hello welcome back the title here is called practice with proportions this is part one you might say why are we doing practice with proportions haven't we already been doing practice with proportion in the last lessons of course that is true these problems will have slightly larger numbers which will cause us to do a little bit more work but really the main reason we have extra practice here is because as we solve these proportions notice that they're all equations so it's really like the first time we're getting exposed to how to solve an equation and i really want to give you extra practice with that because the idea of solving equations actually follows you throughout all of math so we're learning two things here we're learning how to solve proportions which is just a type of equation and then of course we're learning how the rules of how to solve equations so let's take a look at our first example in this lesson what if we have the proportion which is just an equation 1 3 or 1 as compared to 3 is equal to 15 as compared to x so x is the unknown number what we're basically asking ourselves is if we have the relationship one as it compares and relates to the number three then if we have separately the number 15 what number do we have to have down here to have the same relationship we also call that the same ratio so kind of another way to think about it we've been talking about the the example here boys to girls in a room or in a classroom if we have one boy for every three girls then if we encounter 15 boys how many girls should we also have in the group in order to have the same ratio or relationship as what we have on the other side now in order to solve this what we have to do is get the the unknown number x by itself on one side of an equal sign any equation that's all we're trying to do but the letter x here is just a placeholder for a number and it's on the bottom of the fraction so the first thing we have to do is move it to the top so let's rewrite what we have 15 over x here and we need to get this x up to the top so this is dividing by x it's 15 dividing by x we always do the opposite so in order to bring it upstairs we're going to multiply by x and if we do it to this side we have to do the same thing to the other side of the equation all right now we're multiplying by x on both sides we can write that as x over 1 because anything can be written as itself over 1. so what did we have here on the left on the left we have two fractions on the top we have x times 1 which will just be x and on the bottom we have 1 times 3 which is 3. this is just multiplying fractions the idea that there's a letter x here doesn't really change anything you still multiply the numerators x times 1 is just x because anything times 1 is itself and 1 times 3 is 3. now on the right hand side notice what's actually going to happen these x's you have one in the top and one in the bottom and they're going to cancel you're going to learn that when you multiply fractions if you end up having the same thing in the top as you also have in the bottom they cancel because they divide away so to show it a little more clearly if we multiply the tops 15 times x will just be 15x we can't really do the multiplication so we write it right next to each other here and then x times 1 is x but look what we have here on the right hand side here we have an x on the top and an x on the bottom the same thing i pointed out to you here and we can cancel them and the idea of canceling is not magical it's not mystical it's just that when you have something on the top of a fraction and also on the bottom they're dividing away 3 divided by 3 is 1 5 divided by 5 is 1. x divided by x because x is the same thing is just 1. so they kind of disappear there well they basically divide away to 1 and the 1 disappears so what do we have here on the left we have x divided by 3 and on the right we just simply have 15 left over because the x is canceled away now we have x on the top but we have now still have it divided by 3. so we want to get x by itself it's divided by 3 so we need to do the opposite which is multiplying by 3 on the left and then we'll multiply by 3 on the right now on the right it's quite simple because we just have 15 times 3. if you go off to the separate sheet of paper 15 times 2 you'll remember is 30. 15 times 3 you could just go off to the side multiply you'll get that as 45. so 15 times 3 is 45. now on the left hand side notice what we have we're multiplying by 3 but we can of course write it as 3 over 1 whoops not 3 over 3. 3 over 3 over 1 there and we have the same sort of thing going on we have a 3 on the top and a 3 on the bottom let's go ahead and go through it 3 times x on the top will be 3x and on the bottom 1 times 3 will be 3 but you see you still have the 3 on the top and the 3 on the bottom so they go away they cancel right you can cancel them at this step also you can cancel before the multiplication here i'm just showing you kind of every little step here we're doing it after the multiplication there so the only thing left is x and you have it equal to 45. that is the final answer we're trying to find out what is the unknown value of x it has to be 45. so let's go and uh just rewrite the proportion we had it as 1 over 3 or 1 is compared to 3 is equal to 15 as compared to x and we already just found that x is 45 does this make sense well what we're saying is if this is boys to girls a ratio of boys to girls then there's three times as many girls as boys so if we encounter 15 boys there has to be three times as many girls and of course that is three times as many 15 times 3 is 45 so this is the only number that works this problem you could probably just look at it and get the answer i know that a lot of you are looking at this thinking i can see the answer i don't need to do all that stuff but you're going to have to trust me very soon we're going to solve equations that you can't do in your mind you nobody can i mean i can give you an equation right now that none of nobody could solve in their mind so we have to learn the rules and the rules are to get the variable by itself using the opposite doing the opposite operation to get it by itself on one side that's how we solve it all right problem number two we're going to change letters to w as it compares to 24 is equal to 4 over 6. so we're saying the relation the ratio relationship 4 as compared to 6 is the same as the relationship between w as it compares to 24 what does w have to be to maintain the same relationship the same proportion there so on the left we want to get w by itself let's rewrite everything w over 24 is equal to 4 over 6. how do we get rid of the 24 well it's divided by 24 so we'll have to multiply by 24 and then we'll have to multiply by 24 over here now we have fractions here so it's a little easier just to write it as 24 over 1. so we have to do this arithmetic here with the fractions here now the 24 times the w will be 24 w we can't really multiply it so you just write it next to each other and the 1 times the 24 would be 24 and then 4 times 24 on the left you might need to go off to the side and do that 24 times 4 4 times 4 is 16. 2 times 4 is 8 and then you have a 9 here that's going to be 96 right here and then 6 times 1 is 6. but look at what happens on the left because we multiply by 24 now we have 24 divided by 24 which means they go away and so we have w equals here we have 96 divided by 6 let's go over and figure out what 96 divided by 6 is 96 divided by 6. it can go one time 6 times one is six the difference here is three drag down a six and six times something is 36 that's going to be 6 times 6 is 36 remainder 0. so it goes 16 times 96 divided by 6 is 16 and w equals 16 is the final answer so what we're basically saying is 4 as it relates to 6 or compares to 6 the only number that works up here is 16 as it relates to 24. so 16 is less than 24 just like 4 is less than 6 and the relationship between the numbers are the same because the ratios are the same all right making good progress i told you we're just doing more of them because i want you to get in the idea that every time you solve an equation you have to do the opposite and you're trying to get the variable by itself let's take a look 4 as it compares to 10 is equal to some unknown number h as it compares to 25 right so we know it has to be less than 25 because 4 is less than 10 it has to be the same relationship what do we do so let's rewrite the problem 4 as it compares to 10 is h compared to 25. now we want to get h by itself so we're dividing by 25 we must then do the opposite multiply and if we multiply by 25 we have to do the same thing on both sides and we'll write it as 25 over 1. all right on the left multiply the numerators 24 times 4 is 100 1 times 10 is 10. and then what's going to happen here i'm going to start to i don't want to say skip steps but i want you to see what's really going on here we could multiply these h times 25 you get 25h and then you'll have 25 times 1 on the bottom but i want you to start to see that even before you do the fraction multiplication when you see the same thing in the top and the bottom you can cancel those uh kind of ahead of time so the 25 here is going to cancel with the 25 here yes we could multiply you would get something that looks like this uh we had 24 w over 24 and they canceled anyway we're just going to cancel ahead of time and then realize that what's going to pop out on the right hand side is going to be h on the top and 1 on the bottom h divided by 1 anything divided by 1 is itself so all you have is h because it whatever h is if you divide it by one you still have h and the 25s are now gone so all you have is h here now what is uh uh 100 divided by 10 it's 10 and we can flip it around to say that h is equal to 10. this is the final answer to the problem all right so we're saying 4 as it compares to 10 is the same relationship as 10 as it compares to 25 so 10 the ratio of 10 to 25 is the same as 4 as it compares to 10. so as we solve these problems and we do enough of them i'm going to start to do that process of cancellation a little bit i don't want to say i'm skipping a step but i have to teach you how to cancel properly so let's say we have six divided by y is equal to 15 divided by 20. we want to solve this equation for y this proportion for y so y is on the bottom we cannot isolate it if it's on the bottom so we have to move it to the top first we have 6 over y we have 15 over 20 and the very first thing we need to do is bring y upstairs so multiply by y multiply by y we have fractions here so we can write it as y over 1 like this now on the left hand side what's going to happen same thing it'll be y times 6 or 6y and then it'll just be y times 1 y on the bottom it will be the same situation as 24 times w over 20 where they're going to cancel anyway the y's even before you do the multiplication you can see it's going to cancel either way so we'll just cancel ahead of time and on the left we're just going to have a 6 divided by a 1. because the y divided by 1 y also gives us a 1. when you multiply by 1 nothing happens so the only thing left here is six divided by one and so on the left you just have six and on the right you have 15 times y on the top and 20 times one which is 20 on the bottom all right next we have a 20 on the bottom we need to get rid of that and then lastly we will divide by the 15. so we have a division by 20 going on right here the next step is we're going to multiply by 20 to get rid of the 20 on the bottom right we can write this as 20 over 1. here i can write it over 1 if i want but i don't need to because i'm just multiplying numbers now what is 20 times 6 you can just count 20 40 60 80 100 120 and so the answer we get on the left is 120 and on the right same thing i can multiply the 15 times the 20. i mean i could totally do that or i could just recognize and then i can divide by 20 and and i would get a simplified thing on as i do that but you can see that it's going to be way easier to just cancel the 20s ahead of time instead of taking 20 times 15 getting the result and then dividing by 20 to figure out what to do i'm just going to recognize that i have a 20 and a 20. i'm going to kill it get rid of it it cancels the only thing i have left here is 15 times y and i'm dividing by 1 which does nothing so it's just 15y all right now finally what do i have to do let me rewrite where we're at we have 120 is 15 times y here i'm multiplying by 15. 15 times y how do i do the opposite to get rid of the 15 i'm going to divide both sides by 15. and if i divide both sides by 15 what's going to happen well i have a 15 on the top and a 15 on the bottom that's going to go away and on the left i still have to do that division process here so on the right i just have y y is the only thing left what is 120 divided by 15 120 divided by 15 works out to be exactly eight and if you're not sure about that just go over you know to a side board 120 divided by uh 15. it doesn't go into 12 so you have to you really just have to start multiplying to figure out how many times it goes to 120 15 times 8 8 times 5 is 40 and you have 8 9 10 11 12 and it exactly equals 120. so y is equal to 8 and that is the final answer all right next problem now you see why i'm trying to do so many of these because i want it to be second nature i want you to look at this and know exactly what to do what about k over 5 is the same relationship as 12 over 15 the same proportion the same ratios are set equal i want to solve for k k is already on top so all i need to do to get it by itself is to multiply both sides by 5 because i'm dividing by 5 here so we'll do the opposite we'll rewrite everything so we don't get confused all right and then we'll multiply on the left by 5 and we'll multiply on the right also by 5. we'll go ahead and do it over 1 like this on the left you can see what's going to happen even if we multiply the top and bottom all that's going to happen is the 5s are going to cancel anyway so we'll just cancel them ahead of time and on the right hand side well let's back up on the left we'll have k divided by 1 which is just k so all we have left is k we've achieved our goal on the right the numerator is 12 times 5 is 60 and the denominator is 15. so now i have to figure out what is 60 times 15 and that works out to be 4. and if you're not sure about that just grab off to the side you got to figure out how many times it goes in so 15 times 4 5 times 4 is 20 and then 4 plus 2 is 60 and so you can see it does divide and go four times so all we did is multiply both sides by 5 that cancels the 5 leaves k by itself on this side we multiplied the fractions and simplified that's it so it's just fraction arithmetic as usual all right last problem let's do it right here let's say we have uh 5 divided by 2 is equal to 15 divided by c again the variable c is what we're seeking and it's in the bottom so let's rewrite the problem i don't like to scribble over my problem statement then you can't follow what's going on so let's let's bring c upstairs that's always the first thing we need to do we multiply the right by c and then if we do that we have to multiply the left by c now on the right hand side you can see you have a c and a c on the top and the bottom of the fractions that are multiplied so you can just cancel it so on the right hand side all you're going to have is 15 left over because it's 15 divided by 1. that's just 15. on this side c times 5 will be 5c and 1 times 2 is 2. we can't really multiply the 5 times the c so we just write it as 5c when they're written next to each other like this they're multiplied how do we isolate c first step let's move the 2 over here and then lastly we'll get rid of that 5. how do we move the 2. let's rewrite what we have 5c over 2 is 15. let's multiply since this is divided by 2 we're going to multiply by 2 and we'll make a 2 over 1 over here on the left hand side you can see what's going on here we have a 2 on the top and a 2 on the bottom so the 2 just cancels with the 2 it's gone now and all that i really have left on the left side is 5c and it's divided by 1 which means it doesn't really do anything still 5c 15 times 2 is 30. so i have 5c is equal to 30. so let me rewrite what we have here 5c is equal to 30. how do we get c by itself we've multiplied by 5 how do we do the opposite the opposite of multiplication is division right and the reason we're dividing is because now we see we have a 5 on the top and a 5 on the bottom they cancel away so all we have left is a c on the left and what is 50 i'm sorry 30 divided by 5. 30 divided by 5 is 6. and so we get an answer of c is equal to 6. so there's several steps through but i'm hoping that by now you can see that the steps are bulletproof and they're very repeatable but you just have to get practice seeing it because nobody is good at this when they first learn it but after seeing it a little while i'm hoping that you can see the logic and what we're doing we want to get w by itself for instance we have divided by 24 here so we have to do the opposite we multiply by 24 on both sides we have a 24 on the top and the 24 on the bottom which end up canceling of course we've multiplied first here but we could just cancel ahead of time so we have gotten the w by itself as we expect on the right hand side we just multiply by the 24 as a fraction and get the answer and now we know what the variable is equal to if you solve a problem where the unknown value variable is on the bottom then the very first step is to bring it to the top and the way you do that is multiply both sides by the variable itself and then once we get to this step we had uh 5c over 2 we had to get rid of the 2 by doing the opposite of division multiplying by 2. then we got to this step and to get to to unpack and get c by itself here we do the opposite of multiplication which is dividing by five so you see all you're doing is it's like a present and inside of the innermost box of the present is the variable and you have to unwrap the layers to get to the variable and you have to do them by doing the opposite operation if you see multiplication you have to do division if you see division you'll do multiplication later we will solve equations with addition and subtraction and you can guess what you're going to do if you see addition you'll have to do subtraction if you see subtraction you'll have to do addition so we're always doing the opposite here i really would like you to solve all of these and please please do follow me on to the next lesson and conquer those problems as well this is a critically important lesson because we're learning how to solve proportions but we're also learning how to solve equations