 times 7. all right so let's build a little table just to organize these numbers all right so my smallest prime factor is a 2. so when eight I have three of those guys so two times two times two and in 42 I just have one so I'm going to put a blank space there and there and then in 42 I have a 3 . I don't have a 3 in the prime factorization of 8 and then I have a 7. I don't have a 7 in the prime factorization of eight either so I'm going to put some x's here to say that hey these aren't included and we can look to see what's in common and basically they just have one factor of two that's in common right so when we build our LCM when we build our LCM for 8 and 42 we're going to again include every prime factor that occurs between the prime factorizations don't think about two for a second 42 has a 3 and a 7. so I'm going to put 3 times 7. there's no 3 and 8 and there's no seven and eight so I don't have to worry about how many I'm going to put in it's just going to be one occurrence for each number when we get to this number two again if it's something they share you only put the largest number of repeats that occurs between the two prime factorizations and the prime factorization of eight we have three factors of two and the prime factorization of 42 I only have one so the largest number of repeats would occur on the prime factorization of eight or I have three so that's what's going to go on my list so I'm going to put one two three factors of two when I'm building my LCM so now I just need to go through and multiply 2 times 2 is 4 4 times 2 is 8 8 times 3 is 24 and then 24 times 7 24 times 7 7 times 4 is 28. 7 times 2 is 14 plus 2 is 16. so this would be a 168 for my answer all right let's take a look at the LCM of 20 and 35. so 20 is going to factor to 10 times 2 2 is prime 10 is 5 times 2 2 and 5 are both Prime 35 is simply 5 times 7 those are each prime numbers so let's go ahead and just make a table for 20 . I've got 2 times 2 times 5. 2 times 2 times 5. for 35 I don't have any twos so I'm just going to X those out I do have a 5. and then I have a 7 which I don't have up here so I can see that they just have one five in common between the two of them so to build my LCM for 20 and 35 again every prime factor between the prime factorizations of the number has to go in so 2 goes in then another 2 goes in it's just when you have something that's shared you put the largest number of repeats between the two prime factorizations now the largest number of repeats is going to be one I have one five in this prime factorization and 1 5 in this prime factorization so when I build this I only put one five in my list and I don't need two just one right the largest number of repeats when something is shared and then I need to put this 7 in so now everything is accounted for I've got one two I've got a second two I've got only one five because they share it and the largest number of repeats is one and then I've got a seven so now I can just multiply two times two is four four times five is twenty twenty times seven I can do as two times seven that's Fourteen and attach one trailing zero and I get one hundred forty as my answer all right let's look at the least common multiple for 16 18 and then 30. so we're going to factor each number so we've got 16 we could start out with 4 times 4 4 is 2 times 2. of course 2 is prime so we're going to circle all of those for 18 let's do 9 times 2 is Prime and 9 is 3 times 3. 3 is prime for 30 let's do six times five five is prime 6 is then 3 times 2 3 and 2 are both Prime so let's make ourselves a little table so we have 16 we have 18 and we have 30. so our smallest prime factor in any of the prime factorizations is 2. and in 16 I'm going to have four factors of 2. so 2 times 2 times 2 times 2. now in 18 I only have one factor of 2 so I'm going to put a 2 here and then some blank spaces here and let me just X those guys out then in the prime factorization of 30 I only have one factor of 2. and let me put some blank spaces there and X this out so the next smallest prime factor is going to be a 3. so in 18 there's two of those and then in 30 there's only one so I'm going to put a blank space here and here and here and let me just x that out and then the next highest is going to be a 5 and that's only in the prime factorization of 30. so I'm going to space that out and that out let's put an X there and there and let's go down and build our LCM now so the LCM for 16 18 and 30. again we're going to include every prime factor from all of these prime factorizations we're just going to have a special rule when any of them are shared so in this example they share one factor of two everything has at least one factor of two so when that occurs we put the largest number of repeats between any of the prime factorizations 16 has four factors of two so that's what's going to go on our list we're going to put one two three four again don't make the mistake of putting six in there don't say okay I have one two three four five six when there's something that's common you want to put the largest number of repeats that occurs in any of the prime factorizations now the next number we're going to look at is three and you'll notice that 3 is shared between these two prime factorizations so since 3 is shared between these two prime factorizations I want the largest number of repeats in any of the prime factorizations and in this case it's going to be two that occurs in the prime factorization of 18 so that means that we're going to have two factors of 3 when we build our LCM and then the last number is five it only occurs in the prime factorization of 30 so we just throw that guy in there we just need to multiply two times two is four times two is eight times two is sixteen times three is going to be 48 times 3 again is 144 then multiply by 5 you'll get 720. now I don't want you to get confused on this problem some of you might say okay well when we did this one they didn't all have a factor of 3 in it that doesn't matter if you're working with three numbers or four numbers or five numbers if the prime factor is shared with at least two prime factorizations you're always going to go with the largest number of repeats that occurs in any of those prime factorizations in this case we had two factors of 3 that occurred in the prime factorization of 18 that's more than the one that occurred in the prime factorization of 30 so we put two factors of 3 when we built our LCM and again we get 720 as our answer all right let's take a look at one final problem we want to find the LCM for 120 40 and then 200 25 so these numbers are kind of big but not really so 120 it ends with a zero so we know we could do 12 times 10 12 we could do 4 times 3 3 is Prime 4 is 2 times 2 2 is prime 10 is 5 times 2 5 and 2 are both Prime 440 we'll do four times ten or is 2 times 2 2 is prime so we're going to circle those 10 is 5 times 2 5 and 2 are both Prime then for 225 that's 15 times 15 15 is 5 times 3 5 and 3 are both Prime let's go ahead and build our table now so for this I'm going to put 40 first then 120 then 225. so the smallest prime factor that occurs is 2. in 40 you've got three of them one two three in 225 you don't have any and in 120 you have three as well one two three so for 225 I'm going to space these out and just put some x's in here okay so the next smallest prime factor is going to be a 3. and 40 we don't have any in 120 we have one and at 225 we have two so I'm going to space out here and here and here let's put some x's here the next smallest prime factor is going to be 5. in 40 we have one in 120 we have one and in 225 we have two so I'm going to space these out and put an X here and here so now when we build our LCM so the LCM 440 120 and 225 again we're going to include in this list each and every prime factor from all of the prime factorizations if something's repeated between at least two of the prime factorizations we're going with the largest number of repeats for that prime number so when I start out I notice that 40 and 120 both have three factors of two in their prime factorization I always want to go with the largest number of repeats now the largest number of repeats is just three right three factors of two they each have the exact same amount so when I build my LCM I'm going to have 2 times 2 times 2. now the next number that we come across is three now three is shared between 120 and 225 but there's more factors of 3 and the prime factorization of 225. you have two factors of 3 here only one here again when something shared you go with the largest number of repeats so the largest number of repeats would be two that again occurs on the prime factorization of 225 so I'm going to put two factors of 3 when I build my LCM now lastly we have five now everything has at least one five in it so again if it's something that's shared you go with the largest number of repeats the largest number of repeats occurs and the prime factorization of 225 there's two fives there so I put two fives in when I build my LCM and now we just need to multiply 2 times 2 times 2 is 8 times 3 is 24 times 3 is 72 times 5 is going to be 360 and then times five one more time is going to be one thousand eight hundred and again that's going to be our LCM for 40. 120 and 225. hello and welcome to pre-algebra lesson 33 in this video we're going to learn about adding and subtracting fractions so the lesson objectives for today we want to learn how to add and subtract fractions with a common denominator we want to learn how to find the LCD which is the least common denominator of two or more fractions and then we want to learn how to add and subtract fractions without a common denominator all right so let me read this before we get started so when we add or subtract fractions we must meaning it is not optional have a common denominator so when they're exactly the same that's what we mean by a common denominator all right let's just start off with some easier examples where we're already given a common denominator and then later on we'll deal with the more complicated scenario where we don't have a common denominator and we need to get one so let's say you have the problem one-fifth plus two-fifths well I want you to notice that the denominator here is five and the denominator here is also five so we have a common denominator meaning the denominators are exactly the same so that means that I can immediately start and I can add these fractions together but how exactly do I add one-fifth and two-fifths together well let's think about that with this little pizza here let's say that I buy a pizza and I split it up into five equal slices so let's say this is one two three four and then five so I eat one slice of pizza so let me Mark this out so this is the slice that I'm going to eat now we know at this point that we could represent the amount of pizza that I'm going to eat with this fraction one-fifth I'm taking one part out of a total of five equal parts in the terminology G of a pizza I could say I'm getting one slice out of a total of five equal slices let's say that a friend comes over and now he's gonna get two slices out of a total of five equal slices or two-fifths of the pizza so let's say he's going to take let me change my color up a little bit let's say he takes this one and he takes this one so how much pizza did we eat together well that's the addition problem one-fifth plus two-fifths so looking at this easily I can just count and say it's one two three out of a total of five pieces of pizza so that would be the fraction three fifths three slices out of a total of five equal slices how could we have gotten this three-fifths well what we would want to do is have this one-fifth plus two-fifths be equal to we would add the numerators together so my slice plus my friends two slices so that one plus two that's gonna give me the three slices we ate now notice this part right here is not going to change this denominator is going to stay as a 5. I'm not going to add those two denominators together I have a common denominator so I'm just going to leave that alone and it's just going to keep moving forward as I simplify this problem so I start off with one-fifth plus two-fifths this equals one plus two so this numerator plus this numerator over the common denominator of 5 and then it ends up giving me three fifths now when you get your answer you want to think about can I simplify in this case 3 is a prime number and 5 is a prime number so really we can't do anything to simplify So my answer here would just be three fifths let's just blow through a few of these problems if you have a common denominator it's going to be very very easy so we have 3 8 minus 2 8 so again we have a common denominator so all you have to do is perform the operations with the numerators and place that over the common denominator when you're done you're just going to simplify so I'm just going to say this is 3 minus 2 over the common denominator of 8 3 minus 2 is 1 and this is over the common denominator of 8. so this gives me 1 8. now can I simplify no so we'll just go ahead and leave 1 8 as the final answer what about 7 over 13 plus 4 over 13. again what I want to do is add the numerator so 7 plus 4 place that over the common denominator I have a 13 and a 13 so this will stay as a 13. so this becomes what 7 plus 4 is 11. so this is 11 over 13. 11 and 13 are both prime numbers so the greatest common factor here would be one so this is simplified 11 over 13 is your final answer all right let's look at 3 4 plus 1 4. so again just perform the operations with the numerators we're going to go three plus one over the common denominator here is four so this becomes three plus one which is four over four now of course we can simplify because 4 over 4 is 1. any non-zero number divided by itself is always going to be one you could think about this with a pizza if you had a whole pizza and you split it up into four equal slices and you still had all four slices well then you still have one pizza right so four out of four slices would give you one pizza what about three over fifteen plus two over fifteen so this would be 3 plus 2 just add the numerators over the common denominator of 15. so 3 plus 2 is 5. so this is 5 over 15. you could rewrite this let me just do this down here you could rewrite this denominator as 5 times 3 and then you can cancel 5 divided by 5 is 1. remember if your numerator cancels completely you want to write a one a common mistake there is to write a zero so 5 divided by 5 is 1. it's almost like we did the problem like this let's say I did five over five times one over three right if we did the multiplication this would be 5 over 15 which is what we have so now if I cancel this divided by this is one and so basically you have 1 times 1 3 which is one third so that's why we get 1 3 as the simplified answer okay let's look at 7 over 20 minus three over twenty so this would be 7 minus three again just working with the numerators over the common denominator of twenty seven minus 3 is 4 so this would be four over twenty and if I think about 20 it's 5 times 4. so let's go ahead and say this is 4 over 5 times 4. so I'm just going to cancel this with this let me actually use a different color there so it shows up better so cancel this with this and then what I'll do is put this as a one again you do not want to put a 0 there that's very common when you mark this out to see a student leave that and then say okay this is 0 over 5. no this is going to be a one four over four is one so this would be one over five or one fifth okay let's take a look at one third minus two-thirds so we're going to subtract 1 minus two over the common denominator of three so one minus two is actually going to give me a negative so if we think about that one minus two again you can always convert this if you need to you can say this is one plus a negative two and then just use your rules for adding integers so this ends up giving me a negative because the larger absolute value is negative and then you would basically do larger apps value mine is smaller so 2 minus 1 would be one so this would end up giving me negative one a lot of you know that one minus two you have one you take away two you get negative one okay let me get rid of this so this would be negative one over three now it's important to realize that when you have fractions if you have negative one over three this is equal to one over negative three which is equal to the negative of one over three so three different ways to write that a negative divided by a positive is going to be equal to a positive divided by a negative which is equal to just the negative of that guy okay so three different ways to write that it's actually really important that you understand that and we're going to use that in an example coming up okay let's look at this one so we have 31 over 50 minus six over fifty so we're going to do 31 minus 6 over 50. so 31 minus six is going to be 25. so you have 25 over 50 and we know that we could write 50 as 25 times 2. so this would become 25 over 25 times 2. let's go ahead and cancel this 25 with this 25 so again that's going to be a 1 and this will be equal to 1 half okay let's look at one more of these so I'm just going to think about what happens when we have some negatives already so we have negative 1 over 8 plus 5 over negative 8. so here's what I would do notice that you have an 8 and then a negative 8. you can legally move this up here like we talked about a little while ago so what I would do is rewrite this problem as negative 1 over 8 plus negative 5 over 8. that is completely legal again 5 over negative 8 positive over negative is equal to negative over positive you have to have exactly the same denominators so you need eight and then eight you could if you wanted to do negative 8 and negative eight but I think it makes a little bit more complicated to understand so I'm not going to do it that way when you look at it here you have 8 and negative 8. so those are not exactly the same so you need to do something like this here in order to do the addition so now what I would do is say this is negative 1 plus a negative 5 over the common denominator of eight and so here I'm just going to put that this is going to be negative add the absolute values 1 plus 5 is going to be 6. so this is negative 6 over 8. now we're not done because we can simplify I think I grabbed too much there so let me just move this over here and I'll just get rid of that and draw that back in So with this one what I would do is write negative 1 and then times two times three and then for 8 let's write 2 times 2 times 2 and I'm going to cancel a common factor of 2 between numerator and denominator so let's go ahead and do this one with this one you could write a one but it's kind of unnecessary here because the numerator is still going to have something it's really necessary when the numerator is completely canceled out and that's where you're tempted to write a zero in this case you won't have a temptation to do that so I'm going to go ahead and say that the numerator would be negative 1 times 3 or negative 3 and this is over 2 times 2 which is 4. so this would be negative 3 4 so you could write it like this you could write 3 over negative 4 or a lot of people just put the negative out in front so you say negative three-fourths whatever you want to do it's all the same answer all right now let's talk about the more challenging procedure so we're going to do adding and subtracting fractions without so let me highlight that a common denominator so the first step is to find the LCD so this stands for least common denominator this is going to be the LCM which is the least common multiple of the denominators so this right here causes a lot of confusion if someone asks you to find the LCD they are just asking you to find the LCM of the denominators that's all it is then I have here for the next step to write equivalent fractions with the LCD as each denominator I'm going to show you how to do that momentarily then we're going to add and subtract the numerators and simplify so these two steps right here we already know how to do that because we just did a ton of exercises where we already had a common denominator and we would just add or subtract the numerators place that over the common denominator and then just simplify all we need to do is figure out how do we get the LCD and then how do we write equivalent fractions with the LCD as each denominator all right let's jump in and just look at a problem when we get done with this problem I'll go back and show you why this makes sense using some pizzas again so let's start off by finding the LCD so the LCD is the LCM of the denominators I know this is very confusing the denominators here you have three and four so you have three and four so forget about the fact that we said LCD if I asked you for the LCM or the least common multiple of three and four what would you do you would first say okay well 3 is a prime number so I'm not going to factor that and 4 is 2 times 2. well everything has to go in so 3 has to go in and then two factors of 2 those are going to have to go in there's nothing that's common between the factorizations so basically it's just 3 times 4 right it's going to give you 12. so I'm going to erase this and say that this is going to be 12. the next part is a little bit tricky so basically you have one third and you have one-fourth and we've said that the LCD which is the LCM of the two denominators is 12. so what I want to do is I want to rewrite each fraction as an equivalent fraction where 12 is the denominator so how can we do that well I could multiply the denominator here by 4 and I could multiply the denominator here by 3. let me put my addition sign in but is that legal to do that well not if I just do it like this what I would have to do is match this number from the denominator in the numerator so I would do 4 over 4 here and then 3 over 3 here the reason this is legal is because 4 over 4 is going to be 1 right so this equals 1. any non-zero number divided by itself is going to be one and if I multiply by 1 I don't change the number so it's just like I said 1 3 times 1 well that's one-third I am not changing the number I'm just changing what it looks like temporarily so I can do my addition okay so let me get rid of this and we're going to first multiply so remember we're multiplying 1 times 4 numerator times numerator that's 4 and this is over 3 times 4 that's going to be 12. so denominator times the denominator so plus 1 times 3 so that's going to be 3 over 4 times 3 that's going to be 12. so at this point I have a common denominator I have 12 and I have 12. so it's just like the problems we did earlier so you're going to add your numerators so 4 plus 3. put that over the common denominator that's 12 and so 4 plus 3 is 7 and this is over 12. so it's really that simple the only two additional steps you need to do is to First find the LCD which again is the least common multiple or LCM of the denominators so three and four you're looking for the LCM of those two numbers you get 12 then you're going to transform each fraction into an equivalent fraction where the LCD is its denominator so think about what do I need to multiply each denominator by to get to a denominator of the LCD and then you got to match that in the numerator and the denominator so that your multiple applying by one so that it's legal so once you've done that you're at the point where we were at the beginning of the lesson where we had a common denominator and then you could just add or subtract your numerators put that over the common denominator and then simplify all right let me quickly explain why one-third plus one-fourth ends up becoming 4 12 plus 3 12 which gives us 7 12 in the end so I'm going to use an example with pizzas I think it works out kind of nicely so let's assume that I buy two pizzas that are completely identical and when I first get them they're not sliced up so I take the first one the one on the left and I cut it up into three equal parts so you could say I slice it up into three equal slices so one two and then three now if I eat one piece out of the three equal pieces here well I've eaten one third of the pizza so let's say I've eaten this right here so that is going to be represented with a fraction one third then let's say this pizza over here I'm going to cut it up differently so let's say I go one two three four equal parts or four equal slices since it's a pizza and let's say that my friend he's gonna eat one slice here so he eats this slice right here so let's say plus one fourth well if we try to figure out how much pizza we ate we can't just immediately do this problem here we know that we didn't eat a full Pizza even though there's two pizzas here the amount of pizza I ate plus the amount of pizza he ate it's not even gonna add up to one pizza but we can't really figure out immediately what amount of a pizza we ate because if I try to add this one to this one well those are not the same right so this slice is not the same size as this slice because the pizzas are caught up differently obviously looking at the fact that these two pieces are the same size well this one third of the pizza is going to be more than this one fourth of the pizza so what can we do we would have the slice the pizza up in such a way that we can add like parts of a whole amount so what do I mean by that let's start again with this one-third and then plus this one fourth so you see here now I have 12 slices so one two three four five six seven eight nine ten eleven twelve now I don't have two pizzas here because between the amounts that me and my friend ate it's not even going to be a full pizza so I'm just transferring everything in terms of one pizza so nobody gets confused about having two pizzas so what I'm gonna do now is think about okay if I cut this up into 12 pieces here because that's the LCM of three and four how can I transform this amount of pizza to match over here want to multiply this by four over four like we did earlier so that would end up giving me 4 over 12. so if I go through and look at it well I would have eaten one two three four pieces of pizza had it been split up in such a way that I had 12 equal pieces of pizza and you can see that this would be one third right so you go one two three four that's a third then five six seven eight that's another third then nine ten eleven twelve that's another third so I still ate a third of the pizza it just looks different then plus over here I'd multiply this by three over three and so this would give me three over twelve so let's say that I take five six and seven for that and I'll use a different color so my friend he eats five six and seven so now we can clearly see just by counting it's one two three four five six seven that it's going to be seven pieces of pizza that were eaten out of a total of 12 pieces we could have also just done four plus three which again gives us seven over twelve all right let's look at some examples now so we have one fourth plus one half so what is the LCD the LCD is the LCM of the denominators so 4 and 2. so I think you know that that's going to be four again four is going to be 2 times 2 and 2 is a prime number it's not going to factor so basically you want two factors of two you could think about this as one two and here you have two twos so you want the largest number of repeats and that's going to be two twos or four let me get rid of this and basically in this particular case the one-fourth you don't have to do anything with that because the denominator is already the LCD which is four so then Plus for one half you're going to need to multiply this by two over two and so this gives you 1 4 plus 1 times 2 is 2 over 2 times 2 that's 4. so this gives you 1 plus 2 which is three over the common denominator of four so three fourths would be your simplified answer three is a prime number and four is two times two so your GCF is going to be one there so really that's your simplified answer all right let's look at five over twenty one plus three over fourteen so this one will be a little bit of work so the LCD is the LCM of 21 and 14. and so 14 is 2 times 7 and then 21 is going to be 3 times 7. so when you build this it's going to be what you're going to put a 2 in you're going to put a 3 in and then you only put in 1 7. there's 1 7 here and 1 7 here so if it's common just put the largest number of repeats between any of the factorizations it happens once a year and once a year so only one goes in and let's put equals here 2 times 3 is 6 6 times 7 is 42. so here we would say that we have 5 over 21 and then times let me just put this here plus 3 over 14 and then times let me put this here when you think about this you can look at this list and say well what do I need to get to 42 well basically you have 7 times 3 I would need a 2 right so you can say 2 over 2. also you can take this and divide by this so 42 divided by 21 is 2 whatever you want to do sometimes people will erase this and then put this as 42 and they've lost that information so they start freaking out just so you can take this and just divide it by this and 42 divided by 21 would be 2 so I need to multiply by 2 so that I can get to 42. then over here I would need 3 over 3. 14 times 3 would be 42 so this would be equal to 1. 5 times 2 is 10 over we know that the denominators are 42. so you could just write them in and then 3 times 3 is 9 and this is over 42. so now 10 plus 9 would be 19 over the common denominator of 42. so 19 is a prime number and we know 42 is 2 times 3 times 7 we just factored it so the GCF here is one so this is as simple as you can get it 19 over 42 is your answer all right let's look at one that's a little bit tricky so we have negative 7 over 3 plus 2 over negative 15. the idea here is that if you have a negative in the numerator or the denominator what I would say is to put them in the numerator okay so here it's in the numerator I'm just going to move this up here so I'm going to say this is negative 7 over 3 and then plus negative 2 over 15. you don't want to have opposite signs in your denominators you want to make sure that when you get a common denominator they're exactly the same they can't be opposites so you wouldn't want to end up with 15 and negative 15. here when you think about the LCD which is the LCM of 3 and 15 well 3 is a prime number and 15 is 3 times 5. so basically the 3 goes in and the 5 goes in the 3 only goes in once because you have it here and you have it here so you're only putting in one so it's just going to be 15. so this would just be let me erase this and I'll just put that the LCD is equal to 15. we can say this equals we'll have negative 7 over 3. I'm going to multiply this by 5 over 5 because 3 times 5 is 15 and then plus you have negative 2 over 15. so I don't need to mess with this one because the denominator is 15 and that's the LCD so don't do anything there so I'll say this equals negative 7 times 5 is negative 35. this is over 3 times 5 is 15 and then plus you have negative 2 over 15. let me slide down a little bit here so this equals what you would just add the numerators negative 35 plus negative 2. we know that's negative and then add the absolute values 35 Plus 2 is 37 so this is negative 37 over 15. now you could write this as a mixed number if you want it's really up to you because you have an improper fraction here so you could leave it like this 37 is a prime number so you're not going to be able to simplify but you could say that this is negative what so you could say 37 divided by 15 15 goes into 37 twice 2 times 15 is 30. subtract and you get 7. so that's your remainder so you would say that this is negative 2 and then 7 over 15. okay so you could write it like that if you wanted to or you could keep it like this negative 37 over 15. really it's personal preference unless your teacher tells you which one they want all right let's take a look at 11 over 12 minus 3 over 20. so the LCD here is the LCM of 12 and 20. so 12 is going to be 4 times 3 so 2 times 2 times 3 and then 20 is 4 times 5. so 2 times 2 times 5. okay so what I'm going to do here is think about the fact that I have 2 that's common so you want the largest number of repeats and that's going to be two of them so you want 2 times 2 which is 4 and then times you're going to put in a 3 and then times 5. so 4 times 3 is 12. 12 times 5 is 60. so this would end up giving you 60. okay let me put equals here and I'll just go down here so 11 over 12. so what do I need to multiply the denominator by to get to 60 well notice that you have 4 times 3 there that's 12. so you need another 5. so let's go ahead and put 5 here and then 5 up here and then minus 3 over 20. let me make this 3 a little bit better I'm going to multiply this by and that's not much better so let me do that one more time I'm going to multiply this by one so 20 times 3 would be 60. so 3 over 3. so let's say this equals and 11 times 5 is 55 over 12 times 5 is 60 and then minus 3 times 3 is 9 over 20 times 3 that's 60. and now we can just subtract so what is 55 minus 9 so maybe you want to do that as a vertical subtraction let's borrow here this would be four this would be 15. 15 minus 9 is 6. bring down the 4 there so this would be 46. so you could say that this is going to be 46 over 60. now I know that 46 and 60 are even numbers so at minimum I can cancel a common factor of 2. so 46 would be 2 times 23 and 23 is a prime number and then 60 is going to be 2 times 30. so I can cancel that common factor of 2 between numerator and denominator and so this gives me 23 over 30 as my final answer all right what about 14 over 15 minus 17 over 25. so the LCD is the LCM of 15 and 25. so 15 is going to be 3 times 5 and 25 is a perfect square it's 5 times 5. so this would be what 3 goes in and then for 5 you want the largest number of repeats because it's common here it's one here it's two so you're going to put in two of those guys so 3 times 5 times 5 would be 3 times 25 so that's 75. so let me actually erase this and just put 75 like this and let me get rid of this border and I'll put equals here and we'll just come down here so 14 over 15 what do I need to multiply 15 by to get to 75 well 5. so I'm going to do times 5 over 5 and then minus 17 over 25 what do I need to multiply 25 by to get to 75 well 3 so times 3 over 3. all right so 14 times 5 let me do that off to the side it's going to be 70 but let's do that off the side so 5 times 4 is 20. zero down carry the two five times one is five plus two is seven so this is 70. so let's say this is 70 over we know that again this is going to be 75. and then let's get rid of this now you have 17 times 3. let's do that here real quick so 3 times 7 is 21 one down carry the 2. 3 times 1 is 3 plus 2 is 5. so that would be 51. so let's go minus 51 and again we know this denominator is 75. so let's say this equals what I think we could do 70 minus 51 in our head that is going to end up giving me 19 which is a prime number over 75. so this would be simplified again 19 is a prime number we know it's 75 factors into it's going to be 3 times 5 times 5. so nothing you can really do there other than to say that 19 over 75 is your simplified answer all right let's take a look at one final problem so now we have more than two fractions involved so you see we have two-thirds plus one-fourth minus 5 8. so again let me just start by figuring out well what is the LCD this is the LCM of the denominators so 3 4 4 and then eight okay so 3 is a prime number so don't worry about that 4 is going to factor into 2 times 2 and then 8 is going to be 2 times 2 times 2. okay so basically between 4 and 8 you see that you have two factors of two here but three factors of 2 here so you're going with the largest number of repeats so I'm basically going with eight here and then I need the 3 there so it's going to be 8 times 3 which is 24. let me go ahead and just say that this is going to be equal to 24. okay let me write this down here because I'm going to need some room so the LCD is equal to 24. okay let me get rid of this and I'm just going to say that this is equal to 2 3 times what do I need to multiply 3 by to get to 24 well 8. so I'll multiply this by 8 over 8 and then plus we have 1 4 times what do I need to multiply 4 by to get to 24 well 6 so 6 over 6. this is probably not going to fit so let me move this down here let me actually rearrange everything and move this over here and I'll just drag this over here okay then minus we have five Ace times what do I need to multiply 8 by to get to 24 well 3. so this is 3 over 3. all right let me slide down here and just say equals so first let me just simplify everything 2 times 8 would be 16 over we know all the denominators would be 24 then plus 1 times 6 is 6 again over 24 and then minus 5 I'm sorry is 15 and then over 24. so when you have all these different operations going on again if it's addition and subtraction that's worked left to right so I'm just going to think about it like I have this 16 plus 6 minus 15 over the common denominator of 24. that's all I'm really doing so I'm going to add first and then I'm going to subtract last so 16 plus 6 well 16 plus 4 is 20 plus another 2 is 22. so this is 22 minus 15 over 24 and then 22 minus 15 well that's going to be 7. so this ends up giving me 7 over 24. now 7 is a prime number so you're not really going to be able to do anything we know 24 is 3 times 8 right or 2 cubed so you really can't do anything to simplify there it's just going to be 7 over 24 for the simplified answer hello and welcome to pre-algebra lesson 34. in this video we're going to learn about comparing fractions so the lesson objective for today is just to learn how to compare the size of two or more fractions alright so in order to compare the size of two or more fractions one thing we can do is we can convert all of the fractions into equivalent fractions with the same denominator so with this first procedure I'm going to give you you're basically going to figure out what is the LCD or the least common denominator you're going to transform each fraction into an equivalent fraction where the LCD is its denominator and then you're going to compare the numerators so given the fact that you have the same denominators the largest numerator will belong to the largest fraction this is one way to do it it's the slower way but we're going to use it to build up a concept as we get further along in the lesson I'll give you the shortcut which basically involves just cross multiplying all right we want to replace the question mark with either the less than symbol or the greater than symbol and we're going to start with something very easy just to get an understanding of the concept we have 3 8 and then a question mark and then five eighths so to think about this I have a little pizza and the pizza has been split up into eight equal slices so let's say that this is one two three four five six seven eight so let's say that we're at a party and basically I say that I'm gonna eat three slices of pizza so let me Mark these out so I'm gonna get this piece I'm gonna get this piece and I'm gonna get this piece so I've taken 3 8 of the pizza three slices out of a total of eight available and then you say okay I'm gonna go ahead and take five out of the eight slices and just finish the pizza off so let's say you take this one and then this one and this one and this one and then this one well who got more pizza well of course you did you got five slices out of eight whereas I only got three slices out of eight so when we have the same denominator it's very easy to compare which just think about what numerator is bigger 5 is bigger than 3 so we can say that 5 8 is a bigger fraction with this one the 3 8 is on the left so we would say 3 8 is less than 5 8. of course you could flip that around let me write that over here you could say that 5 8 is greater than 3 8 if you were presented with the problem in a different way let's look at one that's a little bit more tedious so now we have two fifths question mark three sevenths so it's not as straightforward here because the denominators are not the same so what we have to do is convert this over to where each fraction has the same denominator again as we get further along the lesson I'm going to give you a shortcut to do this very very quickly for right now let's just build up the concept so we have two fifths and then we have three sevenths so when we think about the least common denominator remember this is the least common multiple of the denominators so a five and then seven so this this is equal to what well 5 is a prime number and 7 is a prime number so just multiply those two together so this is 5 times 7 which is 35. so basically all I have to do is multiply this by 7 over 7 and multiply this by 5 over 5. and so that would give me 14 over 35 so 14 over 35 and then this would be 15 so 15 over 35. so let me actually move this over here like this and this like this so we know what we're talking about and basically now that we have the same denominator so this is 35 and this is 35 then I'm looking at which numerator is larger in this case 15 is bigger so I'm going to put a less than here so 14 over 35 is less than 15 over 35 in terms of the original problem remember two-fifths is now 14 over 35 so I would say two-fifths is less than 3 7 is now 15 over 35 so we'll say 3 7. so to answer this we'll say two-fifths is less than 3 7. all right let's take a look at 7 over 20 and then a question mark 5 over 12. so let me actually write this so 7 over 20. I'm going to leave some space and then I'm going to write this one 5 over 12. the first thing is with this method we're going to find the else CD which is the least common multiple of these denominators for 20 and then 12. let's think about 24 seconds I don't think we need a factor tree 20 is 4 times 5 and 4 is 2 times 2. so this is 2 times 2 times 5. for 12 again I don't think we need a factor of three it's very simple it's 4 times 3 and 4 is 2 times 2. so let's say 2 times 2 times 3. now when you look at these prime factorizations and you're trying to build your LCM if there's something that's common in this case you have a 2 here and a 2 here you want to go with the largest number of repeats between any of the factorizations so here I have two factors of two and here I have two factors of two so the largest number of repeats would be two so I'm going to put in 2 times 2. the big mistake is everybody puts in 2 times 2 times 2 times 2 or they put in 16 there right you only need four then here I want to put in the 5 and then I want to put in the three so everything goes in so if something's common put in the largest number of repeats between all of the prime factorizations okay let me get rid of the scratch work and you can just multiply now so 2 times 2 is 4. 4 times 5 is 20 and 20 times 3 is going to be 60. so this would be 60. to figure out what I need to multiply by all you do is take this LCD here which is 60 divide by the denominator so 60 divided by 20 is 3 so I need to multiply by 3 over 3. and then over here I divide 60 by 12 and I get 5 so I need to multiply by 5 over 5. so this is going to end up becoming 7 times 3 is 21 so this would be 21 over 60. this 5 times 5 is 25 so this would be 25 over 60. so now it's very easy right we have the same denominator 60 and then 60 and I could just compare the numerators so 21 is less than 25 so we would say 21 over 60 is less than 25 over 60. again to put this in terms of the original problem that's how we want to answer so 7 over 20 is going to be less than 5 over 12. all right let's take a look at one that's pretty tedious especially if you're doing it the slow way in the future we're just going to cross multiply so it'll be a lot quicker but let's start out by finding the LCD which is the LCM of these denominators so 14 and then 34. so I would Factor 14. that's 2 times 7 and then I would Factor 34 so that is 2 times 17. so to build up the LCM you would basically put one factor of 2 in again it's common to both of these factorizations so you just put one end the largest number of repeats then times you have 7 and then times 17. you can get rid of this if you want and then basically you can go through and crank this out a lot of you are using calculators at this point if you're not just do 2 times 7 that's 14. 14 14 times 17 is going to be 238 you can always stop and do a quick vertical multiplication so 14 times 17 7 times 4 is going to be 28 8 down carry the two seven times one is seven plus two is nine so this is ninety eight let's erase this and shift down let me actually move this up here and then 1 times 14 is 14 so 4 and then one let's put some addition there bring down the eight and then 9 plus 4 is 13 3 down carry the one one plus one is two so it's 238. okay let's get rid of this now let me put this over here and say this is 5 over 14 times I need to get to 238 now notice that 14 is 2 times 7 so I would need to multiply by 17 to get to 238 in terms of the denominator so I'm going to multiply the numerator and denominator by 17. now you could have also said what is 238 divided by 14 and you get 17. so that's another way to do it and this guy over here let's do 15 over 34 and we're going to multiply by what do I need well 34 times 7 would be 238 so 7 over 7. so now that we have that figured out you can actually get rid of this if you want you don't need it anymore you would just do some multiplication we know that the denominator is going to end up being 238 in each case so let's put that up there what is 5 times 17 well some of you might know that that's 85 again if you don't you can always stop and do a little vertical multiplication five times seven is 35 5 down carry the three five times one is five plus three is eight so that's 85. okay so the other one is going to be 15 times 7 so 15 times 7. 7 times 5 is 35 5 down carry the three seven times one is seven plus three is ten so this would be 105. so if I think about the denominators they're the same so I just want to compare the numerators so 105 is larger so this guy right here this 85 over 238 is smaller or less than 105 over 238 so I'll go ahead and translate that into the original problem let me get rid of this so I have some rum and I will just say that 5 over 14 is less than 15 over 34. all right let's take a look at an example where we have the same numerators but different denominators so we have two-thirds question mark two-fifths so let me write this out so two-thirds and then two fifths let's start with the LCD method and then we'll show you a little shortcut for this one so we have the LCD is going to be the LCM of the denominators so three and then five so because three and five are both prime numbers we can just multiply them together 3 times 5 would be 15. so I'd multiply this by five over five and I'd multiply this by three over three let me get rid of this scratch work here and so this 2 times 5 would be ten over we know the denominator is fifteen here two times three is six we know the denominator is 15. let me get rid of this and get rid of this so given the fact that we have the same denominators now we could just compare the numerators so the larger numerator will belong to the larger fraction so 10 is larger than 6 so we can say that 10 15 is greater than 6 15 which is going to translate into two-thirds being greater than two-fifths now we could have found that in a much quicker way in fact if we have the same numerators all we have to do is look at the denominators and see which one is smaller so whichever one is smaller belongs to the larger fraction so I know that's a little bit tricky the first time you hear it when you have the same numerators you want to look at the denominators and see which one is smaller so the smaller denominator will belong to the larger fraction so here 3 is smaller than five so two-thirds is greater than two-fifths let me make this simple and give you a visual example with some pizzas so let's say that you go to the pizza shop and you buy two identical pizzas you get back to the house and you split them up in different ways with the first pizza or the one on the left you're going to split it up into three equal pieces so one two and then three with the one on the right you're gonna split it up into five equal pieces so one two three four five so let's say this one's for you and this one's Forefront so let's say you decide that you want to eat two pieces of pizza so maybe you eat this piece right here and then this piece right here so you've eaten two thirds of the pizza now when you think about your friend he also wants to have two pieces of pizza so let's say he eats this piece of pizza and then this piece of pizza okay so I'm gonna represent that with two fifths so who ate more pizza given the fact that there were identical pizzas before you slice them up well you did right you had two thirds of the pizza and he had two fifths in other words your two is going to be a larger amount than his two because we're talking about the slices themselves being different right so your slices are larger so when you eat two larger slices that's going to be more than your friend who's eating two smaller slices okay so you're thinking about how the pizzas are split up so that tells me that two-thirds is going to be greater than two fifths all right let's take a look at two problems where we're asked to arrange from least to greatest once we're done with that I'll show you the shortcut and we'll be able to move through these problems more quickly all right so we have five over six eleven over twelve and then three over twenty so we are asked to arrange these from least to greatest sometimes they want you to put in commas like you see here so put the smallest then comma the next smallest then comma you would have the largest because here we have three and then sometimes they want you to use inequality symbols so I'll just give you both ways in case you are asked for one of those on your test so first and foremost I would get the LCD which is the LCM of the denominators so 6 12 and then 20. okay so this equals what for 6 this factors in a 2 times 3 for 12 this factors into 4 times 3 so 4 is 2 times 2. and then 20 is going to be 4 which is 2 times 2 and then times 5. so when I build my LCM again if something is common you want to think about the largest number of repeats so here for 2 it occurs once here twice a year and twice a year so the largest number of repeats is two so I'm putting two factors of 2 in when I build my LCM then I'm looking at three so it occurs once here and once here so I'm putting in one and then I'm looking at five it only occurs once here so I'm going to throw that in 2 times 2 is 4. 4 times 3 is 12 12 times 5 is 60. so this would be 60. so I'm going to say 5 over 6 times in order to get a denominator of 60 I need to multiply by ten so I want times 10 over 10 and then 11 over 12. if I want a denominator of 60 I need to multiply by 5 over 5. and then 3 over 20 if I want a denominator of 60 I'm going to multiply by 3 over 3. okay let me get rid of this we know that in each case the denominator would be 60. so let me just write that in real quick and then we'll do our multiplication so 5 times 10 is 50 then 11 times 5 is 55 and then 3 times 3 is 9. okay let me get rid of this so thinking about how we would order this the denominators are the same so you have 60 you have 60 and you have 60. so I'm just thinking about in terms of the smallest fraction I would have the smallest numerator so that would be 9 so that belongs to 9 over 60. so let's say that this is 9 over 60. and then comma the next smallest I'm looking for 50 here because 50 is smaller than 55 so I would go 50 over 60. and then the largest is going to be the 55 over 60. so if I want to write this in terms of what we had originally which is always what you want to do well if you think about 9 over 60 again that's going to be 3 over 20. so let's list that first so this is 3 over 20. I'm going to put a comma 50 over 60 is going to be 5 over 6. so this is 5 over 6. put a comma and then 55 over 60 is 11 over 12. so 11 over 12. so if you're asked to do this with commas in a range from least to greatest this is how you would do it so 3 over 20 comma 5 over 6 comma 11 over 12. they may ask you to arrange them from greatest to least sometimes they do that so then you would put 11 over 12 comma 5 over 6 comma 3 over 20. a lot of times they'll also ask you to do this with inequality symbols so generally speaking you'll do this from least to greatest but you can also see it from greatest to least so I I would say that this is 3 over 20 is less than 5 over 6 Which is less than 11 over 12. all right let's look at one more of these so this one's pretty simple we have two fifths two-thirds two sevenths and then one-sixth with this one I would not find the LCD it's going to be a little bit faster since we have the numerators two two two and then a one I can just say this is times two over two which is going to give me 2 over 12. so let me go ahead and say I'm just going to temporarily line this out I'll bring it back in a moment if this is 2 over 12. okay so given the fact that we have the same numerators now the smallest fraction is going to have the largest denominator remember if you have the same numerators then the largest fraction is going to have the smallest denominator so that means the smallest fraction is going to have the largest denominator so if I think about this the largest denominator is 12 so that means 2 over 12 or 1 over 6. let me just bring it back that's going to be the smallest so let me say that this would be 1 over 6 and then comma the next largest would be 7. so I'm going to put 2 over 7 and then comma the next largest would be 5 so I'm going to put 2 over 5 and then comma lastly the smallest denominator here is three so I'm going to use two-thirds as the largest so again when I think about these I've already transformed this over let me temporarily put this back so we can think about so this is 2 over 12. going back to my pizza example if I think about having 2 thirds of a pizza I'm gonna eat more of that pizza than if I had two fifths of it here I'm eating two slices and here I'm eating two slices but again with the two-thirds example the slices are bigger with the two-fifths example the slices are smaller so that's why we would say that this one-sixth here which is 2 over 12 is going to be smaller than two over seven which is smaller than two over five which is smaller than two thirds so if you want to use inequalities we'll say 1 6 is less than two sevenths Which is less than two-fifths Which is less than two thirds all right so now let's get into the shortcut so we can determine which fraction is larger by cross multiplying and comparing the results so now we're back to replace the question mark with the less than symbol or the greater than symbol all right so let's start off with this example here so we have four ninths question mark five twelfths so I'm going to build you up to the shortcut we're going to start with what we know so we have that the LCD is going to be the LCM of these denominators so 9 and then 12. what is 9 it's 3 times 3 what is 12 it's 3 times 4 and 4 is 2 times 2. so 2 times 2 times three so when you build this again you look at this and you say okay well there's going to be two factors of two so two factors of two but then I only need two factors of three right because there's two factors of three here and one here go with the largest number of repeats between any of those prime factorizations so I'm going to put in two factors of three get rid of this we know that that's basically four times nine which is 36. so let's put this as 36. so at this point we know using our slower procedure we would say four over nine and then times what do I need to multiply nine by to get the 36 well four so I would do times four over four and then over here I have 5 over 12. what do I need to multiply 12 by to get to 36 well 3 so times three over three and so this becomes 16 over 36 and then this becomes 15 over 36 and now that we have a common denominator we can just compare the numerators so 16 is greater than 15 so you'd say 16 over 36 is greater than 15 over 36 which gives me 4 9 is greater than 5 12. Okay so we've been doing it that way it's a little bit slower than what I'm going to show you now so let me erase this and let's talk about the shortcut in every case you can just multiply the two denominators together to get a common denominator it's not always going to be the least common denominator sometimes it will be it just depends but you can always just get a common denominator by doing that so let's say I erase this and I erase this and I just say okay I'm not going to find the LCD here I'm just going to get a common denominator so I would multiply this by 12 over 12 and I would multiply this by 9 over 9. okay well 4 times 12 is 48 and then 5 times 9 is 45. it actually does not matter what the denominator is because I know it's going to be the same right because I'm multiplying the two denominators together it's just in a different order here it's 9 times 12 and here's 12 times 9. now 9 times 12 or 12 times 9 we know that's 108 so let's just go ahead and write that for completeness at this point I can just compare the numerators so 48 is larger than 45 so you would say 48 over 108 is greater than 45 over 108 okay so what can we learn from this process what I could have done let me get rid of this for a moment I could have just said well if I multiply those two denominators together they're going to be the same so I don't really care what they are I just need to figure out what is this numerator times the other denominator and what is this numerator times the other denominator and just compare those numbers to figure out which fraction is bigger so in other words I can say 4 9 and then 5 12 and I could do something called cross multiplying so take this denominator and multiply it by this numerator 9 times 5 is 45 and that's the exact same thing we got here 5 times 9 is going to be 45 and then you're going to take this denominator which is 12 and multiply it by this numerator which is 4 and that's going to give you 48. so 4 times 12 is 48 done this way so it's just a much quicker way to obtain which fraction is bigger I only need to think about the 48 and the 45. 48 is bigger so I'm going to go with 4 9 the fraction that that is next to as the bigger fraction so I can say that 4 9 is greater than 5 12 using this cross multiplying method okay let's blow through a few examples so we have two thirds and then you have a question mark and four fifths so again you're just going to cross multiply so 3 times 4 is 12 and then 5 times 2 is 10. so 12 is greater than 10 so you can say that 2 3 is smaller right so two-thirds is less than four fifths what about four-thirds question mark six-fifths so don't worry about the fact that you have improper fractions here it's not going to matter you can use the same strategy so let me write four thirds and then six fifths and again we're just going to cross multiply 3 times 6 is 18 then 5 times 4 is 20. so 20 is bigger so you would say four thirds is greater than six fifths let me give you one example with some negatives so these are a bit tricky because everything I said is going to now be reversed when you're thinking about a negative so when I think about negative three fourths and two over negative seven the first thing you want to do if you have negatives involved bring your negative here into the numerator and that's legal because a negative divided by a positive or A positive divided by a negative is a negative I could just as well put the negative out in front or I could put the negative in the numerator or the denominator it's all legal so let me put them both in the numerator and then I would cross multiply so 4 times negative 2 is negative 8 and 7 times negative 3 is negative 21. okay so here's where we've got to be really really careful when we think about a big or negative it is further to the left and it is therefore a smaller number so negative 21 is a big or negative and therefore a smaller number than negative eight so negative 21 would be next to the smaller fraction so you would say that negative 3 4 is less than negative 2 7. if you had gotten the problem three-fourths and then 2 7 notice that the answer would be flipped so you cross multiply here you have 8 and then you have 21 right so 21 is greater than 8. so here you would say that 3 4 is greater than two sevenths but because these guys are negative it's going to get reversed right because now 3 4 becomes negative three-fourths and that's going to be a bigger negative and therefore a smaller number than this negative 2 7. all right let's look at one more example where we arrange from least to greatest so we have 2 15 3 7 and 4 13. so what I do is just start with any two that you want it doesn't really matter so let's start with 2 15 and then three sevenths and just figure out the relationship so 15 times 3 is 45 and then 7 times 2 is going to be 14. so this one is obviously smaller so we'll say 2 15 is less than three sevenths okay so erase this and now you can take this one right here this three-sevenths and compare it to four thirteenths so I'm going to say three-sevenths and then four thirteenths you'll cross multiply so 7 times 4 is 28 and then 13 times 3 is 39. so 3 7 is bigger so I'm going to say that 3 7 is greater than 4 13. so we know that 2 15 is less than 3 7 we could also turn that around and say 3 7 is greater than 2 15. here we see that 3 7 is also greater than 4 13. so I know that 3 7 is actually the biggest number I only now need to compare these two numbers and see which one is larger so let's slide this down a little bit and cross multiply so 15 times 4 is 60 and then 13 times 2 is 26. you can say that 2 15 is going to be less than 4 13. so this would be the smallest number so let me change my color here so this is 2 15. I'll put a comma this one would be in the middle so it would be 4 13 and then the largest would be this three-sevenths if you want to write this using inequality symbols you would typically go from least to greatest in the order of the number line but you could reverse that if you want it's pretty much up to you in most cases you're going to see two fifteenths is less than 4 13 Which is less than three sevenths hello and welcome to pre-algebra lesson 35. in this video we're going to learn about ratios and rates okay so for the lesson objectives for today we want to learn how to write a ratio that describes a given situation and we also want to learn how to write a unit rate that describes a given situation so we're going to kind of just start out with a basic definition here we're going to say that a ratio is a comparison of two quantities with the same units and later on we're going to talk about rates and oh rate is basically a special ratio where the units are going to be different to kind of start off here we're going to use ratios a lot moving forward and they're actually something that will help you in your everyday life as an example let's say that you're at a school dance and at the school dance you talk to The Chaperone and they tell you that there are four boys for every two girls at this dance so four boys for every two girls at the dance well we can describe this relationship by writing a nice little ratio and there's three main ways we do this so if we want the relationship between boys and girls we can write boys to girls like this as a fraction and so boys is in the numerator so I put the number of boys in my numerator when I write the ratio girls is in the denominator so I'd write the number of girls in the denominator when I write the ratio and we'd say it differently right instead of saying 4 over 2 we would say four to two now if we flip this and said girls to boys this would be two to four right it's relative right the number of girls I have is 2 to the number of boys that I'm told that I have is four now there's more ways to write it but I want to show you something before I get into that when you have a ratio you always want to simplify it just like we did when we work with fractions so we know that 4 over 2 or 4 to 2 as a ratio is the same as 2 to 1. if I divide 4 by 2 I get 2 if I divide 2 by 2 I get 1. now one main difference is if I'm working with fractions and I have 2 over 1 I just write that as 2. when I'm working with a ratio I have this relationship between boys and girls so I leave the one there because I'm saying I have two boys two boys for every one girl one girl and then over here I could just rewrite this as one to two some other ways that we could write this ratio I could write boys to girls like this using a colon okay so this would be two to one right because I have two boys for every one girl and then I could also write it using the word two I could say boys two girls like this and I could say two two the word two not the number two one like that so all of these are ways that we can write a ratio most commonly I would say you're gonna see it this way or this way right as a fraction or with a colon this is sort of uncommon right sort of uncommon but still you may see it from time to time so you need to know that we're talking about a ratio when we see something like this so let's kind of continue with this example and I'll show you a common test question for ratios so remember we had four boys for every two girls and we said that this simplified to a ratio where it's boys to girls of two to one now your typical question would say something like if there were 42 kids at the dance how many were boys how many were girls so let's think about that so out of 42 kids we're gonna put boys question mark and then girls question mark how would we figure something like this out well the first thing I would do is I would take my two numbers and my simplified ratio and I would add them together so I have two boys to one girl so two plus one is three so that tells me in one group in one group I will have three kids and I will basically have two boys and I'll have one girl if I have 42 total kids essentially I'm saying how many equal groups of three can I make out of 42 well 42 divided by 3 is 14. so I'm essentially saying I have 14 of these groups so I'll multiply the numbers here by 14. so 2 times 14 would give me 28. so this tells me that for 42 kids I would have 28 boys and I would multiply 1 times 14 and get 14. so I'd have 14 girls and if this isn't clear for you let me just kind of make a little chart so we would have kids and then we would have boys and then we would have girls so if three kids attend we know that two are boys one's a girl and we're going to increase in increments of three here so if we go from 3 to 6 then we're going to have what we've doubled here so we have to double everything here now we're going to have four boys and two girls right all I did was multiply everything here by two two times two is four one times two is two three times two is six right everything has just been doubled well if I now go to 12 kids essentially I'm taking the original amount and just multiplying it by four right so I have four times the amount of kids there so instead of two boys multiplied by four I'd have eight instead of one girl multiply by four I'd have four so we could take this all the way up to let's say 42 and basically that's 14 times the original amount so I just multiply everything here by 14. 2 times 14 would be 28 so I'd have 28 Boys 1 times 14 would be 14 so I'd have 14 girls all right so let's take a look at a different problem so we have a mixture that contains 24 milliliters of alcohol for every eight milliliters of water and we want to State the ratio of alcohol to water so I'm just going to do this using a fraction but again if you want to use a colon you can do that as well so again the ratio of alcohol to water so alcohol to water notice that alcohol the first word or the word that occurs before the word two is in the numerator water the second word occurs in the denominator so I have 24 milliliters 24 milliliters two eight milliliters and again with a ratio you're going to have the same units so you don't really have to write them because you can kind of think of them as just canceling each other out so we could just write 24 over 8 and if we simplify this we know that 24 divided by 8 is 3. so really how to write it as three to one and again very important leave the one there I know generally when we work with fractions we just write that as 3 but with a ratio we're talking about the relationship where we're saying we have three milliliters of alcohol to every one milliliter of water in this mixture now it says to State the ratio of water to alcohol so again if your words are flipped you just flip your ratio so water to alcohol remember when we looked at alcohol to water the simplified version was three to one so when we look at water to alcohol it's going to be one to three meaning I have one milliliter of water for every three milliliters of alcohol in this mixture let's take a look at some follow-up questions so let's suppose that we have a 120 milliliter mixture so I'm going to write here that we have 120 milliliter mixture we want to know how many milliliters of alcohol are in the mixture and how many milliliters of water are in the mixture let me kind of scroll down to the next page and get a little room going again I'm going to write that it's 120 milliliter mixture and let's just write the ratio of alcohol to water alcohol to water and remember that's three to one three milliliters to one milliliter and again I left off the milliliter because you know we could write that in there if you want but again these would basically cancel out you just write it as three to one that's just very common that's what we do all right so if we want to find out how much alcohol is in this mixture and how much water is in this mixture we have to think about what would be in one group so one group would be three plus one or four milliliters that's one group and you basically have three milliliters of alcohol and you're going to have one milliliter of water so how many of these groups can I make out of 120 milliliters well you take 120 and you divide it by four and you would get 30. so I can make 30 of these groups so I would just multiply 30 by 3 to get the amount of alcohol I'm going to have 30 times 3 is 90. so I'm going to have 90 milliliters of alcohol and I would multiply 1 times 30 to find out how many milliliters of water I'd have 1 times 30 is 30. so I would have 30 milliliters of water in a mixture with 120 milliliters with an alcohol to water ratio of three to one you're going to have 90 milliliters of alcohol and 30 milliliters of water so similar to ratios a rate is a special type of ratio that compares two quantities with different units so when we start talking about rates we're generally going to be talking about something known as the unit rate and that's going to be the amount of something per one unit so generally speaking we just divide the numerator by the denominator to get a unit rate or the rate per single unit so as an example we have Jeff earn 63 dollars in nine hours what is the unit rate in dollars per hour so you just kind of set this up just like you would with a ratio so we're going to put 63 dollars in the numerator and we're going to put 9 hours in the denominator and essentially to find the unit rate or the amount of dollars that he earns per one hour you're going to take the number in the numerator which is 63 and divided by the number in the denominator which is not we know that 63 divided by 9 is 7. so when we basically just simplify this we end up with seven dollars per one hour or he earns seven dollars per hour we have that Jennifer drove 125 miles in five hours what is the unit rate in miles per hour so this is a common one you might use on a trip somewhere so we have 125 miles we'll put that in the numerator and we'll have five hours we're going to put that in the denominator we're trying to find out how many miles per one hour so again we take the number in the numerator we divide that by the number in the denominator so 125 divided by 5 is 25 so this numerator is going to be 25 and then miles and in the denominator this is going to be a one and then it's going to be our so she was going 25 miles per one hour so one common thing we use unit rates for is to compare prices and a lot of times when you go to the grocery store you'll see next to the price it's this much per unit maybe this much per ounce or this much per gallon so on and so forth so in our example here we have that five gallons of milk sells for fifteen dollars while 10 gallons of milk sells for twenty dollars which option has a lower cost per gallon so what we would do is take the dollar amount in each case and divide by the number of gallons of milk we're getting in each case that'll give us the unit rate or the price per gallon of milk so the first scenario is fifteen dollars for five gallons of milk so if I take 15 and divide by 5 I get 3. so this is essentially three dollars per one gallon right I could put gallon of milk but just going to kind of shorten that a little bit the next scenario is 10 gallons for 20 dollars so I'm going to set this up as twenty dollars and then in the denominator I'll put 10 gallons of milk and again I'm going to do 20 divided by 10 that's going to give me 2. so my new numerator will say two dollars and then per again one gallon one gallon so obviously this is cheaper so between these two scenarios this one is going to be cheaper right spending twenty dollars on 10 gallons of milk gets you a price of two dollars per gallon and again if you were to spend fifteen dollars on five gallons of milk you'd be paying more per gallon or a price of three dollars per gallon hello and welcome to pre-algebra lesson 36. in this video we're going to learn about proportions so our lesson objective for today is just to learn how to determine if two ratios represent a proportion so in the last lesson we learned all about ratios and we didn't go too much into depth but we learned enough for pre-algebra now we're going to start talking about proportions and this is another topic where we're going to go way deeper once we get into algebra one but before we can start talking about proportions we have to be able to answer a simple question that question is how do we determine if two fractions are equal well we're going to use something called the equality test for fractions and basically what this tells us is that two fractions are going to be equal if they're cross products are equal so for example let's say we take something like one half and I don't know let's say six twelfths we can eyeball this and see that these two fractions are equal I know that if I divide six by six I get one if I divide 12 by 6 I get two one half and six twelfths they're equivalent fractions so the equality test for fractions tells me that I can multiply this denominator by this numerator and this denominator by this numerator right the cross products we're multiplying across and I should get the same answer 2 times 6 would give me 12 and 12 times 1 gives me 12 as well so we have the same value and so we know those two fractions are equal let's try another one let's say we have something like 1 4 and I don't know let's say three-sevenths we know these two fractions are not equal one of the ways we can tell is that these two fractions are reduced to their lowest terms or simplified and so I know that they're not going to be the same value so I can multiply 4 times 3 that would give me 12 and 7 times 1 that would give me 7. 7 and 12 are not equal so these two fractions are not equal now you might be asking how does this little test work how come when we have the same value here we know that we have equal fractions or equivalent fractions well you remember when we're talking about comparing fractions we found that when we cross multiplied if 12 was greater than 7 that tells us that 3 7 is the greater fraction so that stems from the fact that we're comparing the numerators here if we had multiplied each fraction top and bottom by the other fraction's denominator so for example if I have 1 4 and three-sevenths what I can do is I can form a common denominator of this fraction's denominator times this fraction's denominator 4 times 7 would be 28. now you can always do this it's not always going to give you the least common denominator but it will give you a common denominator so if I multiply this fraction top and bottom by the denominator of this fraction well what I get 1 times 7 is 7 that's what we had here and then 4 times 7 is 28. so you get 7 over 28. now if I do the same thing over here so I have 3 over 7 I'm going to multiply this top and bottom by the denominator of the other fraction so times 4 over 4. 3 times 4 is 12. notice that's what we got here and then 7 times 4 is 28. so now it's easy to compare these two fractions they have the same denominator I just look at the numerator and I know that 12 is bigger than 7 so this is the greater fraction so 7 28 is less than 12 28. when we cross multiply we are shortcutting this process I am just saying hey all I need to know is what's the numerator going to be right in this case I know the numerator is going to be a 7 and I know the numerator is going to be a 12 over here 7 is less than 12 so I know this fraction is going to be smaller I don't care what the denominator is going to be because the denominator is going to be the same and again that allows me to compare the numerators only so again when two fractions are equal their cross products are equal so if we have a problem like this 1 4 is equal to 7 28 and I have a big question mark here so we kind of have to prove and say okay is this true or is this not so I would just find the cross products so I'd multiply 4 times 7 that gives me 28 I'd multiply 28 times 1 that give me 28. so yes these are equal yes and one of the things you can do if you have both fractions in their simplest form you can easily tell if they're equal right because if I can't reduce this any further I can't reduce 1 4 any further but if I look at 7 over 28 I can reduce that 28 divisible by 7. so if I divide 28 by 7 I'd have 4 if I divide 7 by 7 I had one so I'd end up with 1 4. what about two-fifths is equal to 6 over 20. again we can just form the cross product so 5 times 6 is 30. and 20 times 2 is 40. so 40 is not equal to 30. so these two fractions are not equal so no what about 3 4 is equal to 9 12. again form the cross products so 4 times 9 is 36 and 12 times 3 is also 36 so this would be a yes these two fractions are equal what about 17 over 12 is equal to 34 over 24. so we'd multiply 12 times 34 two times four is eight two times three is six one times thirty four is thirty four bring down the eight six plus four is ten one plus three is four so this is going to be four hundred eight then we're going to do 24 times 17 so 24 times seventeen seven times four is twenty-eight 7 times 2 is 14 plus 2 is 16 and then 1 times 24 is 24. so bring down the eight six plus four is ten one plus one is two two plus two is four so this is going to be 408 also these two fractions are equal because their cross products are equal so yes all right let's take a look at one final problem we've got 12 over 13 is equal to 48 over 65. so multiply 13 times 48 so 48 times 13 3 times 8 is 24. 3 times 4 is 12 plus 2 is 14 and then 1 times 48 is 48. bring down the four four plus eight is twelve one plus one is two plus four is six so that's 624. and then 65 times 12 is what we're going to do next so 65 times 12 2 times 5 is 10. 2 times 6 is 12 plus 1 is 13 and then 1 times 65 is 65. 3 plus 5 is 8 1 plus 6 is 7. so that's 780 so 780 is not equal to 624 so no these are not equal fractions all right so let's talk about what a proportion is now so when two ratios are equal or you could say when two rates are equal as well they are called a proportion and really you could extend that and say when two fractions are equal they represent a proportion so those examples that we looked at earlier where we said yes those were proportions we start out with this example three shrubs to 12 feet and then we have six shrubs to 24 feet and we're saying they're equal so if this is true then we'd have a proportion now the way you check this if I have shrubs in the numerator here I've got to have shrubs over here I can't have shrubs here and then trees over here that doesn't make any sense and then whatever's in the denominator has to be the same so feet here and feet here so once you've checked that all you need to do is cross multiply with the numbers if the result of the cross products there are the same you have a proportion so in other words here I would take 12 and multiply it by 6. 12 times 6 is 72. then I would take 24 and multiply it by 3. 24 times 3 is also 72 so yes here we have a proportion let's take a look at another one here we have 600 miles to 10 hours and we're saying that's equal to 300 miles to 5 hours now again you're looking at what's in the numerator we have miles here miles here what's in the denominator we have hours and then hours so that part checks out then you just work with the number parts so I'm going to cross multiply I'd have 10 times 300 so basically we just put another zero at the end of 300 that's 3 000. then I'm going to multiply 5 times 600. so 5 times 6 is 30. and then we have two trailing zero so we put one two zeros at the end and in each case we get three thousand so yeah this is a proportion so yes this is a proportion here we have 12 points to 48 games we're saying this is equal to 36 points to 96 games so again we check everything out in the numerator points and points in the denominator we have games and then games so that's all okay then we're just going to cross multiply with the numbers so 48 times 36 and just in the interest of time we'll do that with a calculator it's 1 728 and then we do 96 times 12 and that's 1152. so these two numbers are not the same so this is not a proportion so not a proportion all right for the next one we have two dollars to seven euros we're saying this is equal to five dollars to 18 Euros so again we have dollars and dollars euros and Euros so that's okay now we would cross multiply 7 times 5 is 35 18 times 2 is 36. 36 and 35 are not equal so we don't have a proportion so it's just not a proportion okay let's take a look at one more problem again this is really really simple so we have 13 gallons to four acres and we're saying this is equal to 39 gallons to 12 acres so again we have gallons and gallons and acres and Acres so that checks out so we just want to cross multiply the number parts so 4 times 39 is 156. and then 12 times 13 is also 156. so those two numbers are the same so we do have a proportion so yes this is a proportion hello and welcome to pre-algebra lesson 37. in this video we're going to learn about operations with mixed numbers so the lesson objective for today is to learn how to add subtract multiply and divide mixed numbers so when we add or subtract mixed numbers we're going to add or subtract the fractions first followed by the whole numbers so let's just start out with an example we have three and one half plus six and one fourth so I want you to remember that if we see something like three and one half this is really three plus one half right it's just written this way for convenience right we don't want to take up all the space with a Plus on each time so 6 and 1 4 is really six plus one-fourth so to add these two mixed numbers all we're going to do is just kind of rewrite them as three plus one half and then plus six plus one fourth and then I'm gonna do some reordering remember I can do that with addition because it's commutative so I'm going to put my fractions next to each other I'm going to put one half plus one fourth and then I'm gonna put my whole numbers next to each other so I'm going to put plus 3 plus 6. so now what I'm gonna do again as I said I'm going to add the fractions first and then I'll add the whole numbers last so what if I had one half plus one-fourth what would I do well I'd find my LCD and between two and four I know the LCD would be four right four is two factors of two two does not Factor so if I have one half again plus one-fourth I'd multiply one half by two over two so that I could get the denominator of four so one times two is two over two times two that's four and then we're adding to that one fourth so we'd add these numerators two plus one is three and we place that over the common denominator of four all right so let me erase all this and just kind of go back up to the top now so we know at this point that adding the fractions here gives us the result of 3 4. so I'm going to put 3 4 plus and then again you have 3 plus 6 there now adding the whole numbers are easy right three plus six is nine you can do that in your head so now when I look at it I have 3 4 plus 9. if I just reorder this one more time to nine plus three fourths I can then use my convenient notation that I have up here the way that we just write a mixed number right so this is going to be equal to 9 and 3 4. all right let's take a look at another one we have 6 and 3 8 plus 2 and 1 12. so again what I'm going to do is I'm going to rewrite this I'm going to put 6 plus 3 8. and then plus 2 plus 1 12. so I'm going to reorder this addition to where the fractions are next to each other and the whole numbers are next to each other so I'm going to put 3 8 plus 1 12. and then plus 6 and then plus 2. and again you don't have to do that you can just kind of add these together and add these together and you'd end up with the same result I just do this so that it's convenient to see what we're doing all right so moving on now we're going to add our fractions first and let's just do this down here to make it convenient so if I have 3 8 and I have 1 12 what's going to be my LCD well it's going to be the LCM the least common multiple of 8 and 12. so again how do we figure something like that out well we've been working with fractions for a long time now we know that 8 factors into three factors of two so two times two times two we know that 12 is two factors of 2 and I've already accounted for that and then one factor of 3 so that's got to go in so basically my LCD is going to be 2 times 2 times 2 or 8 times 3 which is 24. so when I'm adding these fractions what I'm going to do is I'm going to first transform the fractions into equivalent fractions where they have a denominator of 24. so I'm going to multiply this by 3 over 3. and I'm going to multiply this by 2 over 2. so what we're going to get is 3 times 3 that's 9 over 24. plus 1 times 2 is 2 again over 12 times 2 which is 24. so now we have our common denominator we can just add the numerators 9 plus 2 is 11. and that's going to be over the common denominator of 24. so let's erase all this and go back up we know what the fraction part is going to be it's going to be 11 over 24. so I'm going to put equals 11 over 24 and then plus now the whole number parts 6 and then 2. we add those together that's going to give us 8. and again I can just reorder this one more time 8 plus 11 over 24. again to write this as a mixed number we just simply drop out that addition sign right just for convenience so this is really going to be 8 and 11 24. all right let's take a look at one final addition problem and then we'll move on to subtraction multiplication and division so we're adding nine and one-fifth plus 13 and 1 4 plus 2 and 7 8. again I'm going to just write this out as nine plus one-fifth plus thirteen plus one fourth plus two plus seven eighths and I'm going to reorder it again to where my whole numbers are next to each other and my fractions are next to each other let me just put my whole numbers first so let's do nine plus thirteen plus two and then plus down here I'm going to put my fractions of one-fifth plus 1 4 plus 7 8. so let's take care of the fraction Parts first and we're going to scroll down and do that so if I have one fifth plus one-fourth plus seven eighths again I need to find the LCD to start so what's my LCD going to be again it's going to be the least common multiple for these denominators so of 5 four and eight so five dozen factors so that goes in four is two times two and eight is three factors of two so because I already have two factors of two in here I just need to add one more in so the LCD is going to be five times two which is ten times two again which is 20 times 2 one more time which is 40. so I need to rewrite each fraction as an equivalent fraction with 40 as its denominator so for one-fifth I'm going to multiply by eight over eight or 1 4 I'm going to multiply by 10 over 10 and for 7 8 I'm going to multiply by 5 over 5. okay so 1 times 8 is 8 over 5 times 8 that's 40. then plus 1 times 10 is 10 over 4 times 10 that's 40. then plus 7 times 5 is 35 over 8 times 5 that's 40. so we have 8 plus 10 that's 18 plus 35 which is going to give us 53. so we'd have 53 over the common denominator of 40. now when you get a result that is greater than one remember you're working with mixed numbers here so your fraction part has to be a proper fraction this is an improper fraction so what you want to actually do here is convert this into a whole number part in a fraction part right and that fraction part is going to be a proper fraction and we remember how to do that right we take 53 and we would divide it by 40. so 40 goes into 53 once 1 times 40 is 40. subtracting we get 13 we would have one and 13 over 40 as our mixed number there so let me erase all of this and we're going to go back up to the top and I'm just going to rewrite this right here 9 plus 13 plus 2 plus I just found out that the result from adding the fractions was 1 and 13 40. so I'm going to put a 1 here to add along with my whole numbers and then plus 13 over 40. this fraction part I don't need to do anything else with so now I'm just going to add these whole numbers together 9 plus 13 is 22 22 plus 2 is 24 then 24 plus 1 is 25. so I would have 25 Plus 13 over 40. and again as a mixed number we just write this as 25 and 13 40. okay now let's look at some subtraction problems and really when you work with subtraction problems they're not any more difficult except for the simple fact that sometimes you'll be subtracting fractions and the fraction you're subtracting from is too small so you have to do a little bit of borrowing and we're going to come across that in the next problem so we're going to start with 23 and 1 3 minus 20 and 1 7. so again what I'm going to do is I'm going to subtract the fraction Parts first and then I'm going to subtract the whole numbers so in other words I'm going to take 1 3 and I'm going to subtract away 1 7. now you have to do in that order here because remember subtraction is not commutative like addition so I'm going to take 1 3 and I'm going to subtract away 1 7. now to do this I need to get a common denominator going and my LCD here would be the product of the denominators 3 times 7. and that's because 3 and 7 are both prime numbers so I'm going to multiply this by 7 over 7. I'm going to multiply this by 3 over 3. and we're going to end up with 7 over 21 minus 3 over 21 and that's going to give me 4 over 21. all right 7 minus 3 is 4 over the common denominator of 21. so now that we know the fraction part we only need to subtract the whole number part and again I have to be specific about how I do it it's 23 minus 20. if I do it in a different order I'll get the wrong answer so 23 minus 20 would be 3. so what we end up with is 3 for the whole number part and then 4 over 21 for the fraction part all right let's take a look at another subtraction one so we have 14 and three-fifths minus five and thirteen fifteenths so we would start this problem out the same way so we do three-fifths minus 13 15. right just subtract your fractions first get that out of the way what's going to happen is you're going to find out that three-fifths is smaller than 13 15. so the result of this is going to be a negative number now we don't want that so what we're going to do is we're going to end up borrowing from the whole number part so let me show you how to do that the first thing I'm going to do is pretend that I don't know that this number is smaller so I'm going to start out by saying okay the LCD between the two fractions is 15. so I'd multiply this by three over three and what would happen is I'd say that okay I have 9 over 15 minus 13 over 15 and I'd say okay this is going to result in negative 4 over 15. you know what do I do now again to correct this I'm going to borrow I'm going to borrow so instead of having 14 there I'm going to rewrite this and say that this whole number is going to be 13. and instead of three-fifths what I'm going to do is I'm going to add five fifths so that's where this one went I took one away and I basically said okay instead of having 14 I now have 13 that 1 where it went I wrote it as five fifths and then I'm adding that to three-fifths so essentially my fraction part now is going to temporarily be eight fifths right so you can just think of this as having 13 and eight fifths and that's not how we write mixed numbers right we always want a whole number and a proper fraction but we're just doing it so that we can perform our operation so if I have this as 13 and eight fifths now when I look to subtract the fractions what I'm going to have is I'm going to have eight fifths minus 13 15. again the LCD is still going to be 15 so I'd multiply this by three over three and what we'd end up with now is we'd have 3 times 8 that's 24. over 3 times 5 that's 15 minus 13 over 15. and you can see that 24 is bigger than 13. so you can do the subtraction and not get a negative number so 24 minus 13 is going to give us 11. so this ends up being 11 over 15 for our fraction part now when we subtract the whole numbers now we have a different whole number to work with because we let one to our fraction we no longer have 14 as our whole number part we have a 13. so we have to keep that in mind so we would do 13 minus 5 and 13 minus 5 is going to give us 8 so 8 and 11 15 would be our answer okay let's take a look at another one we have 12 and 4 9 minus 8 and 7 8. so again when I have a subtraction problem and I'm working with mixed numbers I'm going to try to subtract my fraction part first and then I'm going to worry about the whole numbers so I would subtract 4 9 minus 7 8 4 9. minus 7 8. again you got to do it in that order because subtraction is not commutative now when I look here I see that 9 and 8 they're not going to have any common factors right 9 is 3 times 3 8 is 3 factors of two so really the LCD here is going to be the product of the two denominators or 72. so I'd multiply this by 8 over 8 and I'd multiply this by nine over nine and when you do this you see you have a problem right a times 4 is 32 so this would be 32 over 72 minus 7 times 9 that's 63 over again 72. so when we perform this subtraction we end up with a negative result and although we can do it that way it makes it more complicated so to keep things nice and simple let's erase all this and I just want you to again temporarily think about this we're going to borrow one from this whole number so instead of 12 I'm going to have 11. and so this 4 9 here I'm going to just temporarily add 1 to that 4 9 but just using fractions so instead of 4 9 if I add one in fractional form I'm going to add 9 9 and that's going to give me 13 9. so instead of having four nights there I basically have 13 9 and again this is now 11 for the whole number part so now if we subtract fractions I would do 13 9. minus 7 8 we're going to have the same LCD so I'm going to multiply this by eight over eight I'm going to multiply this by 9 over 9. and let me just scroll down here so we have a little room 13 times 8 is 104. and this would be over 9 times 8 that's 72. then we'd subtract away 7 times 9 that's 63 over 8 times 9 that's again 72 104 minus 63 is 41. and this is over the common denominator of 72. so the result of subtracting the fractions would give us 41 over 72. so this is 41 over 72 for that part and now we just subtract the whole numbers so remember I don't have 12 anymore because I borrowed to do my subtraction with the fractions so I have an 11 here and I'm just subtracting over 8. 11 minus 8 would give me 3 so we'd end up with 3 and 41 over 72 as our answer so now we're going to start doing some multiplication and division and we're going to change something up here now when you add or subtract mixed numbers generally what everybody's going to say is to add or subtract the fraction Parts first then add or subtract your whole number parts now alternatively if you wanted to when you add or subtract mixed numbers you could have converted those mixed numbers into improper fractions perform the additional subtraction and then you can convert them back to mixed numbers if you have to it's totally up to you which way you want to do it it's just usually faster to do it the way that I showed you now when we multiply or divide mixed numbers we're going to convert each mixed number into an improper fraction once we're done if it's required we'll convert the answer back to a mixed number sometimes your teacher won't tell you to convert it back you can just leave it as an improper fraction whatever they want you to do just follow the instructions you're given so we're going to start with five and two-fifths times three and one-seventh so what I'm going to do is convert each of these into an improper fraction 5 times 5 is 25 plus 2 is 27. so I'd have 27 fifths times 7 times 3 is 21 plus 1 is 22. over 7. so again when I multiply I always look to cross cancel first there's no common factors between anything here so I just do the multiplication 27 times 22 is going to give me 594. and this is over 5 times 7 which is 35. so our answer here is 594 over 35. now like I said if you wanted to you can leave this as an improper fraction if your teacher makes you convert it back to a mixed number so let's do one or two where we convert it back and then we'll just leave them as improper fractions moving forward so 594 we're going to divide that by 35. 35 goes into 59 once 1 times 35 is 35. subtract here 9 minus 5 is 4 5 minus 3 is 2. bring down the 4. 35 is going to go into 244. six times six times 35 is going to give us 210 subtract here 4 minus 0 is 4 4 minus 1 is 3. so that's going to give us 34 as the remainder so as a mixed number this would be 16 and 34 over 35. okay for the next one we have 8 11 times 4 and 1 half so I'm going to keep 8 11 unchanged that's already a fraction and I'm going to multiply by four and one half after I converted into an improper fraction 2 times 4 is 8 plus 1 is 9 so we'll have 9 halves here we look to see what we can cross cancel I know that 8 is divisible by 2 8 divided by 2 is 4. so just think about canceling a factor of two that would give me four cancel factor of 2 that would give me one so now I basically have 4 times 9 that's 36 over 11 times 1 that's 11. so I get 36 11 as my answer now again for the last time we're going to convert this into a mixed number again this is something you should already know how to do so if your teacher requires it go ahead and do it but again a lot of teachers will just say however you get the answer as long as it's simplified it really doesn't matter so 36 divided by 11 11 goes into 36 3 times 3 times 11 is 33. subtract and get 3. so if we write this as a mixed number we'd have 3 and 3 11. three and three elevenths now we have four and two-fifths times six and one-fourth so I'm going to convert each into an improper fraction 5 times 4 is 20 plus 2 is 22. so we'd have 22 over 5 times for this one 4 times 6 is 24 plus 1 is 25. and that's over 4. and we can see we can do some cross canceling here I know that 25 is divisible by 5. 25 divided by 5 is 5. so this will cancel out completely and become one this will cancel become 5. 22 is divisible by 2 and so is four so 22 divided by two is eleven four divided by 2 is 2. so now we have 11 times 5 that's 55 over 1 times 2 that's 2. so we end up with 55 halves as our answer let's take a look at a few division problems now so we have 8 and 2 9 divided by seven and one-half again what I want to do is change each of these mixed numbers into an improper fraction then do the division so 8 and 2 9 I'm going to change this 9 times 8 is 72 72 plus 2 is 74. and this is going to be over 9 and we're dividing by seven and one half two times seven is fourteen fourteen plus one is fifteen so that's 15 halves and again if you're dividing fractions the first fraction or the left fraction stays unchanged so 74 9 we don't do anything to that then we're going to multiply by the reciprocal of 15 halves so that's going to be 2 15. and we'll look to see if we can cross cancel anything I can really do here 74 is not divisible by three or five so I can't really cancel anything so we just multiply 74 times 2 is 148 and this will be over 9 times 15 which is 135. So my answer here is 148 over 135. all right for the next one we have negative 6 divided by 2 and 1 8. so we have negative six and since we're dealing with fractions here I'm going to go ahead and write that as negative 6 over 1. and then divided by I'm going to convert this into an improper fraction 8 times 2 is 16 plus 1 is 17. so you'd have 17 over 8. again to divide fractions this leftmost fraction negative six over one will stay unchanged and we'll multiply by the reciprocal of this fraction so times 8 over 17. so nothing I can cross cancel here so I just need to multiply negative 6 times 8 is negative 48 and this is over 1 times 17 or 17. so we get Negative 48 17 as our answer all right final problem we have 2 and 1 5 divided by four and one-third so again I'm going to convert each into an improper fraction 5 times 2 is 10 plus 1 is 11. so this would be eleven fifths and then divided by 4 and 1 3 I'm going to have 3 times 4 that's 12 12 plus 1 is 13. and then over three so eleven fifths times the reciprocal of 13 thirds would be 3 13. nothing to cross cancel 11 times 3 is 33 5 times 13 is 65 so we'd have 33 over 65 as our answer hello and welcome to pre-algebra lesson 38. in this video we're going to learn about complex fractions so the lesson objective for today is to learn how to simplify a complex fraction using two different methods so before we kind of get started let me just give you a basic definition for a complex fraction a complex fraction is a fraction that contains at least one fraction in its numerator or denominator it could have a fraction in its numerator and denominator as well so let's start out by looking at a problem so here's a complex fraction we have one-fifth plus two-fifths over one-third plus four-thirds so notice how we have an operation in the numerator and in the denominator and there's fractions in both the numerator and the denominator so one way you can simplify this is to simplify the numerator and the denominator separately and then you're going to have a main division this is going to be your main division for the complex fraction so if we kind of go through here and let me just erase this real quick if we kind of go through here and simplify the numerator we already have a common denominator so we have one-fifth plus two-fifths we would just add one plus two that's three put that over the common denominator of five and then in the denominator here we have one third plus four thirds again we have a common denominator so we can do one plus four that's five put that over the common denominator of three so at this point we have a complex fraction that's nothing more than a division problem we have three-fifths basically divided by five thirds so if you saw that problem written this way you would know that you take three-fifths and leave it unchanged and you're going to multiply by the reciprocal of the second fraction which is three-fifths as well so you're going to do the same thing here you just got to get used to looking at it differently we have three fifths again divided by Five Thirds so three-fifths is going to stay the same and we're multiplying by the reciprocal of what we're dividing by so the reciprocal of Five Thirds again is three-fifths and we just multiply three times three is nine five times five is twenty-five so we get nine twenty-fifths as our answer all right for the second problem we have kind of an easier one this is basically just a division problem we have 4 15 over 12 13. this is the problem 4 15 times the reciprocal of what we're dividing by so times thirteen over twelve now I can cross cancel here the greatest common divisor between 4 and 12 is 4. so if I divide four by four I get one if I divide 12 by 4 I get 3. nothing I can really do between 13 and 15. so we'd end up just multiplying here 1 times 13 is 13 over 15 times 3 that's 45 so this is going to simplify to 13 over 45. all right let's take a look at 5 plus 3 halves over negative 2. so again we're going to work by simplifying the numerator and denominator separately our denominator here is simplified already so I can just kind of copy that I don't need to do anything for the numerator if I have 5 plus 3 halves I got to get a common denominator going there so I can write 5 as 5 over 1 and the least common denominator I'm going to be able to form here is going to be 2. so I can take 5 over 1 and multiply it by 2 over 2. and I'm going to add that to three halves so continuing 5 times 2 is going to be 10. that's going to be over 2 plus 3 halves and I'm just going to copy this denominator for the complex fraction that's negative 2. and I'm going to continue down here but in the numerator complex fraction I have a common denominator now of 2. so I'm just going to add the numerator so 10 plus 3 is 13 so that's 13 over 2. and basically we could say this is divided by or over negative 2. now to make this easy on ourselves I'm going to write negative 2 as negative 2 over 1. so now I can see this problem as 13 halves divided by negative 2 over 1. so again it's just a division problem at this point this fraction here 13 halves will stay unchanged and I'm going to multiply by the reciprocal of negative 2 over 1 which would be 1 over negative 2 or you quit negative 1 over 2. it doesn't really matter nothing to cross cancel so we just multiply 13 times negative 1 is negative 13. two times two is four so you get negative 13 fourths as your answer all right let's take a look at another one so we have 3 7 minus 5 over 21 that's the numerator for the complex fraction and this is over you have 5 8 plus 2 fifths so this one will be a little bit tedious to work through so let's kind of just simplify the numerator here and then we'll work on the denominator in a minute so we have 3 7 minus 5 over 21. so we got to get a common denominator going and the least common denominator we're going to be able to form is going to be 21 right 21 is 7 times 3. here we just have a 7. so what I'll do is I'll multiply 3 7 by 3 over 3. that'll give me a denominator of 21 there then minus 5 over 21. so 3 times 3 is 9 so we'll have 9 over 21 minus 5 over 21 that's going to give me 4 over 21. so my simplified numerator for the complex fraction is 4 over 21. so I'm just going to write up here that this is 4 over 21 and then over let's figure out with a simplified denominator for the complex fraction is going to be so we have 5 8 plus 2 5. so 5 8 plus two fifths now to find the LCD here 5 is a prime number and eight is three factors of 2. so the LCD is basically found by multiplying eight times five that would give you 40. so what I'm going to do is I'm going to take 5 8 and multiply by 5 over 5. and I'm going to take two fifths and I'm going to multiply by 8 over 8. so 5 times 5 is 25 and of course this is over 40 right 8 times 5 is 40. then plus 2 times 8 that's 16 and again this is over 5 times 8 which is 40. so 25 plus 16 is going to be 41. and then the common denominator is 40 so this is over 40. all right so we have 41 over 40 here so my simplified denominator again is 41 over 40. so now we basically have a division problem we have 4 over 21 divided by or over 41 over 40. so to divide this we take 4 over 21 leave that unchanged we multiply by the reciprocal of what we're dividing by so the reciprocal of 41 over 40 is 40 over 41 and we can see that we can't really cross cancel here because 4 is 2 times 2 41 is a prime number 21 is 7 times 3 and 40 is 2 times 5 times 2 times 2. so nothing I can cancel here so I just multiply 4 times 40 is 160. over 21 times 41 and that's 861. so we're going to end up with 160 over 861 as our answer so let's take a look at 5 minus 1 9 and then this is over 3. so I'm going to write this as 5 over 1 because I'm working with fractions and I'm going to multiply 5 over 1 by 9 over 9 so that I can have a common denominator of 9. so then minus 1 9 and this is all over 3. so 5 times 9 is 45 so this would be 45 over 9 minus 1 over 9 and then this is all over 3. continuing down here in the numerator of the complex fraction I have a common denominator now so I would do 45 minus 1 that's 44. over the common denominator of nine and then this is divided by or over 3. now when we work with fractions I like to have fractions everywhere because it makes it easier to think about what I'm doing so instead of just having a 3 I'm going to write 3 over 1. so I can see the division problem is 44 over 9 divided by 3 over 1. that's easy to do so we have 44 over 9 that stays the same we multiply by the reciprocal of three over one which is one third so four plus four is eight so it's not divisible by three so there's not really anything I can do to cancel here so I just go through a multiply 44 times 1 is 44. 9 times 3 is 27 so we end up with 44 over 27 as our answer so alternatively we can find the LCD of all denominators involved then multiply the numerator and denominator of the complex fraction by the LCD so when I say multiply the numerator and denominator of the complex fraction by the LCD you have to pay close attention to that and I'll show you why in this example so we have four fifths over two-fifths so what's my LCD here well the denominator of this fraction is 5 the denominator of this fraction is also 5. so the least common denominator is going to be 5 right because they both have the same denominator so when we say we're going to multiply the numerator and denominator of the complex fraction by 5. I'm essentially saying I'm going to multiply this part here by 5 and I can do by five over one if it makes it more convenient and then this part by 5 as well right this is the numerator of the complex fraction this is the denominator of the complex fraction and each part gets multiplied by 5. again this is legal because I'm multiplying by 5 over 5 which is one so if you go through what's going to happen is this 5 will cancel with this 5. this 5 will cancel with this 5 and I will be left with 4 over 1 which is 4 divided by 2 over 1 which is two right so I can keep it like this in fractional form or I can rewrite it if I want and just say this is 4 divided by 2. right because this way I would have 4 over 1 times the reciprocal of 2 over 1 which is one half it's the same thing as four over two right so when we do this division four over two or four divided by 2 is 2. all right here's one that's a little bit more complicated we have one fourth plus three-fifths and this is over 3 8 plus 5 12. so look at all your denominators involved you have a four a five an eight and a twelve so what's your LCD here again this is the LCM for 4 5 8 and 12. so if you think about what we're going to put in the prime factorization for the LCM 4 is 2 times 2. 5 doesn't Factor so it just goes in 8 is 3 factors of 2. I already put two factors of 2 in so I only need to put in the largest number of repeats between any of the prime factorizations so I'm just going to add 1 2 in there then 12 is 4 which is 2 times 2 that's already accounted for times 3. so I'm going to put times 3. that's all I need to add to it because I already have three factors of 2 in there so if I multiply this out 2 times 2 is 4 4 times 5 is 20 20 times 2 is 40. 40 times 3 is 120. so now what we're going to do is we're going to multiply the numerator of the complex fraction by 120. and you could do again 120 over 1 if that makes it more convenient for you and you're going to multiply the denominator of the complex fraction by 120 also so what is this going to give us we have to use the distributive property here because you have some addition going on so 120 times 1 4. think about what that would be so 120 basically over 4 and that would be 30. so I'm just going to write this over here this would be 30 plus next you'd have 120 times three-fifths so 120 times three-fifths and let me write this over one to make it a little bit more convenient so I can cross cancel 120 with five I know 120 divided by 5 is 24. so this cancels with this and gives me 24 and then 24 times 3 is 72. so the denominator is gone you can put it over 1 if you want but basically it's gone so we're just going to put a 72 there and again if you wanted to write 30 over 1 plus 72 over 1 you can but it's just completely unnecessary so we'll just write it like this and then our denominator again we're going to do the same thing we're going to multiply 120 times 3 8 and then we're going to add to that 120 times 5 12. so for 3 8 times 120 and again we could put 120 over 1 if we want 120 divided by 8 is 15. so I can cancel this with this and get 15. 3 times 15 is 45 so you'd have 45 over 1 or just 45 and then plus again we're going to multiply 120 times 5 12. so we'll have 5 12 times 120 again you can put that over one if you want 120 divided by 12 is 10. so this cancels with this and gives you 10. 5 times 10 is 50. and again you can put that as 50 over 1 if you want but I'm just going to put 50. so now we have 30 plus 72 and I'm going to write that down here that's going to be 102 over 45 plus 50 and that's going to be 95. so our fraction has been simplified to 102 over 95. okay let's take a look at one more like this we have three fifths plus one-half and this is over 5 6 minus one third so look at your denominators involved you have a 5 a 2 a 6 and a 3. so what's the LCD well it's going to be the least common multiple of 5 2 6 and 3. so again five dozen Factor so that goes in 2 doesn't Factor that's going in 6 is 2 times 3. I already have a two so I'm just going to put a 3 in and then three we already have that in there so 5 times 2 is 10 10 times 3 is 30. so that's going to be your least common denominator and so again what we're going to do is we're going to multiply the numerator of the complex fraction by 30. and I'll put times 30 over 1. and we're going to also do that to the denominator so times 30 over 1. and again if I multiply by 30 over 30 I'm multiplying by 1 I'm not changing the value of the fraction I know what looks more complicated but really you're still doing the same thing you're multiplying by a complex form of 1. so again we're going to do this using the distributive property so we would have 30 over 1 times 3 5. to start so 30 divided by 5 is 6 so I cancel this with this and get a 6. 6 times 3 is 18. so this would be 18 over 1 or just 18. then plus next we're going to have 30 over 1 times 1 half and I can do this down here but basically we know that we would have 30 divided by 2 in that case right because we're just saying what is half of 30 that's 15 right 30 divided by 2 is 15. so I'll put a 15 here and this is over next we have 30 over 1 times 5 6 so 30 over 1 times 5 6. so I know that 30 divided by 6 is 5. so cancel this with this and I'll get a 5. 5 times 5 is then 25. then minus next I'm going to do 30 over 1 times 1 3. so 30 over 1 times 1 3 30 divided by 3 is 10. so this would cancel and become 10 and I'm just left with 10 times 1 or 10 over 1 which we can just write as 10. so simplifying this I have 18 plus 15 which is 33 over 25 minus 10 which is 15. now I'm not done because I always want to report a fraction in its simplest form I know that there's a common factor of 3 between 33 and 15. right 33 is 3 times 11 15 is 3 times 5. so really if I just divide each by 3 I'm going to end up with 11 over 5 or 11 fifths as my final answer hello and welcome to pre-algebra lesson 39. in this video we're going to learn about decimal fractions all right so our lesson objectives for today would be to learn how to change a decimal fraction into a decimal then also we want to learn how to change a decimal into a decimal fraction so basically we're going to go back and forth between a decimal and a decimal fraction so before we get started I want you to think about the very first lesson where we started talking about fractions what I told you there is that although some of you didn't have any experience with fractions in your math class you had seen fractions in your everyday life every time you and a group of friends get together and you split something up so you know if you got together and you split up a pizza or you split up a cookie or you split up an apple or whatever it was you gain some experience with fractions even though you didn't know it it's going to be the same thing for decimals right so although you might not have seen decimals yet in your math class you have worked with them in everyday life one example is that you are absolutely bombarded with decimals every time you go shopping for anything so let's kind of get started with our lesson with a basic definition so when we start talking about decimals kind of the introduction to that would be the decimal fraction so a decimal fraction is a fraction whose denominator is a power of 10. let me reread that a decimal fraction is a fraction whose denominator is a power of 10. so what do we mean by that the denominator is a power of 10. it means you can take the denominator and rewrite it as 10 raised to a whole number that is one or larger so here are some examples of some decimal fractions so we have five tenths here and you can see that this 10 I could rewrite that as 10 to the power of 1. here I have three hundredths I can rewrite 100 as 10 to the power of 2. here we have seven hundred thousandths and I can rewrite this hundred thousand in the denominator as 10 to the one two three four five there's five zeros here so ten to the fifth power so in each case we have a decimal fraction because the denominator is a power of ten and again more specifically that means that 10 is raised to some whole number that is one or larger so to kind of get our feet wet with decimals let's kind of run through a typical scenario so let's suppose you go to the grocery store and encounter a can of tuna priced per can at 1.38 cents so this is typically how you're going to see decimals now let me just skip the dollar sign real quick and just write 1.38 now the first thing I want to draw your attention to you might not know that this is called a decimal point this is a decimal point now what's to the left of the decimal point this one this is just a whole number this is what we've been working with forever so I'm just going to put that this is a whole amount a whole I'm out you know and in terms of money I just have a dollar there now this part that occurs after the decimal point or to the right of the decimal point is part of a whole it's part of a whole again we've already seen this with fractions and if I was to think about 1.38 cents kind of using fractions well okay one dollar is kind of a whole amount it's a one dollar bill just think about 38 cents for a minute and how you could relate that in fractional terms well if I had everything in pennies let's say it takes a hundred pennies to make a dollar or one whole amount so if I only have 38 pennies out of the 100 pennies that are needed really what I have is I have 38 out of 100. or 38 Parts out of the 100 needed to make a whole amount remember when we work with fractions our denominator is the whole amount split up into equal parts so this is me taking a dollar and just splitting it up into equal parts of a hundred that's what would happen if I got a dollar in pennies and then if I only have 38 of those pennies well then I have 38 Parts out of a hundred right so this .38 is the same thing as 38 over a hundred so really we can kind of see already that we can go back and forth and write 1.38 as 1 and 38 hundredths right these would actually be the same here you have a whole amount and then you have part of a whole here with your mixed number you have the same thing you have a whole amount whole amount or your whole number part and then you have your fractional amount your part of a whole you have 38 Parts out of a hundred so kind of the first thing we're going to learn is how to change to a decimal so if you're starting out with the decimal fraction the first thing you're going to do is count the number of zeros in the denominator and then you're going to delete the denominator so if I had something like let's say 56 over 100 I would count one two zeros so two zeros and once I have that information I can just delete the denominator right I'm just going to have the number 56 now and I have 56 and I'm going to know that I have two zeros for that denominator then the next thing in the numerator which is the number 56 that we have move the decimal point to the left by the number of zeros we counted in the denominator so again we had two zeros but some of you at this point might be saying how do I move the decimal point to the left I don't know where it is if you have a whole amount you can always write a decimal point after the rightmost digit so I can write 56 as 56 and then a decimal point and another thing I'm going to teach you is that after this decimal point you can put as many zeros as you want you might see some prices in stores look like this 56 dollars you might see a decimal point with two zeros at the end or you might see 38 dollars like this or twenty seven dollars like this after that decimal point you can put as many zeros as you want and you're not changing the value of the number this is still 56 or in this case since I have a dollar sign it's still 56 dollars so with that being said we're going to move our decimal point to the left by the number of zeros we counted in the denominator we counted two zeros so I'm going to the left one two places so we would end up with point five six now another practice that you might get into when you start working with decimals is if you have a decimal point and you don't have a whole number amount you'll probably put a zero out in front this zero does not change the value of the number it's just kind of used for clarity right so that people when they're eyeballing it will say okay oh I have a I have a zero first well that's a little weird so that draws their attention to the fact that there's a decimal point so that they don't miss it right you don't run a mistake 0.56 or 56. all right let's look at the first one here so we have 40 over 100. so again the first thing I'm going to do is count the number of zeros in the denominator again I have a situation where I have two so I have two zeros and then I'm going to delete the denominator so I don't actually have to do that I'm just going to write over here I'm just going to write the numerator without the denominator so I'm just going to write 40. then the next step is to move the decimal point by the number of zeros that we counted in the denominator again we counted two zeros so again I can put my decimal point here after the final digit of 40 and I'm going to move it one two places to the left and that's going to give me point 4 0. now I want to draw your attention to something if I have a 0 right here it's the same thing as if I erase it and had 0.4 again after you have a decimal point once you have your final non-zero digit you can add as many zeros as you want so point four is the same as point four zero zero zero zero zero zero where people get confused is let's say you have a bunch of zeros but then you have a non-zero digit involved let's say you have a nine at the end this is different than 0.4 those are not the same if I have all zeros then it's the same but if I have something that's not zero that kind of is at the end or even in between let's say I had a 9 here and then a bunch of zeros that's still not point four right you can't have it interrupted in any way it has to be zero all the way down so having said that we can rewrite this answer as just 0.4 all right let's take a look at another one and I'm going to kind of shorten this up a little bit and one of the things I want you to realize here and you're going to use this later on in math when you divide by 10 or some power of 10 and when I say a power of 10 I mean 10 raised to a whole number that's one or larger you move your decimal point one place to the left for every 0 in that power of 10. so just forget about your procedure for a second and just say okay I have 7 divided by 10. so I'm going to write 7 and I'm going to write my decimal point I have 1 0 in this power of 10 so that means I'm moving my decimal point one place to the left and I'll end up with 0.7 as my answer again you can still use that procedure that's kind of a procedure you're going to get in a book count the number of zeros you have in the denominator you have one zero delete the denominator so I would have just wrote this as seven and then move the decimal point one place to the left for every zero that you count in the denominator so it goes over here and again you end up with 0.7 all right let's look at 85 over 10 000. so again I'm going to use that kind of shortcut because when we find a shortcut in math we kind of want to use it so I'm just going to write 85 and then I'm dividing by 10 000. so this is a power of 10 with one two three four zeros so basically all I need to do is move my decimal point four places to the left so I put a decimal point after the final digit in 85 and I go 1 2 and then I see that what do I do here I don't have any more places when that happens I can put zeros in so I can put a 0 in here and here and then 3 and then 4 and I'm able to complete that so now I'm going to end up with point zero zero eight five as my answer what about 5 and 35 over one thousand well what happens is if you have a mixed number and you're doing this just take the whole number part and go ahead and just write that then we'll follow that with 35 over 1000 expressed as a decimal so to do that I have 35 and basically I'm dividing it by a thousand so I have one two three zeros in this power of 10 right 1000 is basically 10 to the third power so all I need to do is move the decimal point 1 2 put a zero in three places to the left I'd have .035 so if I have 5 and 35 thousandths then basically I have five point zero three five the decimal point with zero three five after it is 35 thousandths and then the 5 is just the same okay so now we're going to go backwards right we're just going to reverse what we just did so we're going to be going from a decimal to a decimal fraction now one of the things I want to tell you right away we're not going to be simplifying here and the reason is if you want to report something as a decimal fraction again the denominator has to be specific it has to be a power of 10 right meaning that it's able to be expressed as 10 raised to a whole number that is one or larger so the first thing we're going to do is write any whole number if you have one then you're just going to count the number of decimal places and basically when we say count the number of decimal places we're saying that we're counting the number of digits that occur after the decimal point and these are only digits that are going to add value to the number so for example if I have the number 0.38700 I don't have five decimal places here some people would say okay I have five decimal places I don't I only have three these two don't add any value to the number this is the same thing as 0.387 so I would count three decimal places here so once you know that you take that decimal part so that's the part that occurs after a decimal point and you put it over a denominator that begins with a one and is followed by the same number of zeros as decimal places so for 0.387 basically you write 387 in your numerator you write a 1 in the denominator and you follow it by three zeros and that's because we had three decimal places here if I had let's say made the mistake of including these two zeros here I would end up putting five zeros down here and I would get an incorrect answer this is wrong right because these zeros didn't add any value so I don't have a denominator of one hundred thousand I would take these away and then just say okay I have three decimal places so I'm going to have one two three zeros this is 387 over one thousand all right so we're going to start out with 0.05 and again this zero out here to the left of the decimal that's just for clarity right we don't need to do anything with that so the first thing I'm going to do is I'm going to count the number of decimal places that I have so after the decimal point I'm looking to see what I have I have a 0 and then a 5. so I have two decimal places two decimal places and remember this zero matters because it's followed by a five if I have 3.00 these zeros do not add any value but the second I follow that zero with a non-zero digit let's say 3.03 this zero does matter now so because I have again a 5 that follows the zero this zero does matter so we have two decimal places now the next thing we want to do is write our decimal part so that's the part that occurs after the decimal in this case that's a zero five and the numerator of the fraction and we're going to put the denominator of the fraction as a one followed by a zero for every decimal place we have so we had two decimal places so you get two zeros now a couple things here I don't need to have this zero in front of the 5 because that adds no value to the number when you're working with whole numbers I can always put zeros to the left so I can put as many zeros to the left this is still 5. kind of similar to when I'm working with decimals I put a decimal point I can put as many zeros here it's still 5. what I can't do if I have a whole number I can't just start adding zeros in this number is now five thousand it's not five anymore so the last thing I'm going to draw your attention to is a lot of you will say okay well you can simplify this fraction it's 1 over 20. well yeah we can but it's no longer going to be a decimal fraction right it's just a regular fraction at that point and a decimal fraction has a definition where the denominator again is a power of 10. so we're going to leave it as 5 over 100 and that's exactly what you're going to do if your teacher says that she wants a decimal fraction right do not simplify it all right for the next one we have 21.34 and again if we have a whole number part like we have here we're going to write that first now for the fraction Parts I'm going to kind of make this simple take the part that occurs after your decimal point just write it as the numerator so 34 goes in the numerator put your fraction bar then put a 1 and then you can kind of just mentally look at this and say okay I have one two decimal places so I know I have two zeros so we get 21 and 34 hundredths as our answer and it's really really that simple you can go through the procedure each time but after you work five or six problems you can kind of just do a lot of this stuff mentally what about 0.3975 well I'm just going to take the decimal part again the part that occurs after the decimal point write that in my numerator so 3 9 7 5 or 3975 for my denominator I'm going to put a 1 and I'm going to follow that up with a 0 for every decimal place I have well for decimal places I have one two three four so this is going to get one two three four zeros or ten thousand for the denominator so we get 3975 over ten thousand okay for the last one we have 4.279 and again I'm just going to write this whole number here to start for the fraction part just take this your part that occurs after the decimal point put that in the numerator I got my fraction bar I'm going to put a 1 in the denominator and again I'm going to follow that up with a 0 for every decimal place that I have so I have one two three of those guys so I'm going to put one two three zeros here and end up with 4 and 279 over 1000 or I could say 4 and 279 thousandths hello and welcome to pre-algebra lesson 40. in this video we're going to learn about comparing and rounding decimals so the lesson objectives for today we want to learn the place values to the right of the decimal point we also want to learn how to compare two or more decimals and determine their rank and then finally we want to learn how to round decimals so in our last lesson we introduced decimal fractions and then we learned how to go back and forth between decimal fractions and a number that has a decimal point so in this lesson we'll dig a little bit deeper into decimals before we move on and start doing some operations with decimals so the first thing I want to show you is an expanded place value chart so kind of in our very first lesson where we learned about place value we saw everything to the left of the decimal point right we have the ones the tens The Hundreds the thousands as we move to the left remember each place if I think about the ones places starting with a one every time I go to the left I just multiply by 10 to get to that next place so 1 times 10 gives me 10. now I'm in the tens place ten times 10 gives me a hundred now I'm in the hundreds place 100 times 10 gives me a thousand now I'm in the thousands place and so on and so forth that pattern would continue forever now that we have a decimal point involved we have some place values that go to the right of the decimal point that we need to become familiar with so the first thing immediately to the right of the decimal point is the tenths and I got a lot of students that will yes that this place is called the once right because this is the ones the tens the hundreds and you have a similar pattern over here it's just you don't have a once place right you start off with the tens place and I'm going to explain why in a second so you have the tenths the hundredths the thousandths then you have the ten thousandths the hundred thousands the millions the ten millions the hundred millions the billions so on and so forth and that continues in a usual way right after billions you'd have 10 billionths 100 billionths trillions so on and so forth now one of the things I want to show you right away and just following this pattern that I just demonstrated again if you start off right here in the ones place you have the number one I know that if I go to the left I multiply by 10. so if I go this way I multiply by 10 so times 10 and I get to the tens place 1 times 10 is 10. multiply times 10 again go this way get to 100. but if I was to go backwards and go to the right really I could think about starting out at 100 dividing by 10 to get to 10. so to go this way just think about dividing by 10. another way to think about dividing by 10 is multiplying by 1 10 right we learned about how division of multiplication related if I have 100 divided by 10 it's the same thing as if I said what is 100 times 1 10. all right that's 100 divided by 10. either way it's the same thing so what I'm going to show you here is that as you keep going to the right so as we move from here to here again we're dividing by 10 or multiplying by 1 10. dividing by 10. so when we get to one let me just erase everything to this point when we get to 1 and we pass over the decimal point it's no different so think about having one and multiplying by 1 over 10. well 1 times 1 over 10 is 1 over 10. so that's how we get to the tenths then if I multiply 1 over 10 times 1 over 10 I get to 1 over 100 and that's the hundredths place do it again 1 over 100 times 1 over 10 is one over one thousand that's the thousands place so as I'm going to the right I'm multiplying by one over ten or I'm dividing by ten again either way you want to think about that as I'm going to the left I'm multiplying by 10. so if I take 1 over 1000 and I multiply by 10 I'd get to 1 over 100. if I did that again I get to 1 over 10. if I did that again I'd get to one right I'd pass over my decimal point I'd be at 1 then 10 then 100 then thousand so on and so forth so although you might not memorize the names right away you can kind of put it together by just using this pattern all right so we're going to do a quick little exercise similar to when we worked with place values for whole numbers we just want to State the place value for each digit and let's start out with this number 0.175 and this zero is just for clarity I can put it in the ones place but really there's no point I'm just thinking about these numbers that occur after the decimal point so let's fill in our place value chart I have a one that will go in my tenths place let me kind of make that a little better I have a 7 that's going to go in my hundredths place and I have a 5 that's going to go in my thousandths place and so remember if you have 0.175 you could write this as a decimal fraction has 175 over you have a 1 followed by three zeros right because you have three decimal places one two three and look at your bottom number it's one thousand look at your rightmost digit of this number it's in the thousandths place so now having said that let's go ahead and label everything so the one is going to be in the tenths place the tenth place and notice how everything after the decimal point has a th s right so you have 10 and then you put a THS 100 and then you put a THS 1000 then a THS you know so on and so forth then for seven that's going to be in the hundredths place hundredths and then lastly we have a five that's in the thousandths place the foul Zend s place all right let's take a look at the next one so we have 3.128 so the three is obviously in the ones place and we have our decimal point and then we have a one in the tenths place a two in the hundredths place and then an eight in the thousandths place all right so let's label this so 3 again is in the ones place of course we know that position already and the one is in the tenths the tenths the 2 is in the hundredths on dreads and then finally the eight is in the thousandths the thou Zen all right we have a really long number now so we have 15.77201958 so 15 here's your decimal point you have 7 7 2 0 1 9 5 8. now I want you to notice something again I want to draw your attention to the fact that the rightmost digit here is in the 100 millionths place so what that means is that just thinking about pure numbers forget about the this part of this number if I have a hundred million if I have a hundred million it looks like this so what that tells me is that my denominator here if I was to write this part just the part that occurs after the decimal point as a fraction the denominator would be 100 million right I can take this off now and then go ahead and put a fraction bar the top part would be 77201-958 or the number 77 million two hundred one thousand nine hundred fifty eight and again if you follow your official procedure you'll get the same answer you would write this part right here as your numerator and then you put a one followed by a zero for every decimal place we have one two three four five six seven eight and I have one two three four five six seven eight zeros so that does match up and of course if we were to write this number it does have a 15 out here as a whole number so I would need to put a whole number apart let me just kind of scoot this over a little bit let me scoot this over a little bit and then I could write 15 and 77 million 201958 100 millionths okay that's how you would say that all right but let's get back to what we're doing and of course we want to label the parts of the number and I don't even know if I can fit this on the screen but I'm going to try my best so the one is in the tens place this is the tens the 5 is in the ones the 7 is in the tenths the tenths this 7 appears in the hundredths hundredths then we have a 2 that's going to be in the thousandths then my zero let me like draw an arrow over here this is going to be 10 hyphen thousandths and then my next number which is this one here that is going to be in the hundred the hundred hyphen thousandths then I have this 9 that's going to be the millionths then I have the 5 the 5 is going to be in the 10 put a hyphen and then Millions and then finally I have this 8 and that 8 is going to be in the hundred the hundred hyphen millionths mil yes so if you're already really good with working with place value for whole numbers learning this will be really quite easy because everything is very similar it's just a simple fact that at the end you put a THS you just put a THS so you're going in a familiar way if you can just remember that as you move to the right you are basically multiplying by 1 10 or dividing by 10 you can go through and say okay if I have something in the tenths place you could think about just one over ten to figure out what the next place over is just multiply by 1 over 10. and you'd end up with one over a hundred so you could say okay I'm in the hundredths place now all right so the kind of the next topic we're going to talk about is something that students kind of struggle with so we want to know how can we compare the size of two or more decimals so we're going to kind of think about how to do this through a practice exercise we want to replace each question mark with the less than or greater than symbol and for the first problem we're going to look at we have 0.285 question mark 0.287 so kind of you're looking at these two numbers and it's not completely obvious right away which one is bigger it's not like if we looked at 13 and let's say five I know 13 is bigger than five right away for these two you might struggle a little bit so what I want you to do is just stack the numbers on top of each other so you have 0.285 and 0.287 and just line everything up by place value so this decimal point is going to line up with this decimal point you have the tenths that are in line the hundredths the thousandths and if you had more digits it'd be so on and so forth now what I'm going to do is I'm going to start at the very left of each number and I'm going to compare digits that have the same place value so these zeros are irrelevant I just kind of move on I'm going to look at this 2 and this 2. now those numbers are the exact same so I'm just going to move to the right I have an eight and an eight those numbers are the same so I'm going to move to the right now I have something different I have a five and I have a 7. now we all know that 7 is greater than five so that tells me that the number that has the seven is the bigger number so basically what you're doing is you're just moving to the right until you find corresponding digits that are different the digit that's larger is going to belong to the bigger number so for this relationship here the number 0.287 is bigger so I would put a less than sign in here right to replace that question mark let's go ahead and officially do that we'll put that 0.285 is less than or smaller than 0.287 and again that's because this 5 here that's in the thousandths place is smaller than this 7 that's in the thousands place for this number but again up to that point they look identical let's look at another one we have 12.9251 we have a question mark and then 12.9051 so again these numbers look kind of similar so stack them on top of each other so we'll have 12.9251 and 12.9051 and when we have some whole number parts involved you have to line those up as well the key here is you want to think about lining up the decimal point when you do that everything else is going to take care of itself right I have the tens places lined up the ones the tenths the hundredths the thousandths the ten thousandths and again if I had more digits it would go so on and so forth all right so we're going to start all the way at the left and one thing you have to make sure of is that these numbers have the same number of digits so if I had something like 112.9051 and 12.9251 obviously this one's bigger right because it has another digit to the left but that's not the case here we have 12 and we have 12. so that part is the same so we're going to move on past this decimal we have 9 and 9 and then we have 2 and 0. so once I see something that's different again whichever is the larger digit belongs to the larger number so this number is going to be bigger because 2 is greater than zero right and those numbers are in the same position so that tells me that 12.9251 is going to be greater than 12.9051 and again the reason for that is everything is identical until we get to this 2. this two that's in the hundredths place is bigger than this zero that's in the hundredths place for that number for the next one we have 0.0053051 question mark zero point zero zero five zero zero zero zero now one thing you might notice right away remember in the last lesson I taught you that when you have a decimal point after your final non-zero digit the zeros after it do not add any value I can delete these zeros and it would be the same as point zero zero five these two would be the same value whether I put the zeros on it or not right after your last non-zero digit you can put as many zeros after it as you want you're not changing the value of the number so with that being said let's compare our two decimals so we have [Music] 0.0053051 and zero point again lining up the decimal points zero zero five zero zero zero zero so everything is the same until we get to this right here we have a 3 here and a zero here the digit that's larger 3 is greater than zero so that tells me that this number right here is going to be the larger number so we can go ahead and say that 0.0053051 is larger than 0.00 and I can just put five here I don't need to add these zeros at the end you can do it if it makes you feel more comfortable but it's the exact same number all right let's take a look at one final one we have 15.489146 question mark 15.489276 so again I'm just going to stack the numbers on top of each other 15.489146 15.489 276. and at this point you can kind of just start eyeballing things I can see that everything is the same going down until I get right here and let me highlight that in light blue so I have a one and a half a two two is larger right two is greater than one so that tells me that this number right here is going to be bigger let me just kind of erase that this is going to be the bigger number and so that number is over here so I'll put that 15.489 146 is less than 15.489276 and the reason for that is my digit in the ten thousandths place and this number is a one my digit in the ten thousandths place and this number is a two two is greater than one so that means this number is the larger number so the last thing we're going to talk about is rounding decimals and basically it's the exact same procedure as you used for rounding whole numbers there's one slight difference though and as we're working through the problem I'm going to explain that to you and why it makes sense so we want to round 26.13 to the nearest tenth so the procedure again starts out the same so you have 26.13 . so find the digit in the round off place so where's the tenths place that's right here so this is my digit in the round off place that's a one then I look one digit to the right that's this three and you look for the rule right the rule is if it's four or less okay 4 or less if you leave the digit in the round off Place unchanged if it's 5 or greater you add 1 to the digit in the write-off place now this falls in the category of four or less so I'm going to leave one unchanged now when we round whole numbers we're told that we replace everything after the round off Place digit with a zero when you're working with decimals they're just going to tell you to cut everything off after the round off Place digit and why does that make sense well if I had followed my procedure for whole numbers I would have put a 0 here and that zero doesn't add any value to the number so I can just get rid of it it's different than if I was rounding 2613. let's say I wanted to round this number to the nearest ten well this is the round off place I would look at three three falls in the category of four or less so I would keep one unchanged and I would change three into a zero in this case the zero adds value to the number right the number zero at the end makes it two thousand six hundred ten if I just erase that I'd only have 261. well it's not the same when you're working with decimals if I have 26.10 or if I have 26.1 it's the exact same number all right let's take a look at another one we want around 572.91557 to the nearest hundredth so 572.91557 again to the nearest hundredth so where's the hundredths place this is the tenths place here's your hundredths place and again if you forget where your places are either kind of work with a place value chart or again knowing that this is the tenths or one over ten I can then move to the right by multiplying by 1 over 10 and 10 times 10 is 100 so I know I would get to one over a hundred or I'm in the hundredths place all right so now that we're in the hundredths place we look at the digit immediately to the right that's a five so that falls in the category of five or greater so that means I'm adding 1 to the digit in the round off place so I'm going to have 572.9 1 plus 1 is 2. and then again I would replace all of these with zeros but they don't add any value to the number so I just leave them off right I can just not put anything else after that too so my final answer here would be 572.92 okay let's take a look at one final problem we want to round 0.85791356012 to the nearest millionth so let's rewrite this long number 0.85791356012 so where's the millionths place again we have the tenths once we know that spot you can kind of just go in a familiar way tenths hundredths thousandths ten thousandths hundred thousands millionths so here's your millionth place this is the Millions place all right so once we find the digit in the round off place we look at the digit to the right of it in this case it's a five five is in the category of five or greater so I'm going to add 1 to that three it's going to become a four so I would have 0.857914 . and again when we round whole numbers at this point everything after it becomes a zero the zeros here if I change everything to a zero I'm not changing the value of the number so I can just delete them right I don't need to write that step because again 0.857914 is the same as 0.857914000 right I don't need to put those five zeros there they add no value to the number hello and welcome to pre-algebra lesson 41. in this video we're going to learn about adding and subtracting decimals all right for the lesson objective for today we want to learn how to add and subtract decimals and this is something where we can kind of just jump right in it's really really easy to add and subtract decimals especially if you've already mastered vertical Edition and vertical subtraction so to add or subtract decimals basically all we want to do is write the numbers vertically and line up the decimal points so you recall when we did multi-digit addition or multi-digit subtraction meaning vertical addition or vertical subtraction with whole numbers we stacked the numbers on top of each other and we line them up by place value well it's no different here we just need to make sure that we line up the decimal points when we stack our numbers on top of each other the next thing we want to do is bring the decimal point directly down into the answer so just go straight down don't deviate then lastly we're going to add or subtract starting with the rightmost column and working your way to the left so just like we did again with vertical addition or vertical subtraction so for the first problem we have 0.17 which is basically 0.17 plus 0.3 which is 0.3 remember these zeros here don't add any value to the number we just kind of write them to make it clear that we have some decimal points so I can just write .17 on top of 0.3 and put my addition sign out here and my horizontal line and I am ready to add now for some people it makes them more comfortable they have the same number of digits when they're adding so remember that if you have something like 0.3 or 0.30 it still has a value of 0.3 right after a decimal point when I have a non-zero number like three after that number I can put as many zeros as I want and I won't change the value so I could keep putting zeros if I want it's still going to be 0.3 so with that being said I can write a 0 in here and then I can start my addition remember we always start at the rightmost column and we work to the left so I would do seven plus zero that's seven and move to the left 1 plus 3 is 4. and then I told you in the lesson to bring this down to start it really doesn't matter when you do it it's just a matter of remembering to do it and when you first kind of start working with decimals sometimes people will forget they'll just go okay the answer is 47. just make sure that at some point in this procedure you bring that decimal point straight down into your answer and so we're going to end up with 0.47 or again for clarity if we want to we can write 0.47 right there's no change in the value of that number all right for the next one we have 72.65 minus 31.521 so again you're just going to stack these numbers on top of each other so 72.65 that has to go on top because remember with subtraction it's not commutative and then on the bottom you're going to have 31 points notice how I'm lining up the decimal points five two one okay so I'm going to write a horizontal line and then my subtraction symbol and now I'm ready to go well let me go ahead and put the decimal point down and again you can do this at any time I know in the procedure I put that you do it before you start your operation really it doesn't matter just remember to do it now when we start on our rightmost column you're going to find that you have a problem if I was doing addition here if I change this to addition I can just go ahead and bring down the one say okay I have one there and keep going with subtraction I need to have something that I can take one away from right so I need something minus one well remember I can put a zero at the end here and not change the value of the number and I can't do 0 minus 1 without getting a negative so what I would do is I would just borrow so I'd borrow from this five this would become 4 this would become ten and now I can do 10 minus 1 that's going to give me 9 4 minus 2 would give me two six minus 5 will give me one two minus 1 would give me one and then 7 minus 3 would give me four now since I already put my decimal point down there I really don't need to think about it again I can just write this number as my answer it's 41.129 all right now we're going to look at some addition with three numbers involved so we have 0.8274 plus 3.6 Plus 12.007 . again just stack the numbers on top of each other just make sure you line up the decimal points so I'm going to put point eight two seven four I'm going to put 3.6 and then I'm going to have 12.007 now this is one of the situations where you might want to put some zeros in so that you have the same number of digits for everything after the decimal point so I can put three zeros in here and a zero here again I'm not changing the value of the number so when I start my addition problem again I'm going to put this down here now to start so I don't forget I have 4 plus 0 plus 0 that's 4. I have 7 plus 0 plus 7 that's 14. so put a four down and carry the one again just like you would if you were adding some whole numbers then I have one plus two that's three plus zero plus zero that's still three and I have eight plus six that's fourteen plus zero is still fourteen so put a four down carry the one one plus three is four four plus two is six and I can just bring down that one so I get 16.4344 so let's write this up here sixteen point four three four four so what if we have some negatives involved well really we just kind of use the rules that we use when we added integers together just figure out what the sign is going to be first so we have a negative plus another negative that's going to give me a negative and then we can kind of just work with the absolute values so I can think about 6.13 Plus 7.85 so now I would just start in the rightmost column and work my way to the left 3 plus 5 is 8 1 plus 8 is 9 and 6 plus 7 is 13. and again notice how I forgot to bring my decimal point down so if I wrote 1 398 obviously I would get the wrong answer so you got to make sure that you're bringing this down at some point and put it there to where we know we have 13.98 as the answer now drag this back up here and put it next to that negative to say we have negative 13.98 as our answer all right for the next one I have negative 7.21 plus 3.64 so just think about the fact that if I was adding negative seven and three what would I do here well again I would use the sign from the number with the larger absolute value if I kind of get rid of this negative here and think about both numbers as if they were positive 7.21 would be larger so we're going to use its sign right its sine is negative and now we're going to do a subtraction we're going to take 7.21 and we're going to subtract the way 3.64 and let me go ahead and just write this down here before I start so I don't forget like I did in the last problem so we'll borrow from the two it will become a one this will become eleven eleven minus 4 is 7. then I'd have 1 minus 6 I need to borrow again so bar from the seven this becomes six this becomes eleven eleven minus six is now five and then six minus three is three so we get 3.57 drag that up there and attach the negative to it and my answer is going to be negative 3.57 okay for the next one we have 128.209 minus 34.55813 . so again just stack the numbers on top of each other because of subtraction this 128.209 has to go on top and then 34.55813 is going to go on the bottom put my minus sign out here and my horizontal line and I'm going to bring down my decimal point now again we have a problem because when we go to start subtracting in the rightmost column I don't have anything to subtract 3 away from again when you have this situation remember you can put zeros here to kind of fill in these spaces you're not adding any value to the number right it's the same number as it was and now I can go through and borrow but I have to go and borrow from the 9 9 becomes 8 this is going to become 10 then I'll borrow from the 10 10 becomes 9 and this 0 becomes 10. so now I can start cranking this out 10 minus 3 is going to give me 7. 9 minus 1 is going to give me 8. e minus 8 is going to give me 0. now I have 0 minus 5 so I need a borrow so I'm going to cross this 2 out and put a 1 this 0 becomes 10 10 minus 5 will give me 5. and now I have a 1 over here so I need to borrow again this 8 will become seven this one would become 11 11 minus 5 is 6. then I'll have 7 minus four that's three I need to borrow one last time so I'll borrow from the one it will become zero this will become 12 12 minus three is nine so we end up with 93.65087 so 93.65087 . all right now we have addition and subtraction mixed so I have 0.685 plus 0.312 minus 5.24 minus a negative 7.035 so the first thing I want you to do is just rewrite this we have 0.685 plus 0.312 minus 5.24 minus a negative is plus a positive so I'm going to put plus positive 7.035 . so even though we're not working with integers anymore we have some decimals involved the same rules are going to apply I'm subtracting away a negative I'm adding a positive so I'm just going to work left to right here and I'm going to start out with 0.685 Plus 0.312 bring down the decimal point five plus two is seven eight plus one is nine six plus three is nine so we get 0.997 I'm gonna put equals 0.997 let me just put a zero out here just for clarity so that took care of those two and then I'm subtracting away 5.24 and I'm adding 7.035 so the next thing I'm going to do is 0.997 minus 5.24 so let me write this over here 5.24 . so because this is going to result in a negative when I do this subtraction kind of the easy way to do this would be to reorder everything just think about having .997 plus a negative 5.24 what will we do in this situation well we know that the answer would be negative and then what we do is we would just think about the absolute values for a second then we would subtract 5.24 minus 0.997 so I'm going to put a zero here to start again that doesn't add any value to the number and I'm going to borrow from this 4 this will become 3. and then this is going to become 10 10 minus 7 is 3. I have a 3 minus 9 so I need to borrow again so I'm going to borrow from the 2 it becomes one this becomes 13 13 minus 9 is 4. then I have 1 minus 9 I need to borrow again this becomes 11. 11 minus 9 is 2. again I forgot to bring down my decimal point when I started but again as long as you remember to do it at some point you'll be okay then lastly I have the four nothing to subtract away so I just bring it down and I've got 4.243 now remember this is negative I already wrote my negative sign out here so I'm going to have a negative 4.243 okay so there's one more operation to carry out here and I'm just going to be adding to this 7.035 so negative plus positive again we think about what the sign is going to be and 7.035 is bigger in terms of its absolute value than this negative 4.243 so our answer is going to be positive and now we just need to subtract so I'm going to do 7.035 minus 4.243 so 5 minus 3 is 2. so now I want to do 3 minus 4 and I need to borrow I can't borrow from the zero so I got to go to this seven seven becomes 6 the 0 becomes 10 then 9 then finally this 3 becomes 13 13 minus 4 is 9. then I'm going to have 9 minus 2 that's 7. and again I forgot to bring down my decimal point when I started and so I bring it down now and then I'll have 6 minus 4 that's 2. so I'm going to get a positive 2.792 . so again when you're adding and subtracting decimals it's not really any more difficult than when you were just performing vertical addition or vertical subtraction with whole numbers a couple of things we sprinkled in here some negatives and some situations where we had to you know write some zeros in but nothing that's really too complicated hello and welcome to pre-algebra lesson 42. in this video we're going to learn about multiplying decimals so our lesson objective for today is just to learn how to multiply decimals so before we get started I just want to mention that multiplying decimals is no more difficult than multiplying whole numbers if you've already mastered multi-digit multiplication meaning vertical multiplication then basically at this point you can already multiply decimals they're just one additional step that you're going to have to deal with and that's going to be where you're going to put your decimal point at so here we go multiplying decimals the first thing we want to do we want to ignore any decimal points and multiply the resulting numbers so essentially what I'm saying here is that you're going to pretend like your numbers are whole numbers so if I had something like 5.5 times 3.7 I would start out by pretending I had 55 times 37 right I would find the answer for that now in the next step I want to move the decimal point in the answer and when I say the answer I mean the answer to the multiplication I just got where I was using whole numbers so I want to move this to the left by the number of decimal places between all factors okay so between all factors so let's jump in and look at an example so we're going to begin today with 5.15 times 0.32 so I'm going to start out I'm going to ignore any decimal points that I have and so this number here will be 515 just going to treat it like it's a whole number and this number here will be 32. and I just find this product first and then I'll figure out where to put the decimal point so 2 times 5 is 10. zero down carry the one two times one is two plus one is three two times five is ten okay I'm going to erase this remember we're moving to the left here because we're starting over here in the tens place now three times five is fifteen five down carry a one three times one is three plus one is four and then three times five is fifteen and so now we're going to add go ahead and bring down this zero three plus five is eight zero plus four is four one plus five is six and then bring down that one so what I get is sixteen thousand four hundred eighty but that's obviously not my answer because I haven't figured out where to put the decimal point yet so just go ahead and come up here drag this up one six four eight zero and now we're going to figure out where to put the decimal point at now what I told you is that we're going to move the decimal point to the left by the number of decimal places we have between all the factors so if I look at these two factors here and 5.15 I have one two decimal places two decimal places and in 0.32 I also have two decimal places okay we're talking about what's after that decimal point right here I have two and here I have two so if I combine those two two plus two was four so between all of the factors here we have four decimal places so that tells me we're going to have to move our decimal point four places to the left now when we don't have a visible decimal point like in a whole number like we have here we can always write one at the very end of the number so at the very right part of the number and now I'm free to move my decimal point four places to the left so I'm going to go one two three four places to the left so my decimal point is going to go right there and we have an answer that's 1.6480 but remember I don't need this zero here I can erase it because that zero does not add any value to the number after your decimal point or to the right of the decimal point after your final non-zero digit you can add as many zeros as you want so I could put zero zero zero zero zero it's still 1.648 right these zeros do not change the value of the number at all all right so again our final answer here will be 1.648 okay for the next one we have 0.00153 times 2.71 so again I'm going to ignore the decimal points and I'm going to pretend like I had 153 153 times 271. 271. and I just multiply so 1 times 3 is 3. 1 times 5 is 5 and 1 times 1 is 1. now I'm going to the tens place I need to move to the left of my answer 7 times 3 is 21. put a 1 down carry the 2. 7 times 5 is 35 Plus 2 is 37 7 now carry the 3. 7 times 1 is 7 plus 3 is 10. and move over one more time we're going to the hundreds place two times three is six two times five is ten zero down carry the one two times one is two plus one is three all right so let's add now get rid of these arrows bring down the three five plus one is six one plus seven is eight plus six is fourteen four down carry the one one plus zero plus zero is one and then one plus three is four so we end up with forty one thousand four hundred sixty three but again that's not our answer right we can just drag this up to the top here and just to get a little room going this is going to erase this so again it was forty one thousand four hundred sixty three my next step is to figure out where I'm going to put the decimal point in this number if I look between the two factors here I need to count how many decimal places I have in this number I have one two three four five decimal places right five decimal places and this number over here I have two one two two decimal places so between all the factors or between both factors I have five plus two or seven decimal places so that tells me I need to move my decimal point seven places to the left let me erase this real quick and we'll put a visible decimal point to the right of that 3. remember if you have a whole number you can just put it at the end of the whole number and we're going to move the decimal point seven places to the left so one two three four five and do I stop now no I put some zeros in right so I put a zero in here so that I can do six and then one more zero so that I can do seven and so I put my decimal point there can erase all this at this point and say that my answer is .0041463 all right let's take a look at one with a negative involved so we have negative 3.01 times 1.8 so remember when we worked with integers we learned that a negative times a positive is a negative the rules are the same for all real numbers so when you come across a negative times a positive you're always going to get a negative doesn't matter if you're multiplying fractions or decimals or whole numbers or integers whatever you're doing negative times positive is negative now once we figure out the sign remember we can just work with the absolute value so I could just pretend like I had positive 3.01 times positive 1.8 and when we multiply decimals remember the first step is just to ignore the decimal point and pretend like we had whole numbers so 301 times 18. so a times 1 is 8 a times 0 is 0 8 times 3 is 24. now we're moving to the left because we're going to the tens place we're doing 1 times 301 that's 301 so I'm going to write a 1 a 0 and a three and now we're going to add so just bring down this eight zero plus one is one four plus zero is four two plus three is five so I get five thousand four hundred eighteen and again this is not my answer I'll write this up here next to this negative five thousand four hundred eighteen and I can go ahead and erase this now let's just figure out where we're going to put our decimal point at again I'm looking for how many decimal places I have between the factors so here I have one two decimal places two decimal places here I have one one decimal place so what that means is that between all the factors or between both factors here I have three decimal places so my decimal point is going to go three places to the left so I'm going to put it right there to start and then I'm going 1 2 3 places to the left so I'm going to end up with negative okay don't forget the negative out here 5.418 as my answer okay let's take a look at one final problem and then we'll kind of jump into a section where we'll talk about multiplying by 10 or power of 10. something that's going to help you speed up your multiplication all right so we have negative 19.06 times 0.0051 times negative 2.02 now what we're looking at here is we have a negative times a positive times a negative now we know from working with integers that if we multiply more than two numbers and we have an even number of negatives we're going to get a positive result right each pair of negatives makes a positive so I'm just going to go ahead and write that the answer is going to be positive and now we can just think about these numbers as if they were all positive right we just think about the absolute values now the next thing we want to do is multiply any of the two numbers together and so I'm going to start out with 19.06 times .0051 so to do that I'm going to pretend this is 1906. times 51. so 1 times 1906 is 1906. then I'm moving to the left here so I move to the left when I put my answer down 5 times 6 is 30. zero down carry of the three five times zero is zero plus three is three five times nine is 45 put a five down carry the four five times one is five plus four is nine so now let's add bring down the six zero plus zero zero nine plus three is twelve two down carry the one one plus one is two plus five is seven bring down the nine so I get ninety seven thousand two hundred six so let me write that here ninety seven thousand two hundred six now at this point you can figure out where the decimal point would be based on these two right here or you can keep going and multiply this number times this number here as if it was 202. now both ways will work I think it's a little faster if we just figure out where the decimal point is going to be in the end so let's go ahead and do that so I'm going to think that I have ninety seven thousand two hundred six times 202 for right now okay so 97 206 times 202 and once we figure this out we're gonna put our answer here we're gonna go back and figure out where the decimal point should be so 2 times 6 is 12. put a two down carry the one two times zero is zero plus one is one two times two is four two times seven is fourteen four down carry the one two times nine is eighteen plus one is nineteen now I know zero times anything is zero so that would have started here I'll just put one zero there then we're going to the two choosing the hundredths place so I'm moving over one more time my answer is going to start there two times six is twelve two down carry the one two times zero is zero plus one is one two times two is four two times seven is fourteen put a four down carry of the one and lastly 2 times 9 is 18 plus one is nineteen so now we're going to add go ahead and bring down the two one plus zero is one four plus two is six four plus one is five nine plus four is thirteen put a three down carry the one one plus one is two plus four is six bring down the nine bring down the one so we end up with 19 million 635 612. so let's erase this we're going to write that up top so now we want to just figure out where our decimal point should be in this answer so to do that we count the total number of decimal places between all of the factors so if I come here I have one two decimal places so two decimal places here I have one two three four so four decimal places and here again I have one two decimal places so two plus four is six six plus two is eight so that means I need to move my decimal point eight places to the left so I'm going to start it right here and let me kind of make this number a little bit straighter so again put the decimal point here and we're going eight places to the left so one two three four five six seven and finally eight so that decimal point is going to go out in front and we have point one nine six three five six one two as our final answer so for this next section I want to talk a little bit about multiplying by ten or a power of 10. and I want to say power of 10 let me just explain that I'm talking about an exponent on 10 that's going to be a whole number that's one or larger so for example you'd have something like 10 to the first Power which is just 10. 10 squared which is a hundred ten cubed which is a thousand ten to the fourth power which is ten thousand 10 to the fifth power you know so on and so forth right I'm not talking about raising it to 10 to the power of negative four right I'm only including these values here so for the trick essentially what you're going to do is you're going to move your decimal point one place to the right for every zero in the power of 10. so let me give you something to start with that you're already familiar with let's say you had the number five and you multiply it by 10. well we already know that this is 50. but one of the ways you can figure this out is you can put a decimal point behind Five right you can do that with a whole number and I can put a zero behind that decimal point perfectly legal if I'm multiplying by 10 really all I need to do is take that decimal point and move it to the right one place because there's one zero here so that tells me that the answer is 50. if I took 5.0 and multiplied it by 100 okay by 100 there's two zeros in one hundred so five point zero move this one put another zero in two places to the right and I'd get five hundred I could do the same thing with a thousand or ten thousand so on and so forth now really a hundred is just ten squared so I could replace this hundred with ten squared if I want and I know that if the exponent is a two I'm going to have a 1 followed by two zeros right so it's one followed by two zeros this tells me how many zeros I'll have so without even converting ten Square to a hundred I know I just take 5.0 and move that decimal point one two places to the right to get five hundred and so if I ran across something like 5.0 times 10 to the ninth power I know I would do five point and then I would basically just move this nine places to the right so let's do that so one two three four five six seven eight nine so because this is all the way out there I'm gonna use some commas now and we'd end up with the number 5 billion all right so let's try some basic ones here we have 0.015 times 10. so I'm going to take this decimal point move it one place to the right and so I'd end up with what I would end up with point one five or I could put a zero out in front I have 0.15 in that case what about in the case of 0.015 same numbers above times 10 squared well really all I need to do is move this one place to the right from here because I have one additional zero here I have one zero here I have two right I can just look at the exponent so I can take this number 0.15 basically move it one place to the right and I'd have 1.5 or you can do it from here and just say okay I have one two places to move my decimal and again I get 1.5 in this case I'm moving the decimal four places to the right all right I'm multiplying by 10 to the fourth power there's going to be four zeros in that power of 10. so what I want to do is go one two three add a zero and then finally four so that's going to give me one five zero or a hundred fifty right because if I kept the zero out in front it would add no value to the number zero one five zero is the same thing as a hundred fifty right so that's how we would read that number here's an example where this would really come in handy let's say you get this on a test negative 23.2815 times 0.001 times 10 squared if you do this without using any technique it might take you a little while but the first thing I do is I kind of eyeball and I see that okay 10 squared times .001 I can just move this one two places to the right and have point one so essentially this problem just becomes negative 23.2815 times 0.1 or for clarity we can do 0.1 now I know that negative times positive is negative so I know the answer is negative so if I were to pretend these are whole numbers I would do 232 815 times 1. well that's just going to give me two hundred thirty two thousand eight hundred fifteen at that point I just have to figure out where to put my decimal point and I'd say okay well I have one two three four five decimal places between both factors here so the decimal point is going to go one two three four five places to the left and then end up with negative 2.32815 all right for the final problem we have 0.012456 times a hundred times a thousand so a hundred most of you should know at this point is 10 squared right that's a power of 10 and the exponent is a whole number that's one or larger 1000 is also a power of 10. it's 10 cubed right so same scenario so we can use our trick here and you basically can count the number of zeros between both of these you have one two three four five because if I was to multiply these two numbers together let's say I started out by saying what is a hundred times a thousand well using my trick for trailing zeros I know that I would do one times one that's one and then I would attach five trailing zeros right so one two three four five I'd get the number one hundred thousand or we could write that as 10 to the fifth power so let me put this over here 0.012456 and now to do this multiplication all we need to do is move our decimal point five places to the right right because we have one two three four five zeros in this power of 10. so let me copy this number we're going one two three four five places to the right and I can delete this 0 and this zero I don't need them anymore they're not going to add value to the number and so we would have 1245.6 hello and welcome to pre-algebra lesson 43. in this video we're going to learn about dividing decimals so our lesson objectives for today we want to learn how to divide a decimal by a whole number and also we want to learn how to divide a decimal or whole number by a decimal so before I started talking about multiplying decimals I told you that it was really no more difficult than multiplying whole numbers you just have an extra step involved well it's really the same thing here some cases you're going to have one extra step in some cases you'll have two extra steps just depending on what the situation is but if you can divide whole numbers using long division you can divide decimals it's no more difficult so let's start out by saying to divide a decimal by a whole number which is the easier scenario we just want to set up the long division normally and place the decimal point in the answer directly above the decimal point in the dividend now I know we haven't talked about division in a while so just remember if you have something like 6 divided by 3 equals two the six the amount you start with is the dividend the dividend 3 the amount you're dividing by is the divisor and two the answer that you get or the result is the quotient so once you've performed this step we'll replace the decimal point and the answer directly above the decimal point in the dividend you just divide normally right you can just ignore the decimal point for 10 lines like it's not there because you already know where it's going to be in your answer all right let's start out with 1.5 divided by 3. so we set up our long division normally so I take my dividend which is 1.5 and that goes underneath I take my divisor which is 3 and I put it off to the left now the first step we're going to do is we're going to take this decimal point and we're going to bring it straight up into the answer make sure it lines up properly otherwise you won't get the right answer now once I've done that I know where the decimal point is going to be and I can just pretend I have 15 divided by 3 right so I would say what is 3 going into one well we can't do that so how many times would 3 go into 15 without going 5 times so I put a 5 here and then I multiply 5 times 3 is 15 and I subtract and I get 0. so my answer for this is going to be 0.5 or I could put 0.5 now remember we can check division with multiplication so we already know how to multiply with decimals I could do 0.5 times 3 right I could go backwards and I should get 1.5 so what is 0.5 times 3. well we already know that we do 5 times 3 that would give us 15. and I would look to see how many decimal places I have between the factors well this has one decimal place this really doesn't have any so I would move my decimal point one place to the left and I would end up with 1.5 1.5 which is exactly what I started with as my dividend right so we know that this answer is correct 1.5 divided by 3 is 0.5 let's take a look at the next one we have negative 2.17 divided by 7. now again if you're multiplying or dividing with negatives the same rules that we learned with integers are going to apply negative divided by positive is negative once you find out the sign you can just pretend like all the numbers are positive you don't need to think about it anymore so I'm just going to set up my long division as 2.17 and we're dividing by seven now bring your decimal point straight up into the answer and then you're done with it you don't have to think about it anymore you can just divide like if you had 217 divided by 7. so 7 is not going to go into 2 but 7 will go into 21. it goes in exactly three times three times seven is 21. subtract and get zero bring down the seven seven will go into seven one time one times seven is seven subtracting gets zero so we have 0.31 as our answer but remember we have this negative up here so let's attach that and we'll say that we have negative 0.31 as our answer and again when you first start doing this it's good for you to check right take your quotient your answer multiply it by the divisor you should get the dividend so let's do one more where we check it and then in the future I'll leave it up to you to check your work but it's good practice to check because you just learn how to multiply decimals and so it can kind of reinforce what you learned so you can take negative 0.31 and we're multiplying this by 7. so what is 7 times 31 so 31 times 7 7 times 1 is 7 7 times 3 is 21. so we would have 217 there remember it's a negative times a positive so that gives a negative and then we just need to figure out how many decimal places we have between the factors we have one two right not over here so this would go one two places to the left we'd have negative 2.17 which is exactly what you have here as your dividend so again you know your answer is correct right our answer here is negative 0.31 okay for our next one we have 44.95 divided by 31. so again let's set this up 44.95 divided by 31. bring this up into the answer so you can forget about it and now you just divide normally so 31 will not go into 4 but it will go into 44. it'll go on once 1 times 31 is 31. subtract here and you would get 13. right 4 minus 1 is 3 4 minus 3 is 1. now bring down the next number how many times will 31 go into 139 well let's see I know that I know that 31 times 10 is 310. 310. let's say I was to mentally cut that in half think about 410 as 300 Plus 10. cut 300 in half you'd have 150. cut 10 and a half you'd have five so half of this would be 155. that's too big so my guess is going to be for 4. so what is 31 times 4 4 times 1 is 4 4 times 3 is 12. That's 124. So that's as close as we can get without going over so we're going to put a 4 here and we're going to multiply 4 times 31 again is 124. now we're going to subtract 9 minus 4 is 5. 3 minus 2 is 1. so we'd have 15 there and then we're going to bring down the 5. now we figured out earlier that 31 times 5 was 155 right we just saw that so I know that 31 will go into 155 five times exactly so this is going to be a 5 5 times 31 again is 155 subtract and we get 0. so we get 1.45 as the answer and let me just scroll back up here and write this next to this so equals 1.45 so now we're going to kind of use this information to kind of tackle something that might have been confusing you for a while I know that a lot of you will do division on a calculator let's say you have something like 17 divided by 5. so if you perform this operation using long division 5 doesn't go into one five goes into 17 3 times 3 times 5 is 15 you subtract and you get 2. so kind of up to this point we've written three with a remainder of two as our answer now your calculator won't give you three with a remainder of 2. it's going to tell you that the answer is 3.4 so how do we go from this format to the 3.4 format well that's what we're going to get to right now so let me erase this real quick and kind of give you the back story so I want you to recall that after a decimal point a zero does not add value unless it precedes a non-zero number so something like 5.000 is still five right where you run into problems if you had something like 5.00001 well these zeros do matter now right because they come before a non-zero digit they precede a non-zero number okay but if you don't have anything that's out there other than zero you can put as many as you want it doesn't matter you're not adding value to the number so knowing that let's explore that 17 divided by five again and see if we can get 3.4 as our answer so 17 divided by 5. so the first thing I'm going to do is I'm going to write 17 as 17 point and I'm going to put 1 0 behind it I'm going to put this decimal point up here in the answer and now I'm going to divide like I normally would 5 won't go into 1 so we say 5 goes into 17 3 times 3 times 5 is 15. subtract and get 2. now instead of stopping and saying that's my remainder I can bring down the zero 5 goes into 24 times 4 times 5 is 20 subtracting at zero so now you see what your calculator is doing it's just continuing the division right so our answer here is 3.4 it doesn't make it wrong if you say you have 3 with a remainder of 2 it's just a little bit more precise to use this method and say okay well our answer is 3.4 let's try another one so let's say we had 6.2 divided by 5. so 6.2 divided by 5. so we already have a decimal point here let's bring that up into the answer and let's divide 5 goes into 6 once 1 times 5 is 5. subtracting you'll get one bring down the two five goes into 12 twice 2 times 5 is 10 subtracting get 2. so we're not going to say that we have 1.2 with a remainder of 2 we're going to continue the division so remember if I have 6.2 I can legally add as many zeros after that 2 as I want not changing the value of the number so let's continue our division let's put a zero here and then let's bring it down so 5 will now go into 20 exactly 4 times 4 times 5 is 20. subtracting get zero so our answer here is going to be 1.24 so we're going to get 1.24 this is basically what we're going to do from here on out we're going to continue the division until we get an answer or sometimes we're going to get a digit that repeats or a pattern of digits that repeat and we'll get to an example of that in a minute all right let's take a look at another one we have 65.98 divided by 50. so 65.98 divided by 50. so bring your decimal point up into the answer 50 is not going to go into 6 but it will go into 65 it'll go on once 1 times 50 is 50. subtract 5 minus 0 is 5 6 minus 5 is 1. and go ahead and bring down the 9. so now how many times will 50 go into 159 well 50 times 3 is 150 and that's about as good as we can do so let's put a 3 here 3 times 50 as I just said is 150 subtracted we get 9 bring down the eight and let me scroll down get a little room going here so I know that 50 is going to go into 98 once only once 1 times 50 is 50. subtract and we'd have 48 right 8 minus zero is eight nine minus 5 is 4. so we continue right we put a zero after the eight and we bring that down so now we're going to ask how many times will 50 go into 480 well I know 50 times 10 is 500 so I'd have to do 50 times 9. so 9 times 5 is 45 attaches zero it's 450. let me kind of go back and forth here because we're not going to be able to fit all this on the screen so 450 subtracting you'd have 30. so I can keep adding zeros until I have something so I'm going to add another zero here and drag that down I'm going to drag that down so now I have 300. now let me scroll up so 50 goes into 300 exactly six times so put a 6 here and 6 times 50 again is 300 and when we subtract we're going to have zero so there's no remainder go up here and our answer is going to be 1.3196 so let's take a look at this other scenario that you're going to run into occasionally we're going to get a digit or series of digits that repeat forever so when this occurs we place kind of a bar over the part that repeats so if I got an answer that was 4.333333 and that three just kept continuing forever what I do is I just put a bar over the three or if I had three digits like three six seven three six seven three six seven I would just erase these and put a bar over the three six and seven right the part that repeats let's look at 1.3 divided by 6. so 1.3 divided by 6. so bring this up into the answer 6 doesn't go into one it goes into 13 twice 2 times 6 is 12. subtract and get one put a zero here to continue the division bring that down 6 is going to go into 10 once 1 times 6 is 6. subtracting get 4. put a zero to continue the division bring that down 6 is going to go into 46 times 6 times 6 is 36 subtract and get four put a zero to continue the division bring that down 6 goes into forty six times six times six is 36 subtract and get four now a lot of you at this point can already see there's an issue every time you continue this if I put another zero here and bring it down I'm going to have 40 again I had 40 here then here then here it's just going to keep going because every time you're going to ask how many times will 6 go into 40 well it's going to go in six times so I'm going to have another 6 up there I'll multiply 6 times 6 is 36. I'm going to subtract 40 minus 36 and I'm going to get a 4. bring down another 0 I'll have 40 again this will continue forever and ever and ever so once you recognize that there's a pattern going on you just stop and you say okay the six will repeat forever so my answer is .216 with a bar on top of the six okay a bar on top of the six that way we know that that 6 continues forever all right let's take a look at another one where you'll have a pattern that continues forever so you have 51 0.2 and we're dividing this by 11. so bring that up into the answer and 11 will not go into 5. it's going to go into 51 four times 4 times 11 is 44. subtract here I would need to borrow this would become 4 this would become 11. 11 minus 4 is 7 4 minus 4 is 0. so I'll go ahead and bring down this 2 here 11 will go into 72 6 times 6 times 11 is 66 subtract I need to borrow 12 minus 6 is 6. so I'm going to put a zero here so I can continue and I have 60. 11 goes into 60 five times 5 times 11 is 55. subtract then I get 5. go ahead and put a zero here and bring that down 11 goes into 54 times 4 times 11 is 44. subtract and you're going to get 6. so we're going to bring down another zero and we're going to have 60. remember the divisor is 11. I don't want to scroll back and forth the divisor is 11. so you'd say how many times does 11 go into 60 well it's going to go in five times it's going to go in 5 times so this will be a 5 5 times 11 is 55. 55 and subtract you can already kind of see what pattern is going to emerge right you have 60 minus 55 here and then you have it here again so you know the result is going to be this next right underneath it I could basically just write 50 and 44 and minus and keep going right that's the pattern but let me erase this real quick and say that okay this will be 5 this will be 10 10 minus 5 is 5. and again I'd bring down a 0 from all the way up top let me do that and we'd have 50 again 11 would go into 50 only four times only four times so you can really see that it's five four five four five four that's the pattern that's going to continue forever so 4 times 11 is 44. subtract and you get six so again you're going to start out with 60 again right you have 60 here and then you go to 50 then 60 then 50 and then that's exactly what's happening again you'd have 60 you go through that again you'd have 50. so on and so forth it would just continue forever and ever and ever so when you notice your pattern you can go ahead and just stop and say okay I have 4.654 now the 5 4 is what's repeating forever so I'm going to put that bar over the five and the 4 only but make sure you cover both digits whatever's repeating so I'm basically demonstrating that the 5 and the 4 will continue forever so in other words 4.654545454 you know so on and so forth all right so now another important fact now when we talked about multiplying decimals I kind of work into the situation when we multiply by 10 or a power of 10 and when I say a power of 10 I mean a whole number that is one or larger so we're talking about 10 to the first Power 10 squared 10 cubed 10 to the fourth so on and so forth we saw that we'd move the decimal point one place to the right for every 0 and the power of 10. so if I did you know 96 times 10 cubed well 10 cubed is a power of 10 with three zeros right it's 1 000. this is basically 96 times 1000 right so I have one two three zeros there so I'm going to move the decimal point three places to the right I'd go one two three and ended up with 96 000 as an answer when we divide by 10 or a power of 10 when I say power of 10 again I'm saying a whole number that's one or larger we're going to kind of do the reverse we're going to move the decimal point one place to the left for every 0 and the power of 10. so multiplying it's going to the right dividing it's going to the left so if I have 96.12 divided by 10 the shortcut is just to say okay I have one zero going to the left by one that would be 9.612 if I have 96.12 divided by a hundred the shortcut is to say okay I have two zeros so I'm going to go one two places to the left so I'd have 0.9612 if I have 96.12 divided by ten thousand I have one two three four zeros in this power of ten so I'm going to have 96.12 I need to put two zeros over here I'm gonna go one two three four places to the left and I'll end up with .009612 as my answer okay let's talk about the other scenario now so our other scenario involves dividing by a decimal and basically what we're going to do here we're going to move the decimal point in the divisor by enough places right to make it into a whole number now once we've done that our kind of next step is to match that movement with the decimal point in the dividend so as a quick example let's say I'm dividing 6.3 by I don't know 3.27 what I want to do is I want to start out with my divisor remember that's the number you're dividing by and I want to make it into a whole number so I'm basically going to move this decimal point one two places to the right I just have to match that movement in my dividend so for 6.3 I'm going to put a 0 behind the 3 and I'm going to move this one two places to the right as well so I'm basically turning this into the division problem 630 divided by 327. now you might stop for a minute and try to figure out how that's legal the best way I can explain it to you is if you put it in fractional form you can kind of think about having 6.3 in the numerator over 3.27 in the denominator and let's say I just multiply the numerator and denominator by a hundred remember that's just like multiplying by one so we would end up with again 630 630 over 327. so let's take a look at a few examples so if I have 91.35 divided by 6.09 this is the divisor this is the divisor and essentially what I'm doing is I'm saying okay I need to move this one two places to the right to make it a whole number to make it the number 600 knot I just need to match that over here move this two places to the right basically I'm just multiplying each by a hundred right we learned earlier if you multiply by 10 you go one place to the right if you multiply by 10 squared or 100 there's two zeros there so you go two places to the right that's all we're doing in fractional form you can kind of think about this set up this way if I just multiply the numerator and denominator here by a hundred [Music] what I'm going to end up with is what I'm going to end up with 9135 over 609 right that's all I'm basically doing I'm matching the movement that was made in the divisor that made it into a whole number in my dividend all right so that's the first thing we want to do so once we have that I'll just rewrite this problem I'll just say it's 9135 divided by 609 and we'll just crank that out real quick so 609 will go over here 9135 will go here let me scroll down and get some room going and so 609 will go into 913 right it's not going to go into 9 or 91 but 913 once 1 times 609 is 609 we would subtract here and I'll borrow here 13 minus 9 is 4 0 minus zero is zero nine minus six is three so that's 304 bring down the 5. so we have 609 how many times is that going to go into 3045 well you can eyeball this and see that it's going to work out perfectly right think about 609 is 600 Plus 9. If I multiply 600 times 5 I get 3 000. If I multiply 9 times 5 I get 45. so that tells me that what I'm looking for here is a 5. right 5 times 609 is going to give me 3045. 5 times 9 is 45. five down carry the four five times zero is zero plus four is four five times six is thirty so again three thousand forty five so again 5 times 609 is three thousand forty five subtracting gets zero so our answer here is just fifteen so let me erase this and we'll scroll up and write that down that our answer is going to be 15. let's take a look at one more all right so we have 183.4272 and we're going to divide this by 21.23 so again I'm looking at my divisor my divisor and I want to make it into a whole number so I need to move this one two places to the right basically I'm multiplying it by a hundred I need to match that movement over here in the dividend so I'm going to move this one two places to the right a lot of students get confused at first they can't remember what they need to move remember it's always the device what you're dividing by that you need to change it to a whole number so I'm going to write this problem now as 18 000. 342.72 divided by 2123. so now let's crank this out I have 18 342.72 again I'm dividing this by 2123 and when you work with really big numbers like this it can be very very tedious and I doubt you get something like this on a test if you did it would probably be like your final question or something like that I just like to do some examples that are a little bit challenging so that you get experience working with something that's very tedious so 2123 it's not going to go into one it's not going to go into 18 it's not going to go 183 it's not going to go into 1834. the first thing we could check is 18 342. so we got to make some guesses I would guess with a 9 that's probably going to be too big so 2123 times 9 9 times 3 is 27. 9 times 2 is 18 plus 2 is 20. pi times 1 is 9 plus 2 is 11. 9 times 2 is 18 plus 1 is 19. so that's 19107 that's too big so we're gonna go with eight so I'll put an eight here and let's do that multiplication real quick so we can get our product going eight times three is twenty four eight times two is sixteen plus two is eighteen eight times one is eight plus one is nine eight times two is sixteen so I'm gonna get sixteen thousand sixteen thousand nine hundred eighty four as my answer there sixteen thousand nine hundred eighty four we're going to subtract and I'll need to borrow here this will become 3 and 12 12 minus four is eight I'm gonna borrow again this will become two this is thirteen thirteen minus eight is five it'll borrow again this will become seven this will become twelve twelve minus nine is three seven minus six is going to give me one and then one minus one is zero so I get one thousand three hundred fifty eight one thousand three hundred fifty eight and I'm gonna bring down my seven so now I'm working with thirteen thousand five hundred eighty seven so again I'm going to make a little guess so what I do is I think about okay two thousand times six would give me twelve thousand let's just try that out first so two thousand one hundred twenty three times six let me scroll down get a little room going six times three is eighteen 6 times 2 is 12 plus 1 is 13. six times one is six plus one is seven six times two is twelve so twelve thousand seven hundred thirty eight that's as close as we're gonna get twelve thousand seven hundred thirty eight is as close as we can get so 2123 will go into 13 587 6 times 6 times 2123 again is twelve thousand seven hundred thirty eight so now we want to subtract and let me just move this over a little bit so I'm gonna borrow here to start this will become seven this will become Seventeen Seventeen minus eight is nine seven minus three is four borrow again fifteen minus seven is eight two minus two is zero one minus one is zero so basically I have 849 and I'm going to bring down this two here so I get 8492. so many times will 2123 go into 8492. well I'm gonna guess four right because 2 000 times 4 will be eight thousand so let's check that out what would 2123 times 4 give me 4 times 3 is 12. four times two is eight plus one is nine four times one is four four times two is eight so two thousand one to twenty three will go in eight thousand four to ninety two exactly four times four times two thousand one of twenty three is eight thousand forty ninety two subtract and we will get zero and we are done our answer here is going to be eight point six four let's go back up to the top and write that in so this is going to be 8.64 as our final answer hello and welcome to pre-algebra lesson 44. in this video we're going to learn about changing fractions to decimals so our lesson objective for today is just to learn how to change a non-decimal fraction into a decimal and the procedure for this is very very very simple to convert a non-decimal fraction remember a decimal fraction is a fraction whose denominator is 10 or a power of 10. so again to convert a non-decimal fraction into a decimal we divide the numerator by the denominator very very easy to do so let's start out with something like 1 4. so all we need to do is take the numerator which is one and divide it by the denominator which is 4. now in the case of a proper fraction like you have here your numerator is smaller than your denominator so what you're going to have to do is put a decimal point at the end of your dividend bring it up into the answer and then put a zero after that decimal point now we can do our division so we would ask how many times will 4 go into 10 well it will go in twice multiply 2 times 4 you're going to get 8. subtract here you will get 2 and I could put another zero so I can continue my division go ahead and bring that down 4 goes into 25 times 5 times 4 is 20. subtracting you would get 0. so 1 4 is the same as 0.25 or you could write 0.25 all right let's take a look at another we have two-thirds so again I'm going to divide the numerator by the denominator so 2 is going to be divided by 3. now 2 is smaller than 3 so I know that I have to put my decimal point after 2. bring that up into the answer put a 0 behind it and now I can start my division 3 will go into 26 times 6 times 3 is 18. subtracting you would get two I could put another 0 and bring that down 3 goes into 20 again 6 times 6 times 3 is 18. subtracting you would get two and I could keep doing this but obviously at this point most of you have recognized that you're going to have an infinitely repeating digit right every time I'm going to bring down a zero and get 20 and I'm going to ask that same question how many times does 3 go into 20 you're going to keep getting six you're going to multiply 6 times 3 you're going to get 18. you're going to subtract you're going to get 2 again so it's kind of an infinite Loop right it will never stop so as I taught you in the last lesson where we covered dividing decimals basically what you're going to do is you're going to put 0.6 or you could do 0.6 and you're going to put a bar a horizontal bar on top of that 6 to denote that that 6 is a repeating digit right it will repeat forever okay for the next one let's look at 2 11 or 2 over 11. so again I want to divide the numerator by the denominator so 2 is going to be divided by 11. and again 2 is smaller than 11 so I'm going to put a decimal point after 2 bring that up into the answer and put a 0 after that decimal point and now I can just ask how many times will 11 go into 20 it's going to go in once 1 times 11 is 11. subtract here and you get 9. so I could put another zero so I can continue my division go ahead and bring that guy down and you'll have 90. 11 goes into 98 times 8 times 11 is 88. subtracting you get 2. now to continue the division I put a zero here and bring that guy down now notice how I started out with the question 11 goes into 20 and I got 1. again I'm asking that question 11 is going to go into 20 once so you can already see that you're going to have a pattern that's going to repeat it's going to be one eight one eight one eight so on and so forth because 11 is going to go into 20 again we put 1 there we multiply 1 times 11 is 11. when we subtract we're going to get 9. we're going to put another zero here and bring this down and again we're going to ask how many times does 11 go into 90 that's going to be 8. 8 times 11 is 88. subtract and we're going to get 2. all right so it's the same it's the same pattern over and over again where I have basically 20 then 90. then this would be 20 again then you'd have 90 again you know so on and so forth so we can stop at this point and go ahead and give an answer it's going to be .18 where the 1 and the 8 repeat forever so we'll put a horizontal bar on top of the one and the eight very important and if you want you can put a zero out here just for clarity so our answer is going to be 0.18 or the 1 and the 8 repeat forever all right let's take a look at another one we have 25 over 44. so I'm going to take 25 my number in the numerator and divide that by 44 my number in the denominator and again 25 is smaller than 44 so I'm going to begin by putting a decimal point there bringing that up into the answer and then putting a 0 behind that decimal point so now how many times will 44 go into 250. a lot of us don't really work with 44 that often but just think about 44 times 10. 44 times 10 that's 440. obviously that's too big but if you cut it in half you're going to get a number that's just right cut 400 in half you get 200 and then cut 40 and a half you get 20. so 44 times 5 would be 220. that's going to be perfect for us because if I go up one more to 6 I'm going to go over 250 right so this is the perfect amount that we need so we're going to write here that the answer is 5. multiply 5 times 44 again that's 220. subtract 250 minus 220 is 30. and then we'll put a 0 here so we can bring that down and continue our division so 44 is going to go into 300 how many times Well 220 plus 44 is going to be 264. that is as close as I'm going to get to 300 without going over so that's going to mean that we're going to have an answer of 6 there 6 times 44 as I just said is going to be 264. now we're going to subtract I'm going to borrow all the way from this three here this will be two this will be 10 then 9 then this will be ten ten minus four is six nine minus six is three and then two minus two is zero so I'm going to put another zero here and bring this down and I'm going to have 360. so when I did 44 times 10 we saw that that was 440. so then times 9 I could just subtract 44 I'd be at 396. that's still too big because I'm trying to get to 360. if I take 44 more away then I'm going to be just right 9 is too big but 8 is going to be just right so I'll use eight and then what is 8 times 44. let's do that real quick 44 times 8 8 times 4 is 32 2 down carry the 3 8 times 4 is 32 plus 3 is 35 so 352. so we're gonna write 352 here and we're going to subtract 360 minus 352 is 8. and we put a 0 here and bring that guy down so what is 44 into 80. well it's only going to go in once all right it's only going to go on once 1 times 44 is 44. 44 and we're going to subtract so I'll borrow here this is 7 this is 10 10 minus 4 6 7 minus 4 is 3. so I got 36 and when I bring down my 0 when I bring down my 0 I'm going to have 360. so I can already see that there's a pattern for me I had 360 already here if I have it again here it's just basically going to repeat so in other words the result there was eight I know I'd get eight again then I get one then eight then one so on and so forth right a times 44 would give me 352 again I'm basically repeating this part right here right if I subtract I'm going to get 8. if I brought down another 0 I'd have 80 and I'd be at this step right here I'll be at that step right there so at this point we can go ahead and just answer right we know that the answer would be 0.5681 or the 8 1 would repeat forever so I'm going to put a horizontal bar over the eight and the one again to show that those two digits are going to repeat forever okay for the next one we have a mixed number and when you have a mixed number basically what you're going to do is you're going to take your whole number part and just write that so that's going to be the part that comes to the left of your decimal point you can basically just write a decimal point after that because after that decimal point you're going to put what the fraction is convert it into a decimal right so you just kind of treat them as two different things for right now and then you merge them together in the end so what is 3 8 right in decimal form we're going to do 3 divided by eight put your decimal point down and bring it up into the answer and I'll put a zero after that decimal point it goes into 33 times 3 times 8 is 24. subtracting you we get six put another zero and bring that down 8 goes into sixty seven times seven times eight is fifty-six subtracting you get four put another zero and bring that down 8 will go into forty exactly five times five times eight is forty subtracting gets zero so we get point three seven five point three seven five and basically you're just merge these two parts together if I have five and three eighths I basically have five point and then three seven five so five point three seven five right this part right here is your part of a whole amount and it's the same over here after the decimal point we're working with part of a whole amount right as a decimal fraction that would be 375 over one thousand right 375 thousandths all right let's take a look at another mixed number we have one and two-ninths so again I'm just going to write this whole number part here this one I'm going to put my decimal point down and basically after the decimal point I'm going to have this part of a hole right this 2 9 expressed as a decimal all right so 2 divided by nine I'm gonna put my decimal point here after the 2 and bring it up into the answer put a zero and now we can start our Division 9 is going to go into 20. twice 2 times 9 is 18. subtracting we get 2. put another zero bring that down you can already see that there's going to be an infinitely repeating pattern right 9 goes into 20 twice 2 times 9 is 18 subtracting you get two right you'll keep getting this over and over so you'll have this 2 repeating forever so we can go ahead and stop and say that the part that comes after the decimal point here is just going to be a 2 with a horizontal bar over the two to say that it's going to repeat forever so 1 and 2 9 is equal to 1.2 again where the 2 repeats forever all right now let's take a look at an improper fraction so with an improper fraction again I'm just going to divide the numerator by the denominator so 31 divided by 20. the only difference is I don't need to put a decimal point out to start right I'm going to have to do it at some point in this problem but I don't have to do it right away because my numerator is bigger now than my denominator so 20 will not go into 3 but it will go into 31 once 1 times 20 is 20. If I subtract I'll get 11. now I can put a decimal point there and bring it up into the answer put a zero behind it and bring it down 20 will go into 110 five times five times twenty is one hundred is again 100 subtracting we have 10 . go ahead and put another zero and bring that down 20 goes into 100 exactly five times five times twenty is a hundred subtracting gets zero this is 1.55 and if we were to write this as a mixed number remember what we would do is just not continue the division right we would have had 31 divided by 20. this would go in once here 1 times 20 is 20. we'd subtract and get 11. so really we could also say this is equal to 1 and 11 20 right all of these forms are the same we have this as an improper fraction we have this using a decimal and then we also have it as a mixed number all right let's take a look at one final problem we have 17 8. so again all I'm going to do is I'm going to take 17 and I'm going to divide it by 8. so 8 is not going to go into one but we'll go into 17 twice 2 times 8 is 16 subtract and get one and in the past we'd stop and say okay our mixed number is 2 and 1 8. but here we can continue we can get a result using a decimal so I'm going to take that decimal point and bring it up into the answer put a zero here and bring this down so 8 is going to go into 10 only once 1 times 8 is 8. subtract and get 2. we'll put a zero here and bring this down so 8 is going to go into 20 twice 2 times 8 is 16. subtracting get four put a zero here and bring this down 8 is going to go into 40 exactly 5 times 5 times 8 is 40. subtracting gets 0. so 2.125 2.125 17 8 says 2.125 or again we could have also written this as a mixed number as 2 and 1 8. right all of these are the same we have our improper fraction we have it as a decimal and then we have it as a mixed number hello and welcome to pre-algebra lesson 45. in this video we're going to learn about percents so our lesson objectives for today we just want to learn how to convert a decimal to a percent and vice versa meaning we also want to learn how to take a percent and convert it into a decimal and then we want to learn how to convert a fraction to a percent as well so a percent is just another method used to describe a part of a whole amount basically it's just another fraction so a percent is a fraction whose denominator is 100 okay so if you see a fraction and the denominator is 100 you're dealing with a percent so as a more convenient way to write a fraction with a denominator of 100 we can use this percentage symbol to denote a percent and I'll show you how to do that in a second but one thing you need to understand is that the literal meaning is parts per 100. all right so let's start out by looking at kind of the easier scenario so let's say you start off on your test you're having a percent test and you have this 17 over 100. your teacher says write this using a percentage symbol well the first thing you'd want to do is take your numerator which is 17 and just write it and then all you have to do is follow it with that percentage symbol so 17 over 100 is equal to 17 percent and that's all you need to do when your denominator is 100. it's nice and clean and simple you have 17 Parts out of 100 Parts kind of something you can relate this to let's say that you take a test and you fail the test badly right you just completely bomb it so you have a hundred questions on this test and you only get 17 of them correct so out of 100 questions I got 17 correct so my score on the test is going to be 17 out of 100 or 17 percent let's take a look at 3 over 100. so again if the denominator is 100 to write this using a percentage symbol I'll just take that numerator which is 3 and I'll follow it with a percentage symbol so 3 over 100 is equal to 3 percent as another example I have 96 over 100 again as long as the denominator is 100 I just write the numerator whatever it is in this case it's 96 and I follow it with a percentage symbol so 96 over 100 is equal to 96 percent now we're looking at 512 over 100 and it's okay that this number is greater than 100 it doesn't matter we're still just going to take the numerator which in this case is 512 and we're going to follow it up with a percentage symbol so 512 over 100 is equal to 512 percent okay what about .01 over 100 again it's the same thing I take my numerator which in this case is .01 and I just follow it up with a percentage symbol so .01 over 100 again the denominator is 100 is equal to 0.01 percent all right let's take a look at something a little bit more challenging still going to be pretty easy though so we're looking at decimal two percent right so I'm taking a decimal and I'm going to change it to a percentage so to write a decimal as a percent we just move our decimal point two places to the right and place the percentage symbol at the end let's look at this quick exercise change each decimal to a percent so as I said we're just going to move our decimal point two places to the right and then we're going to put a percentage symbol at the end so we're going to start out with 0.032 so I'm going to say this is equal to I'm going to move this guy 1 two places to the right so that would give me 3.2 and I'm going to put that percentage symbol after it so 0.032 is equal to 3.2 percent now some of you might be saying where did those zeros to the left of the three go remember if I have two zeros to the left of the three that does not add value to the number so we can just get rid of them all right for the next one we have 0.115 and again all I want to do is take this decimal point and move it one two places to the right so I end up with 11.5 and then I put my percentage symbol after it so 0.115 is equal to 11.5 percent next we're looking at 6.23 again move the decimal point two places to the right so that's going to be 623 and then put a percentage symbol at the end so 6.23 is equal to 623 percent all right now we're looking at 0.85 again take your decimal point and go one two places to the right so I'd have 85 and then put the percentage at the end so 85 percent 0.85 is equal to 85 percent all right now we're looking at 0.00037 again this goes one two places to the right so this will be equal to .037 and then percent and you can put it at zero out in front if you want just for clarity doesn't really matter 0.00037 is equal to 0.037 percent okay let's look at two more we have 0.014 again this decimal point is going one two places to the right so that's going to be 1.4 and then just throw your percentage symbol at the end so 0.014 is equal to 1.4 percent and then for the final one we have 312.25 again this decimal point goes one two places to the right so that's going to be 31 000 225 and then put your percentage symbol so 312.25 is equal to 31 225 percent okay so let's say we want to reverse this process and take a percent and go to a decimal so essentially you just reverse what you just did so to write a percent as a decimal we move our decimal point two places to the left remember we're going two places to the right now we're just reversing that and going two places to the left and then we're just going to delete the percentage symbol right we're just reversing what we just did so we're going to change each percent to a decimal and I'm going to start out with 4.27 percent so let me rewrite this over here I'm going to delete this thing I don't need it and I'm just going to write the number 4.27 . now I'm going to need to add 1 0 in here to make this work and this is going to go one two places to the left so I'm going to have point 0 4 2 7 right so 4.27 percent is equal to 0.0427 or I can put a zero out in front for clarity again 4.27 percent is equal to 0.0427 all right for the next one we have 39.867 percent I'm just going to rewrite this number 39.867 and again I don't need this percentage symbol so I can just leave it off now I'm going to move this decimal point one two places to the left and so I'm going to end up with a decimal point there let me put a zero out in front for clarity and we'll say that 39.867 percent is equal to 0.39867 all right for the next one we have 215.37 percent again I'm just going to rewrite this number 215.37 I'm not going to write the percentage symbol we don't need it and I'm just going to move this decimal point again two places to the left two places to the left and that'll go right there so 215.37 percent is equal to 2.1537 okay for the last one we're going to look at we have 0.003591 percent and of course I'm just going to rewrite the number leave the percentage symbol off I don't need that anymore and the decimal point is just going to go two places to the left so one and then two places to the left so that would be right there and if I want to I can put a zero let me kind of scoot this over a little bit I can put a 0 out in front for clarity so 0.003591 percent is equal to zero point zero zero zero zero three five nine one all right so now let's talk about the most tedious scenario you're going to come across and basically a more tedious situation comes up when we're asked to convert a fraction to a percent there's kind of two ways you can think about doing this the first way let's say you have something like four-fifths well you could divide the numerator by the denominator basically take the fraction and put it into a decimal then take that decimal and convert it into a percentage now the kind of the faster way would be to transform the fraction into an equivalent fraction with a hundred as its denominator but you're going to see that that's not always so simple so for four-fifths let's do it both ways the long way would be to say okay 4 you know divided by five put my decimal point here and bring it up and put a zero here five is not going to go into four but we'll go into 48 times 8 times 5 is 40. subtracting at zero so we know that four fifths is equal to 0.8 and then to convert this to a percentage we'd move this two places to the right so one put a zero there two so the decimal point would go there and then I would add a percentage symbol to the end so four-fifths is equal to 0.8 and it's also equal to eighty percent the other way you could have done this which is much faster in this case is to take four fifths and just realize that if I multiply 5 times 20 I get to a hundred right and if you didn't know that you could take a hundred and just say okay if I divide 100 by 5 I get 20 so that means If I multiply 5 by 20 I get a hundred it's whatever I need to multiply this by to get to 100. so if I multiply this by 20 my denominator is 100 and I know I have a percent If I multiply the denominator by 20 I also have to multiply the numerator by 20 so that I have an equivalent fraction so 4 times 20 is 80. over again 5 times 20 that's 100 so now I have a percent a fraction whose denominator is a hundred and when this occurs we just simply take the numerator which in this case is 80. and we follow it up with a percentage symbol so we get the same answer either way eighty percent it's just a different kind of way to do it and in most cases this is going to be a lot quicker but sometimes you're going to run into problems using this method and I'll show you an example of that in a minute right so what about three-fourths again there's two ways we can kind of do this we can take three and we can divide it by four put my decimal point here and bring it up into the answer 4 is not going to go into three but we'll go into 37 times 7 times 4 is 28. subtracting get two bring down a zero four goes into 20 exactly five times five times four is twenty subtracting gets zero so 3 4 is 0.75 0.75 now again to turn this into a percentage I just move that decimal point two places to the right so it's going to go one two places to the right that gives me 75 and then I just add my percentage symbol in so 3 4 is equal to 0.75 and it's equal to 75 percent now kind of again the quicker way to do this would have been to say okay I have 3 4 and I can multiply four times what to get to 100. well if you don't know the answer that is 25 again take 100 and divide by 4. 100 divided by 4 is 25 so therefore 4 times 25 is a hundred you can always use little tricks like that to get what you're looking for so we would multiply by 25 over 25 since that would give me a denominator of 100 and 3 times 25 is 75 so now I have a percent right I have a fraction whose denominator is 100 and essentially all I need to do is take that numerator and rewrite it follow it up with a percentage symbol so 3 4 is going to be equal to 75 over 100 and that's equal to 75 percent okay what if we had 11 over 20. so again we can take 11 and we can divide it by 20. so I'll need to put a decimal point here and bring it up into the answer so 20 is not going to go into one it's not going to go in 11. it will go into 110 5 times 5 times 20 is 100. subtract and we'd have 10. go ahead and put a 0 there and bring it down 20 goes into 105 times 5 times 20 is 100. subtracting at zero so 11 over 20 or 11 20 is going to be equal to 0.55 as a decimal and then as a percentage we just move this two places to the right so that would be 55 put our percentage symbol so 55 percent right so 11 20 is equal to 0.55 which is equal to 55 percent now again kind of the quicker way to do this I think everybody at this point knows that I can take 20 multiply it by 5 to get to 100 so I could have said okay 11 over 20 I can multiply this by 5 over 5. again I just need to figure out what I need to multiply the denominator by to get to a hundred and once I figure that out I just do it to the numerator and denominator so that it's legal 11 times 5 is 55. over 20 times 5 that's a hundred again now I have a percent I'd have a fraction whose denominator is 100 so to write this as a percentage just take the numerator which is 55 follow it up with a percentage symbol 11 20th is equal to 55 over 100 which is 55 percent okay let's look at one that's a little bit more challenging so I have 23 over 46. now let's say I started out and I said okay what number can I multiply 46 by to get to 100. well it's not going to be a whole number so that's where it kind of gets a little challenging right that's where you might want to save some time by just doing the division right you might want to say in this case okay well 23 divided by 46 and a lot of you already know this is going to be 0.5 right because 23 is half of 46 but I'll put my decimal point here and bring it up into the answer put a zero here 46 is not going to go into two it's not going to go into 23 but it will go into 230 exactly 5 times 5 times 46 is 230. subtract and we get 0. so as a decimal we would write this as 0.5 now to convert it to a percentage again we just take that decimal point and move it one two places to the right it's going to be right there and I can just get rid of it right 50 has a decimal point after the zero but we don't need to write it and then put a percentage symbol after the zero so it's 50 percent right 23 over 46 is equal to 0.5 that's also equal to 50 percent now let's say that we wanted to go through and try to do it using the other method so 23 over 46 times what over what will give me something over 100. so we know from basic rules that 46 times some number equals 100 100 divided by some number will give me 46 or 100 divided by 46 would give me this number so let me just go down to a scratch sheet here so 100 divided by 46. what is that well we can divide it out and kind of get a decimal or we can keep it as a fraction if we actually go through and get the division we're going to get a number that we don't want to mess with right it's very very messy so let's just keep this as a hundred over 46 and I'll show you how you can kind of use it this way so let's go back up to the top and say that 46 times 100 over 46 would give me 100 and you can go ahead and check that if we were to multiply these two together this would cancel with this and just leave me with 100 so that does work now if I do it to the denominator I've also got to do it to the numerator now you see how complicated this has become so I know that if I multiply 23 times 100 over 46 let's just do that problem down here so 23 times 100 over 46. so I know that 46 divided by 23 is 2. so if I cancel this with this I'm going to have a 2 here and I have a hundred up here what is 100 divided by 2 that's going to be 50. so I can just cancel this with this and put a 50. so my numerator is going to be 50. now in the denominator as I just explained this would cancel with this right 46 over 46 is 1 and I'd just be left with a hundred so that's where I get that there so I end up with 50 over 100 which again as a percentage is just 50 percent right I just take what's in the numerator and write it follow with my percentage symbol so you can see in this case it's probably not quicker to go through and deal with all this I probably could have just did a division faster but it just kind of depends I mean it's fun to know both methods because in some cases it's quicker to do it this way and in other cases it's quicker to do you know the division and then convert that decimal into a percent right you just have to kind of know the situation hello and welcome to pre-order lesson 46. in this video we're going to learn how to find the mean median and mode for a group of numbers all right so again the lesson objective for today is just to learn how to find the mean median and mode for a group of numbers all right so let's start out with one that you're probably already familiar with even if you've never heard it called the mean so the mean otherwise known as the average and that's probably what you've heard it as of a group of numbers is found by adding all of the numbers for the group then dividing by the number of numbers in the group so it's very very simple to find and average all right so find the mean and again you could say find the average if you want those two words are interchangeable so we have our numbers here we have 6 12 13 5 and 9. so all I need to do is find the sum to start so let's just do this using a vertical addition I'm going to put the two digit numbers on top so I'm going to put 13 then 12 then we could do 9 then 6 then 5. so again you can add in any order I just did this because it's kind of a way for me to organize it in my head all right so put in my addition operator out there and let's start adding three plus two is five five plus nine is fourteen fourteen plus six is twenty twenty plus five is twenty five I'm going to put a 5 down carry a two two plus one is three plus one is four so I've got 45 as my sum okay so let's erase this real quick and I'm just going to write here that the sum is 45. now the next piece of information is we need to know how many numbers are in the group so I can just go through and count that's pretty easy I just go one two three four five there are five numbers in the group five numbers in our group so to get the average I just take the sum of the numbers in the group which is 45. and I divide by the number of numbers in the group which in this case is 5. now I don't need to do a long division for this I know 45 divided by 5 is not so the average here or the mean is going to be 9. okay let's take a look at another one let's say we have 28 34 115 7 2 and then 6. all right so let's find the sum first and again I'm going to take the numbers with more digits and put them on top and again just to keep this organized I'm going to take and just list them in descending order right starting at the top with 115 and then just moving down so to 34 then 28 then 7 then 6 then 2. okay so let's start our addition I'm going to scroll down a little bit so 5 plus 4 is 9 plus 8 is 17 plus 7 is 24 plus 6 is 30 plus 2 is 32. so let me put a 2 down and carry a 3. let me scroll up a little bit so I can put this 3 up here and three plus one is four plus three is seven plus two is nine then I can just bring this one all the way down and I have 192 for my sum so the sum is 192. now how many numbers do we have in our group well we have one two three four five six we have six numbers in the group so I just take the sum for the numbers in the group divided by the number of numbers in the group and then I have my average or again the mean so what is 192 divided by 6. well I don't really work with those numbers very often so I don't know that off the top of my head how we need to crank out a long division so let's do that real quick so what is 192 divided by 6 6 will not go into one it will go into 19 three times three times six is eighteen subtract and we'll get one bring down the two six goes into 12 twice 2 times 6 is 12. subtracting gets zero so 192 divided by 6 is 32 and again 192 divided by 6 is 32 and so this is our answer right our average or our mean is going to be 32. all right let's take a look at one final problem for finding the mean or again the average I have 125 17 22 and negative 7. so let's add all these numbers together and what I'm going to do since we have a negative involved let's add the positive numbers together first and then let's do a final operation where we add the result to that negative so what is 125 plus 22 plus 17. 5 plus 2 is 7 plus 7 is 14. four down carry the 1 1 plus 2 is 3 plus 2 is 5 plus 1 is 6. bring down the one so I've got a 164 I'm going to have 164 plus a negative 7. so obviously 164 is larger in terms of its absolute value so the answer is going to be positive now since I just have a subtraction problem here 164 plus negative 7 is the same thing as 164 minus 7. so if we do 164 minus 7 borrow here this will be 5 this would be 14 14 minus 7 is 7. bring down the 5 bring down the one so I get positive 157 as my answer so let's write that the sum that's right that the sum is 157. and then the number of numbers is four we're going to have one two three four so the number of numbers in the group is four so now I would just want to take that sum which is 157 and divide by 4 right the number of numbers again something I don't work with a lot so I have to do a long division and so 157 divided by 4 . 4 will not go into one it'll go into 15 3 times 3 times 4 is 12. subtract we get 3 bring down the 7. so 4 is going to go into 37 9 times 9 times 4 is 36 subtracting we get 1. so to continue this I can put a decimal point and bring it up into the answer put a zero here and bring that down 4 goes into 10 twice 2 times 4 is going to give me 8. subtracting we'd have 2. I'll put another zero and bring that down four goes into 20 exactly five times five times four is exactly 20. subtract and we get zero so we get 39.25 so we get 39.25 as our answer again this is the mean or what we call the average all right so another way to think about the middle value for a group of numbers is to use the median now a lot of times when you hear data on people's salaries or on house prices or things like that things that are affected by very high and very low values you'll hear people talk about the median so the median is not affected by these kind of outliers these large large values that kind of pull your average up or these low low values that kind of pull your average down for example if you thought about salaries let's say you had five salaries and it was like fifty two thousand you know let's say another one would be like fifty eight thousand let's say you had one that was 64 000. uh let's say another one was you know sixty thousand so these are all relatively close together so your average should be somewhat representative of you know what these values are it would be close but let's say for my fifth one I throw on something like 18 million dollars well what happens is when I take the average here I'm going to get a number that doesn't really accurately represent what the salaries are right it's inflated from this kind of outlier here this was way above what everything else was and it pulled my average up when we look at the median we're going to correct for that so we're not going to be affected by these very large or very small values and essentially to find the median for a group of numbers we just arrange the numbers from smallest to largest so the smallest goes to the left and then we just keep going to the largest the median is going to be the middle number when the number of numbers in the group is odd or the average of the two middle numbers if the number of numbers in the group is even okay and that's a lot to think in you might want to write that down as a note just use it as you're practicing but once you do a few examples of this you'll kind of have it down it's not very very hard all right so we want to find the median in these examples and we're going to start with a pretty easy one we have the numbers 3 9 8 1 and 12. so again the first thing you would do is arrange the numbers from smallest to largest so I'd start out with one then three then eight then nine and then 12. okay so smallest number goes all the way to the left and then we just the next smallest the next smallest the next smallest and then finally get to the largest value now we have an odd number of numbers here we have one two three four five numbers and so the median is simply the middle number in this case that's going to be eight right it's going to be eight and you could tell that's the middle number because I have the same number of numbers that come before it right I have one two numbers that come before it as that come after it I have one two numbers that come after it so eight is the middle number and therefore the median all right so for the next one we have 150 25 negative 2 14 18 3 and then 9. so we're going to start out with the smallest number which is negative 2 all the way to the left then we'll go to the next smallest that's going to be positive 3. and then we're going to have 9 then 14 then 18 then 25 then 150. now how many numbers do we have in this group we have one two three four five six seven so again 7 is an odd number so we know that it's just going to be the middle number here so that's going to be right here where this 14 is I have one two three numbers that come before it and one two three numbers that come after it so 14 is your middle number or your median and when you get to your statistics class or if you're reading a statistics book you might see a formula in there that tells you that if you have an odd number of numbers the immediate position is that n plus 1 divided by 2. n is just the number of numbers in your group so you have one two three four five six seven numbers in the group so you'd replace that with a seven seven plus one is eight eight divided by two is four so essentially you would be looking for the number that occurs in the fourth position well this is the first the second the third this is the fourth and there's your median right and that works when you have an odd number of numbers all right let's look at some scenarios where you have an even number of numbers and you're going to see it's a little bit more tedious so we have negative 2 4 6 and 5. so to arrange the numbers from smallest to largest I need to flip the order here I'd have negative 2 4 5 and then 6. now there is no middle value here right if you kind of look okay 4 looks like it's kind of in the middle it has one that comes before it but two that comes after it and if you look at five okay that kind of looks like it's in the middle two that come before one that comes after it so neither neither of those is going to work and essentially what you need to do is just treat these two numbers here four and five as if they're in the middle okay four and five as if they're on the middle you can see you have one number that comes before and one number that comes after now because it's two different numbers here what they want you to do is find the average of those two numbers right kind of the average of the two middle numbers so if I average those numbers I would just sum four and five four plus five is nine and then I would divide by two what is 9 divided by two two goes into nine four times four times two is eight subtracting get one put the decimal point there and bring it up put a zero and bring it down two goes into ten five times five times two is ten subtracting gets zero so the average of these two numbers four and five is four point five and that's your median your median is 4.5 again the average of the two middle numbers all right let's look at one more here we have negative 8 14 2 155 30 7 5 and negative 2. so again I'm going to arrange the numbers from smallest to largest so negative 8 is the smallest then negative 2 then positive two then we'd have 5 then 7 then 14 then 30 then 155. and if you count the number of numbers here again we have one two three four five six seven eight so again it's an even number of numbers there is no true middle number if you kind of think about okay well if I have eight numbers that means that kind of these two numbers here that occupy the fourth and the fifth position so this is the first the second the third and I'm kind of messing up the writing on this so let me kind of make that clear so the second the third the fourth and the fifth if you look at these two numbers you have three numbers that come before and three numbers that come after if you just kind of block those two off so those are your two middle numbers they're your two middle values here and you just want to average them to find your median so you would add five plus seven you get 12 divide that by two and you get six so the median here again is the average of the two middle numbers and that's going to be six all right let's talk about probably the easiest one to find and this is called the mode so the mode for a group of numbers is the number that occurs most often that's it you just have to find the number that occurs most often in your group of numbers and it's important to note here let me highlight this that we can have more than one mode so if you had a group of six numbers and each number appeared three times let's say you had something like I don't know three three three seven seven the mode here is both three and seven you have two modes because each number three and seven both appear the same number of times and so you would have two modes it's different than if you had something like three three four six eight two here the number that occurs most often is three and so you have one mode just three right three occurs twice every other number occurs just once all right so we're gonna find the mode for each group of numbers and basically just look through the numbers and see what occurs most often so you have 82 and 86 104 82 again 75 and then 30. so your mode is 82. the mode is 82. and why is that the case because 82 occurs twice in this group of numbers and each other number occurs only once so again the mode is the number that occurs most frequently and 82 occurs twice whereas every other number occurs only once so that's why our mode is 82. all right here we have 103 27 negative four two twenty seven three two twenty seven and 104. so I know 103 only occurs once so I'm just going to kind of cross that out that's not the mode 27 occurs once twice three times and let me use a different color here negative 4 only occurs once 2 occurs once twice three only occurs once 104 Acres one so the mode here is going to be 27. you have one two three occurrences two came close that had two occurrences but three has the most and so that's the mode so your mode is 27. okay for the last one we have 333 75 81 75 621 79 and 333. so you can see here just by eyeballing this that you have 333 and 333 you have 75 and 75 so you have two modes here 333 and 75 each occur twice everything else occurs only once so we'll say that there are two modes and those are 75 and 333. hello and welcome to pre-algebra lesson 47 in this video we're going to be converting between US units of measurement so our lesson objective for today we want to learn how to change between US units of measurement all right so we're just going to start by looking at the relationship between a few common units of measurement we're not going to cover all the possibilities here but just a few of the basics so that we can kind of get our brain thinking about converting units all right so we have here length so here are three common ones we have that 12 inches equals one foot we have that three feet equals one yard and we have that 5280 feet equals one mile all right let's look at weight we have that 16 ounces equals one pound and we have that two thousand pounds equals one ton for volume we have that two cups equals one pint two pints equals one quart and four quarts equals one gallon and then lastly for time we have 60 seconds equals one minute 60 minutes equals one hour 24 hours equals one day and seven days equals one week now based on the circumstances of your life you may know all of these you may know none of them so try not to stress out too badly about memorizing these right away as you work a lot of practice problems you will memorize these relationships very very quickly and a lot of them you probably already know for example 60 seconds equals one minute I think everybody knows that or 60 minutes equals one hour 24 hours equals one day you know so on and so forth so the ones that you don't know as you work through the practice problems you'll pick it up very quickly so let's start out our lesson by just asking a simple question if we are asked to convert 5280 feet into miles it's pretty simple so we know that 5280 feet is equal to one model and we saw that earlier when we're looking at the provided relationship between units but what happens when we want to convert 6 600 feet into miles so if I said 6600 feet equals how many miles well let's go through the process so that we can get the answer here so our first step is just to set up something called a unit fraction and let me highlight this because this is going to be important for you to know a unit fraction shows the relationship between units and has a value of one so that part is very very important the fact that it has a value of 1 because remember when we multiply by one even if it's a complicated form of one like we saw with fractions it does not change the value of the number right if I take the number 4 and I multiply by 3 over 3 right this is a complicated form of one so 4 times 3 is 12. over 3. 12 over 3 is the same thing as 4. right if I do the division 12 divided by 3 I get 4 back so it's going to be the same thing here we're going to be multiplying by a complicated form of 1. for example if I have 5280 feet and that's equal to one mile well if I write 5280 feet over one mile this is going to have a value of 1. right it's the same thing over itself right it's the same distance it just looks different because it has different units involved right this has feet and this is based on Miles so now that we understand what a unit fraction is to actually perform the conversion again you identify the relationship between the units so in our case again 5280 feet is equal to one mod if you didn't know that from memory you'd have to look that up and then gain that relationship now the next thing you want is to write the unit fraction with a unit of measure you want to end up with in the numerator so recall that we had 6600 feet and we wanted to know how many miles how many miles so the unit I want to end up with is Miles this is what I'm trying to get to so I'm going to write the unit fraction again with the unit measure you want to end up with in the numerator so because I want miles in the numerator I'm going to have one mile over 5280 feet this is my unit fraction remember this has a value of 1. and it's just about taking the relationship between the units and putting again the units that you want to end up with in the numerator I want to end up with miles in the numerator so that's why I wrote it this way if I was doing something where I was converting a feet and I wanted to end up with feet I would just flip this I would have 5280 feet on top I would have one mile on the bottom all right then we want to multiply what you're trying to convert by the unit fraction so let's go ahead and do that real quick and see what we come up with so we had 6600 feet and we're going to multiply this by the unit fraction again it was one mile on top over 5280 feet on the bottom now the reason we want the units that we want to end up with on top is because this is what we're going to end up with the units that we're trying to get rid of are going to cancel out when we write it this way feet over feet will just cancel so we don't have to worry about them basically what I have here is forget about this unit for a second that's going to just carry itself over so we'll put miles over here and we're just going to think about okay what is 6600 times 1 over 5280. well basically that's just a division problem right it's six thousand six hundred divided by 5280 and we would just divide this we know that this will only go into six thousand six hundred once and multiply 5280 [Music] and we're going to subtract zero minus zero is zero I need to borrow here so this will become five and this will become ten so now we can do 10 minus eight that's two five minus 2 is 3 6 minus five is one now we're going to continue the division so let me scroll down here so we have room to do this I'm going to put my decimal point here and bring it up into the answer put a 0 here and bring it down so for 5280 how many times can I go into 13 200. well if you just eyeball this if you just approximate this number to five thousand five thousand times three would be fifteen thousand that's too big so the best we can do here is two I'm going to put a two here and I'm going to multiply two times five thousand two hundred eighty let's do that down here so 5280 times two two times zero is zero two times eight is sixteen six down carry the one two times two is four plus one is five two times five is ten so ten thousand five hundred sixty so ten thousand five hundred sixty so again let's subtract zero minus zero is zero and we need to borrow again over here so I'm gonna borrow from this two this will become one this will become ten ten minus six is four and then we need to borrow again this will become two this will become eleven eleven minus five is six and then 2 minus 0 is 2 1 minus 1 is 0. so I have two thousand six hundred forty let's put another zero so we can continue the division and bring that down and so we'll have twenty six thousand four hundred now so if I kind of eyeball this again I know five thousand would go into 25 000 five times so I know it's at least a five let's think about this 280 parts so let me go down here again so 5280 times five what would that give us five times zero is zero five times eight is forty so zero down carry the four five times two is ten plus four is fourteen four down carry the one five times five is twenty five plus one is twenty six so this is exactly what we're looking for twenty six thousand four hundred so we know that the answer here for 5280 goes into twenty six thousand four hundred that's going to be five so five times five thousand two hundred eighty again is twenty six thousand four hundred subtract and get zero so I just need this part this 1.25 I can erase everything else and basically at this point I could just answer this by putting 1.25 here and our unit conversion is done right it's actually that simple probably the harder part is just memorizing the units if you're required to do that if your teacher gives you like a little cheat sheet this is actually a piece of cake right you're just multiplying with fractions basically so six thousand six hundred feet is equal to 1.25 miles and again all you do is you multiply what you start out with by a unit fraction where the relationship you want to end up with in this case it was Miles so we have mile up here is in the numerator if you can just remember that this is no harder than just multiplying with fractions all right let's take a look at another one so we have 360 seconds to minutes so what's the relationship between seconds and minutes well I know that 60 seconds 60 Seconds equals one minute so again this is always your first step just show the relationship between the units how much of one is equal to how much of another now I'm going to set up a unit fraction so I'm going from seconds to minutes all right so I want to end up with minutes so that means I'm going to write this where I have minutes in the numerator or in this case because it's singular I'm going to have one minute in the numerator so I'm going to write 360 seconds times again one minute is going to be in the numerator over 60 seconds that's going to be in my denominator let's scroll down a little bit and get some room going now once you've set this up you've basically done all the hard work because seconds over seconds is going to cancel I don't need to worry about the units anymore because I know this is what's going to be left over and that's what I was trying to get right I wanted this in terms of minutes so I'm just going to write minutes over here and I'm done with that part now I can just think about what is 360 times 1 over 60. well it's basically the same thing as 360 divided by 60. and a lot of us can do that in our head by this point right 36 divided by 6 is 6 and basically we have a trailing zero on each one right so you could just think about this as 36 times 10 up here over 6 times 10. I kind of break that down in your head this over this is 6 this over this is one so basically we're going to have six minutes but again if you can't do that at this point I don't want to lose anybody we can just do our basic division right 360 divided by 60. 60 doesn't go into three it doesn't go into 36. it goes into 360 again six times six times sixty you can do 6 times 6 that's 36 and then put a zero at the end that's 360. subtract and get zero so again we're going to get six minutes for our answer here right 360 seconds is going to be equal to 6 minutes all right let's take a look at another one we have 15 quarts 2 gallons so what's the relationship between quarts and gallons well this is one that a lot of us know because we buy milk right four quarts is equal to one gallon so four quarts four quarts is equal to one gallon so now that I know the relationship between the units I want to end up with gallons right so because I'm going from quartz two gallons so when I write my unit fraction I'm going to put gallons so in this case one single gallon in the numerator so I'm going to have one gallon in my numerator and I'm going to have four quarts four quarts in the denominator and that's the trick right just remembering hey the units I want to end up with that's always going to go in the numerator if you can remember that again you're going to be good to go and we're multiplying this by 15 quarts 15 quarts what we're starting with and again the units here are going to cancel this cancels with this and we're just going to be left with this and again that's what we want right we're going two gallons so now I basically have 15 times 1 4 which is the same thing as saying what is 15 divided by 4. so let's do that down here what is 15 divided by 4. 4 doesn't go into 1 but it will go into 15 three times three times four is 12. subtracting at 3 . let's put a decimal point here and bring it up into the answer put a zero here and bring that down 4 goes into 37 times 7 times 4 is 28. subtracting get two put another zero and bring that down 4 goes into 25 times 5 times 4 is 20 subtracting gets zero so 3.75 is our answer so 3.75 and of course we have gallons and you can kind of think your way through that one if four quarts gives you one gallon Well yeah if I had 12 quarts that's three gallons that's this part right here and then I have three additional quartz right 15 minus 12 is three that's not quite enough to make another gallon because I need four quarts to make another gallon so I just have 0.75 or 3 4 of a gallon remaining right so that's how I end up with this 3.75 gallons when we convert over from 15 quarts so another thing you need to know is how to do multiple conversions in one shot so I have here that you can also carry out multiple conversions in one shot let's suppose that you don't know the relationship between cups and gallons so let's say that you're told to convert 25 cups to gallons so what you know is that two cups equals one pint two pints equals one quart and four quarts equals one gallon so using this information we're going to go from cops two gallons okay we're going to go this way so let me just write these out real quick two cups two cups is going to be equal to one pint one pint and then two pints two pints is going to be equal to one quart one quart and then lastly four quarts four quarts is going to be equal to one gallon one gallon so if I'm starting out with cups which is right here and I want to go to gallons which is right here I would first just think about converting cups to pints so let's start out here we have 25 cups and I'm going to multiply this by one pint one pint over two cups two cups now why did I do that well again I want to convert the pints to start if I want to convert to pints it's two cups equals one pint so one pint is in the numerator right because I'm going I want to go two pints and two cups is going to be in the denominator now this right here now would be in terms of pints after I do this multiplication it'll be in terms of pints so I'm going to multiply again now I'm going to go to quarts so I have the relationship between pints and quarts I want to go to quartz so one quart's going to be in the numerator going to be in the numerator and two pints is going to be in the denominator again just put the units you want to end up with in the numerator so this is going to put it in terms of quartz now the last step is to put it in terms of gallons so I have 4 quarts equals one gallon so I'm going to multiply one more time again gallons will be in the numerator so one gallon will be up top and four quarts will be on the bottom in our denominator okay so this will put it in terms of gallons and that's all it is you're just doing multiple conversions at once and if you're not comfortable with this just do it individually first just to a part where you go okay 25 cups I'm going to convert that to pints stop then convert it into quartz stop then convert it into gallons but once you get good you want to just do it all in one shot like this okay so let's go through and you can cancel out all the units you're going to see that cups will cancel with cups pint will cancel with pints quart will cancel with quartz all you'll be left with is gallon right there and let me I'm going to make that a little better I cover up that one so you're just gonna be left with this in terms of gallons so now all we need to really think about is what numbers are involved because I know my units I'm going to end up with gallons and so I would just think about 25 times one half times one-half again times 1 4. what's that equal to well I know that I could just write this over one just think about multiplying the numerators 25 times 1 times 1 times 1 is just 25 and for the denominators 1 times 2 times 2 times 4 is the same thing as 4 times 4 that's 16. so we have 25 over 16 that's just a simple long division so 25 divided by 16. 16 will go into 25 only once 1 times 16 is 16. subtract here we would get 9. right if I borrow from the 2 I put a 1 there and a 15 here 15 minus 6 again is 9. put my decimal point here and bring it up into the answer let's put a 0 here and bring it down how many times will 16 go into 90. well 5 times 16 is 80 that's as close as we can get so let's put a 5 here again 5 times 16 is 80. if we subtract we get 10 put another zero and bring that down 16 goes into a hundred I know 16 times 5 is 80. so 16 times 6 would be 16 more or 96. again that's as good as we could do so we'll put a 6 here 6 times 16 again is 96. subtract we would get four right if I go through and do the official procedure borrow here this will be 10 and then it'll be 9 and this will be 10 10 minus six is four my minus that is 0 and bring down that zero we just have four so I'm going to put another zero here and bring that down they give me 40. so how many times will 16 go into 40 well only twice only twice 2 times 16 is 32. subtract and we're going to get 8 right borrow from the 4 this would be three this would be 10 10 minus 2 is 8. put a zero here and bring that guy down now I have 80. and we know that 16 goes into 80 exactly five times five times sixteen is eighty is 80. subtracting gets zero so we have our answer of 1.5625 so 1.5 let me scroll over a little bit so I can get some room going 625 and then gallons and it really is just that simple we started out with let me see if I can fit all this on the screen we started out with 25 cups way over here and then we went through when we converted basically one at a time but in one shot right we converted the pints then we took pints and we converted to quarts then we took quartz and we converted to gallons and then we ended up with our answer which was again 1.5625 gallons all right let's take a look at one final problem and we want to convert 4.5 days to minutes so let's start out by looking at the relationship between minutes and hours how many minutes are in an hour we know that 60 minutes 60 minutes equals one hour and then how many hours are in a day well 24 hours are in one day so then now 24 hours equals one day so we're going to go from minutes to days so not really that complicated here if we start out with 4.5 days we're going to multiply by a unit fraction where ours is going to be in the numerator because our first step is to convert from days to hours right we're trying to get to minutes so I'm going to put 24 hours in my numerator in my numerator and I'm going to put one day in the denominator again always what you want to end up with in the numerator so this will put this in terms of hours now I'm going to multiply one more time to get it in terms of minutes right now I need to go from hours to minutes so I want to end up with minutes so minutes will go in the numerator so 60 minutes in the numerator and this will be over one hour that will be in the denominator and now this will be in terms of minutes which is what we want right we want to end up with minutes let's scroll down here and you can see everything's going to cancel except for minutes this will cancel with this this will cancel with this and this is what I'm left with for my units and basically I just have 4.5 times 24 times 60. right it's just a multiplication problem at this point so let's go down to a scratch sheet what is 24 times 4.5 so 24 times 45 forget about the decimal point for now 5 times 4 is 20. with a zero down carry of the two 5 times 2 is 10 plus 2 is 12. so erase this 4 times 4 is 16 6 down carry the one four times two is eight plus one is nine we'll add here bring down the zero two plus six is eight one plus nine is ten so that's one thousand eighty let me go back up here real quick 1080 so I'll put 1080 and I have one decimal place between these two factors that I did so that means I'm going to move the decimal point one place to the left so basically that's going to give me the number 108. now I still need to multiply by 60. but to save myself a little bit of time remember I have a trailing zero here so I can just basically do 108 times 6 and then attach a trailing zero to the end of that so 108 times six six times eight is forty eight eight down carry the four six times zero is zero plus four is four six times one is six so that's 648 and then we're going to put a trailing zero at the end it's number 48 put a zero at the end it's six thousand four hundred eighty so let me erase this and we'll put this is equal to 6480 and then the units again will be minutes [Music] hello and welcome to pre-algebra lesson 48. in this video we're going to learn about converting between metric units so again the lesson objective for today would be to learn how to convert between metric units and for most of you watching this video you're going to live in the United States like I do we don't really deal with the metric system that often but it is important for us to get a general understanding of the metric system and how to convert between the units because at minimum it's going to come up on one of your tests in the future all right so let's start out by looking at the commonly used metric units so we have three that we're going to cover today we have the gram that's for weight we have the liter that's for volume and we have the meter that's for length now one of these I know that all of you have seen the leader is used when you buy a 2-liter of Coke or Sprite or Pepsi or whatever soda you'd like to consume now one thing that's different about the metric system when we have values that are larger or smaller than the basic units we use a prefix so when we talk about prefix that's a word that's going to come before the basic unit right so we put a word before we say gram and here are our prefixes so the first three we're going to look at represent larger values than the basic unit so we have kilo which stands for a thousand we have hecto that stands for a hundred we have Deca that stands for 10. now when we start talking about values that are smaller than the basic units we have deci that stands for 0.1 or 1 10. one tenth we have centi which stands for .01 or 100th and we have Millie which stands for .001 or 1 000. now let's see an example real quick of how we could kind of use this prefix if we know that kilo stands for 1000. if I had one thousand meters instead of writing 1000 meters I could just say I have a kilo meter so we could say that one thousand 1000 meters is equal to or is the same as one kilo meter and that's all you're doing just taking this prefix and putting it before the basic unit meter if you had a hundred meters if you had a hundred meters you could say that you had a hectometer so you had one hecto I'm just putting that word hecto before I put meter and let's skip a few we'll come down here to let's say Millie so if I had one thousandth of a liter let's say I could say I have a milliliter so one thousandth of a liter or in decimal form you could put .001 of a liter is going to be equal to one milliliter so I'm going to write m i l l i and then follow that with leader so the prefix is not too difficult to understand you just take that part and put it before your basic unit all right so let's look at a basic conversion so we want to convert 5 kilograms to grams now I'm going to kind of show you the slow way to do this and then I'm going to kind of show you the trick which makes metric conversion an absolute joke right once you see this trick you're going to be like wow this is easy so to go from five kilograms to grams let's use our knowledge of what we did in the previous section when we converted between US units of measurement let's identify the relationship between kilograms and grams I know that kilo stands for a thousand so what this is saying is that I have a relationship that is one kilogram one kilogram is equal to one thousand one thousand grams so if that's the relationship and I'm trying to go from 5 kilograms to grams remember we write a unit fraction with the units that we want to end up with in the numerator so I want to end up with grams so I'd put a thousand grams in the numerator thousand grams and I put one kilogram in the denominator and I just multiply by what I'm trying to convert so I have five kilograms here five kilograms and the units I no longer want are going to cancel kilograms and kilograms and basically all I need to do is multiply 5 times 1000 and then my units will be grams so 5 times 1000 is of course five thousand five thousand and then grams so that's pretty easy to do again when I show you the simplified method just going to be even easier one thing you can kind of look at and rationalize here if if each kilogram is 1000 grams and I have five of those well yeah then of course I have five times one thousand right five thousand grams so an easier method is to look at a prefix chart so a prefix chart is a chart that has everything in order based on the number 10. let me explain that so we started out at the unit if I started out the unit and I just put the number one as I go to the right I'm looking at smaller values and so I'm dividing by 10. so to go from the unit to deci I take 1 and I divide by 10 and I end up at 1 10 or you could write 0.1 then to go to the right one more time to go from deci to centi I divide by 10 again if you think about 1 10 divided by 10 or 1 10 times 1 10 that's going to give you one over a hundred right or again in decimal form .01 and then if I go to the right one more time and end up at milli I would divide by 10 again or multiply by 1 10 and I'd end up with one thousandth or again I could put .001 now as I go to the left notice I'm just multiplying by 10. so this times 10 will give me this this times 10 would give me this this times 10 will give me this and once I start going to the left over here and working with whole numbers it's really simple 1 times 10 is 10. so Deca remember that means 10 multiplied by 10 again to get the hecto remember hecto means a hundred multiply by 10 one more time to get to kilo remember kilo means one thousand so now we can see that this is based on the number 10 right if I'm moving to the right I'm dividing by 10 if I'm moving to the left I am multiplying by 10. and that should be familiar to you from working with place value right if I come out and put okay the ones the tens the hundreds as you go to the left you're multiplying by 10 as you go to the right you're dividing by 10. same thing here now when we encounter a problem like let's say what we just worked with here convert 5 kilograms to grams you're going to look at your prefix chart and you're going to identify the starting prefix so in this case if we're going from 5 kilograms to grams I'm starting out right here at kilo and I want to end up at grams which is the unit right the basic unit so what I want to do is match the movement and the direction on the prefix chart with my decimal point in what I'm trying to convert so for example if I'm going from kilo to the basic unit right for grams I'm going to go 1 two three places to the right so with five kilograms I'm going to put five I'm going to put my decimal points and then I'm gonna go one I'm gonna put a zero in two I'm gonna put another zero in three places to the right and put a final zero in there and I can put a decimal point at the end or I can leave it off the number is five thousand and then I would just write my new units that I'm going to right which would be grams in this case so no complicated unit fraction involved no multiplication involved I'm simply looking at a prefix chart and matching the movement and the Direction with my decimal point for what I'm trying to convert it's that simple when you work with the metric system the only thing you might struggle with is memorizing this part right here and a lot of students will remember King for kilo Henry for hecto died for Deca unexpectedly for units drinking for deci chocolate for Senti and milk for Millie so King Henry Died unexpectedly drinking chocolate milk so hopefully you can remember that if not again if you just kind of write out the prefix chart a few times while you're doing some examples you will memorize it okay for the next one we want to convert four millimeters to decameters now all I need to do is write this number 4 down write the number four down and I'm going from millimeters from millimeter so this is where Millie is that's your prefix to decameters so Deca Deca is right here I'm going to match the movement and the direction on the prefix chart with my decimal point so I'm starting here at Milli and I'm going 1 two three four places to the left so I just need to match that over here let me just kind of back this up so let me put three zeros to the left of this four so one two three zeros my decimal point for the number four starts here and then again I'm just going to move it four places to the left so one two three four places to the left so it will go right there and then once you're done with that just write in your new unit so we're converting two Deca meters so that's what I'm gonna write deca Deca meters so four millimeters is going to be equal to .0004 decameters okay let's take a look at another one we have 3.8 kilometers and we want to convert that to millimeters so look at your prefixes again you're starting out with kilo and you're going to milli so here's kilo here's milli so you're going all the way across the map so let's write that number 3.8 and we're going to move our decimal point one two three four five six places to the right so that means I'm going to need to place five zeros behind the eight one two three four five this is gonna go one two three four five six places to the right and let's just erase all that put some commas in and we'll get three million eight hundred thousand and then we convert it to millimeters so we'll put Milla meters behind that all right let's take a look at another one we want to convert three milligrams to hectograms so again look at your prefixes you have Milli here you have hecto here so here's Milli here's hecto let me go ahead and write this number here we have three and we're going to be going to the left let me kind of write that a little further over here so that I have some room and we're just going to match the movement and the direction on our prefix chart that's all we need to do so we're going to go one two three four five places to the left so I would need to go and add in four zeros here to the left of the three one two three four so I can move this one two three four five places to the left and I'll end up with .0003 and then hectograms so hecto grams all right for the final problem we're going to look at we have 5.42 centiliters to Deca liters so you look at your prefixes you have Senti and you have deca so here's Senti and here's deca so let's write down our number it's 5.42 and I'm starting at centi and I'm going to Deca so I'm going one two three places to the left let me kind of move this over to the right a little bit and I'm going to put in two zeros to the left of the five and so this is going to go one two three places to the left and I'm going to end up with .00542 and then Deca leaders deca leaders