Transcript for:
5th Grade Math Review

What's up? I'm Vin and I'm going to go through this fifth grade practice test. And I'll leave a link to a copy of this test in the description below. So, first up, question one. What is 550 and 306,000 written in standard form? So, for questions like this, I like to break them down piece by piece. The first part here, 550 is the number to the left of the decimal. When you hear the phrase end, that's representing the decimal point. So, we're going to have the word end represents the decimal. And then after that we have the number to the right of the decimal. Okay, so we have 306,000. So if we break this down piece by piece, 550, we could just write that down. The word end represents our decimal point. And then we have 306,000. Okay, so we're going to have 306,000 like this. And when we scan the answer choices, this will match up with choice A. The problem with choice B is the 550 is correct. But if I throw in another zero here, this would be 360,000. Okay? So that's why it's not matching up with the second piece here. We have 306,000. So we need to put a zero here in the hundreds place. If we just need to review this real quick, we have the tenth's place. Then after the tenth's place, we have the hundreds place. Okay? So the hundreds and then we have the thousandths right after that. Okay? So these are the important places to know when we're going to the left. We have the ones, the tens, the hundreds. But when I'm pronouncing these numbers here in verbal form here, I'm just making sure that I say all these terms correctly. So B is out. The problem with choice C is that this would be 550 and 36,000. Okay? So that's why this one is no good. And then the last one, 505. The first part is just wrong right away. So D is out. Definitely choice A. Question two. Which statement about the quotient of 763 and 52 hundreds divided by 10^ the 2 power is true? So for this one, what we could do is we could just move the decimal two spaces to the left. And the reason we're doing that is we're dividing by 10 to the second power. So when you divide by 10 to a power, the power tells you how many spaces to move the decimal to the left. So we're just going to do that over here. So we're going to have 763 and then we have 0.52. But I'm just going to move this decimal. So let's just see where it's going to go. So we're going 1 two spaces to the left. And this tells us the quotient is going to be 7.6352. The decimal point is located between the seven and the six. So choice C is our answer. Now if instead the question had said what is 763.52 * 10 the 2 power in this case we would move the decimal two spaces to the right. Okay so if I rewrite the original number here we're just going to take that decimal point and go two spaces to the right. And since there's no digit to the right of the decimal I'm just going to put a zero here as a placeholder. So in a case like this, if we were multiplying by 10^ the 2 power, then we would have the decimal point is located to the right of the two. Okay? But that's not what was going on in this question. We were dividing by 10 to the second power. And one extra thing just to point out here, if we're unsure what is the meaning of 10 to the second power, just know 10 to the second power means to do 10 * 10. The power tells you how many times to multiply that number by itself. Okay? Okay, so I have 10 * 10 is equal to 100. If I had something like, let's say, 10^ the 3 power, this would be 10 * 10 * 10. Or I could just think of this as a 1 with three zeros. Okay, so multiplying and dividing by 10 has some nice properties to it. It allows you to do the math very, very quickly. Okay, but um for this question here, we have choice C. Question three, what is 37.2067 rounded to the nearest 100th? So for this question, it helps to know the following. When you have a five or higher, you round up. Okay? So five or higher tells you to round up. When you have a four or lower, you round down. So these are the rules for rounding to know. And we're going to apply it here to get to the answer right away. So we're rounding to the nearest hundredth. That's the space that we're going to be looking at. So I look at the decimal and the hundredth's place is two spaces to the right of the decimal. So we have a zero in the hundredth's place. And when you round, what you do is you always look to the number to the right. Okay? So we look one digit to the right is a six. And then we ask ourselves, is six in the five or higher club or is it in the four or lower club? So six is definitely a number that is five or higher. So this tells us we're going to be rounding up for this question. So now I'll go ahead and apply that that this is going to work out to we're going to have 37 point and we get two and then the zero we're going to round up to one. So this is going to be our answer and this will match up with choice B. But let's say they asked us to round this number to the nearest 100th. Let's say I had the number 37.2047 instead. If this were the case and we were rounding this number to the nearest h 100redth, then we look to the hundred's place and we have a zero. But now when we look to the right, we have a four. And when you have a four or or lower, you round down. So this number would round down to 37.20. And that would bring us over to choice A. Okay? But a is not going to be the answer here because we're going to be rounding this number to the nearest 100th. So this is doing the question by using the rules. But if you want to understand the concept, the concept is actually really nice to see when you use a number line. So I'm going to draw out a nice number line like this. And I'm going to start at let's say 37.2. And I'll put three zeros here. And we're going to be counting by hundreds. Which means what I think about in my head for something like this is I just look at the numbers after the decimal. We have a 2,00. And I'm going to pretend here I'm just going to add a 100 to this. So, I have 2,100 and then I have 2,200. So, what I'm going to write next, I'm going to write 37 point and then I'm going to write 2,100 after the decimal. So, this would be the next number after 37.200 if I'm counting by hundreds and then next I would have after this 37.2200. So for this question, the concept of what's happening is we're trying to see which number 37.2067 is closer to. We want to see is it closer to this number or is it closer to this number. So if you're having trouble seeing where this would land between these two numbers first, I'll put the halfway point. Halfway between 2,00 and 2,100 is 250. And I'm going to put that number over here. We're going to have 37.2. 250. So now when we have to decide where does this number land 37.2067 you see how the 67 at the end is more than 50. It's more than 50. 67 is greater than 50. So this number would be somewhere around here. Okay. 37.2067 is to the right of 37.2050. So now you have to ask yourself which number is 37.206 2067 closer to is it closer to this number or to this number and we could see that it's more than halfway. So this number would round up to 37.2100. Okay, so that's why this number rounds to 37.21. Just know that the trailing zeros at the end of a decimal we could just erase. So this number here 37.21 is the same as 37.210. They're the same exact number. Either way though, choice B is our answer. Question four, which value makes the comparison below true? And we have some mystery number is greater than 0.47. And we have these four options to choose from. Well, for one, we have to know that this symbol means greater than. And how are we going to remember that that symbol means greater than? Well, I remember going back now 30 years, my teacher showed me that this symbol here is less than. And the way to remember it is that it kind of looks like an L, but the L was turned on its side. And 30 years later, I still remember that. So hopefully 30 years from now, you'll still know that to be true using that exact same method. So this one is greater than and this one is less than. Another fun way is to say, how about we draw out this Pac-Man guy here? Okay. And Pac-Man was a popular character way back in the day. You may still know who he is, but here's Pac-Man. And Pac-Man only likes to eat the bigger number. So you could see this is the inequality symbol here in Pac-Man's mouth. So if we have 0.47 over here, then the bigger number has to go here because Pac-Man wants to eat the bigger number. And a nice way to compare these numbers here is to get all of them to have the same amount of digits after the decimal. So notice that we have three digits after the decimal for these two choices. But then we have two digits and one digit. So we are allowed to throw in zeros after our last digit when we are to the right of the decimal place. So here I could put one zero and here I'm going to put two zeros after the four. So now I have a bunch of threedigit decimals. And now the question is 0.47. That's the one we're comparing to. So I'm going to put an extra zero for this and over here. So now we're just looking at a bunch of three-digit numbers. And now it's just basically the same exact thing as asking yourself which number here is the biggest 470 400 470 471 or 47. The biggest number here is 0.471 because that's going to give us we're going to have 471,000. But the nice way to do this once again is to get all of your decimals to have the same amount of digits after the decimal. And then this question is much much easier to answer. So here choice C is going to be the number here that is going to be greater than 0.470. Question five. For which values of M would the product of M over 5 * 10 be greater than 10? And for a question like this, it really helps to have a strong number sense. And if you don't have a strong number sense, the good news is you could develop it. But you just have to practice things like your times tables, you know, your adding, subtracting, stuff like that really helps because when you get to a question like this, there's a lot you just have to think about when you're working through. So, I'm going to do my best here to help you think your way through this. So, I'm going to write out let's say I start with 10 x 1. And why am I doing 10 x 1? Because they're mentioning a product of some number time 10. So, I'm going to say 10 x * 1 is 10. If I do 10 x 2, that's equal to 20. If I do 10 x 3, that's equal to 30. And if I keep increasing the number that I'm multiplying by 10, my product is going to keep getting bigger. But now, let's say I go backwards. What if I do 10 times something smaller, like 1/2? Well, 10 time a half is the same thing as taking half of 10. And half of 10 is equal to five. And if you have to see this with the fractions over here, 10 * a half, what we're doing is we're doing 10 over 1. I'm allowed to write any whole number as that whole number over one. And now I'm just multiplying these two fractions. And when you multiply fractions, you multiply the numerators and you multiply the denominators. Okay? So here I'm just doing 10 * 1 / 1 * 2. And now this works out to I'll write it over here. We would have 10 * 1 is 10 over 1 * 2 is 2 and 10 / 2 is equal to 5. So this product checks out. And I'll do one more. If I do 10 * 1/5, I get 2. And how I'm getting this so quickly is I'm doing 10 / 5. I could show all the work that I just showed over here, but the shortcut is I'm just doing the whole number here divided by whatever number is in the denominator. Okay? Okay, so I'm just doing 10 * 1 over 5 is the same thing as 10 / 5 is 2. But what we should see here is that when we multiply by numbers that are greater than one, the whole number here is going to increase. See how it goes from 10 to 20, from 10 to 30, it's because we're multiplying by numbers that are greater than one. So over here when we multiply by numbers that are greater than one the whole number here is going to grow and if we look 10 * 1 is equal to 10 this is the turning point after this we get numbers that are greater than 10 but when we go less than this when we multiply by numbers that are less than one we get a product that is less than 10. Okay so for this question we're looking for the scenario where the product works out to be something greater than 10. So, I'll color code this. We want to be looking at this last part over here. And in this part over here, we're multiplying by numbers that are bigger than one or greater than one. Okay, so this is the key detail that the numbers we're multiplying by are greater than one. So now, how does that help us with this question? Well, this tells us that m over five, this part of the product here, has to be greater than one. But now we still need that strong number sense. What type of fraction works out to something greater than one? That would be an improper fraction. And when you have an improper fraction, that tells you that your numerator is more than your denominator. So we need m to be greater than five for this fraction to be greater than one. So that when we multiply by 10, when we multiply 10 by something greater than one, we get a number that is greater than 10. Okay? So, what we're looking for here is the answer choice that says any value of m greater than five. Choice C is our answer. But another cool thing we could do here is we could just check the answer choices here and see which one works out. So, if I pick a value of m that's greater than five, like let's say six. If I do m= 6, and I plug in, I would have instead of m over 5, I would have 6 over5* 10. And for this one, I'll use a nice shortcut. I could use this technique over here or I could say 10 / 5 is 2 and then 6 * 2 gives us 12. And notice that this product here worked out to something greater than 10. So choice C is definitely our answer. Question six, which expression has a value greater than one? And we have these four products to work through. Well, just know when you're dealing with fractions, when your numerator is greater than your denominator, your fraction is going to be greater than one. Okay? And this is when your numerator and denominator are both positive numbers. When the numerator is greater than the denominator, the actual value is going to be greater than one. So, that's where my head goes for a question like this. And the next thing I look at though is when we multiply fractions, we multiply the numerators together and we multiply the denominators together. So that's what I'm doing for all of these answer choices here. I'm just going to do we're going to have 3 * 2 on top over 4 * 3 and that's going to give us 6 over 12. And this number is not greater than one because see how the numerator is less than the denominator. This actually reduces if I divide the top and bottom by six to 1/2 and 1/2 is not greater than one. So a is out. If I write out the product for b, I would have 4 * 5 over 5 * 4. And this is going to give us 20 / 20 which is equal to 1. So b is no good. Just know there's a nice little shortcut too. When you are multiplying fractions, you are allowed to cross. So I could say 5 / 5 is equal to 1. And I could say 4 / 4 is also equal to 1. So when everything cancels out like this, everything just works out to one because we're just multiplying 1* 1 ultimately. So B is not our answer because we once again want the product that is greater than one. When we look at choice C, so choice C, we have 5 * 1 over 6 * 2 and this gives us 5 12ths. So this one is definitely no good. And then the last one here has to be it. But let's just verify. So I'll write that over here. We're going to have four * 4 in the numerator. So we're multiplying these two fours. And then we have 5 * 3 in the denominator. And we get 4 * 4 is 16 over 5 * 3 is 15. And notice the numerator 16 is definitely greater than 15. So this one fits the description here and works out to choice D. If I had to convert this into a mixed number, I could do 16 over 15 like this and say 15 goes into 16 once. 1* 15 is 15. I subtract 15. I get a remainder of one. So I have 1 and 115. Okay, which is definitely a number greater than one. Choice D is definitely our answer. Question seven, Mr. Lewis writes the following number in expanded form on the board. and we have 5 million plus 200,000 + 30,000 plus 400 + 9. And we want to know which shows this number written in standard form. So for this one, you can try to squeeze everything together just by looking at it like this and then match it to the answer choices. But if you really want to play it safe here, which I do recommend, write this out vertically like this. Okay, you could do it the fast way, but then I would definitely check my answer here to make sure I didn't make a silly mistake. So, when I add these numbers together, I'm not going to put the commas in, but I am going to make sure that everything is lined up to the correct place value. So, 200,000, I'm going to start by going right to left. I have three zeros, and then I have another two zeros. And then I have my two. Okay? So, 200,000 I'm going to write like this. And then 30,000, we have four zeros. I line up the ones place like this. I write my four zeros. And then I throw on the three. And then we have 400 would go here. And then nine would go here. And I'm adding all of this together. Okay. So, let me just move this over here. So, we have the space to add. So, I'm going to move this over here in this space. And when we add, we're going to have just a nine here in the ones place. In the 10's place, we don't have anything. So, I'm just going to put a zero. And then we have a four. We're just doing 0 plus 0 plus 0 + 4. And then we have 0 plus 0 plus 0 is 0. And we've got 0 plus 0 + 3. 0 + 2 is two. And our five goes down here. And now I could throw in the commas. So I like to start all the way to the right. I go 1 2 3 and I put a comma. 1 2 3. And I put a comma. So we have 5,230,49. And this matches up with choice C. Question eight. Which of the following shows 7,42 written in expanded form with exponents? So for a question like this we have to be able to use the powers of 10. So we should know that 10^ the first power is equal to 10. And then when I write 10 to the second power 10^ the 2 power is equal to 10 * 10 which is equal to 100. And if I go one more we have 10^ the 3 power is equal to 10 * 10 * 10 which gives us 1,000. So, when we're writing a number like this in expanded form with exponents, what we do first is we could rewrite this as 7,000 plus I'm not going to write a 100 term here because there's a zero in the hundreds place. But then next we would have 40 and then we would have two. So when we want to turn this number here in the form of the answer choices now what we do is we replace the th00and here in 7,000 with 10^ the 3 power. So if you want to see a nice in between step what I could do is I could rewrite this as 7 * 1,000. Okay? So 7,000 is 7 * 1,000 and then I have plus 40 is the same thing as 4 * 10 and the plus two at the end. Notice that a bunch of the answer choices have just plus two. So now I'm just going to make a substitution. The 7 * 1,000. I'm going to replace 1,000 with 10^ the 3 power. So in the next line here, we're going to have 7 * 10^ the 3 power. And then we would have plus 4 * 10^ the 1st power. If there's no power there, the invisible power is 1. And then we have our plus two. So now we scan the answer choices here and we need 7 * 10 the 3r. So that eliminates choice B. But at the end we need plus two. So that eliminates choice D. And then we need 4 * 10^ the 1st power. So that's going to tell us here that our answer is choice A. The problem with choice C, choice C is a very good answer, but it's wrong because this number here, so I'll just write this one over here. This number would be 7,000 because I'm doing 7 x 10^ the 3 power plus when I do 4 * 10^2 I'll write that over here. 4 * 10^ 2 is equal to 4 * 10 * 10. And that's going to give us 4 * 100 which is equal to 400. So that would be 7,000 + 400 + 2. And this would give us 7,4002. So the four is in the wrong place here. This would be a four in the hundreds place instead of the 10's place. So the correct answer here is a question nine. This hundredth's model is shaded to represent the multiplication of two numbers. Which equation can be represented by the shaded parts of the model? So for this question, what I notice right away is that we have light gray, we have medium gray, and then we have dark gray shaded squares. Okay? So we have these three types. And the light gray if we see there's 100 squares in total but the light gray go over 1 2 3 four columns and each column has 10 squares. So altogether that would be 40 light squares out of 100 squares in total. Okay. The math that I'm doing in my head here is I'm seeing once again we have four columns going this way for the light gray and each column goes down 10. So when I do four * 10 I get 40. But in total, we're going over 10 down 10. So in total, we have 10 x 10 or 100 squares. And you might be thinking, wait, these are not light squares. But the dark squares here represent the overlap of the light gray and the medium gray. So if we see in these multiplication problems here, we have the light times the medium is giving us the dark. Okay, that's how I'm seeing this going on here. And you'll see this is going to get us to one of the answers that we need. And that's going to be our correct answer. So now for the medium, if we focus on the medium next, notice there is 1 2 3 4 five rows. So for the medium, we have five rows and each row has 10 squares. So if we do this multiplication in our heads, we have 5 * 10 is equal to 50. And once again in total we have 10 x 10 or 100 squares. So 50 out of 100 squares are medium squares. But now all we have to do here is multiply this stuff together. But what I'll do is I'll turn these fractions into decimals. We have 40 hundreds. That's the same thing as 0.40. And then 50 hundreds is the same thing as 050. So now we could focus next finally on the overlap here. The dark, the dark is the overlap. And if we look, we have four going across and we have 1 2 3 4 5 going down. So for this one, we could just count it out here or multiply. We have four by five is 20. Or I could just count these. There are 20 dark squares. And that's going to be the result of the product. But remember altogether we have 10 * 10 or 100 squares. Okay. So the dark is going to be we're going to have 20 out of 100. And if we look through this well first 20 out of 100 I could say is 0.20. But we get 0.40 * 050 is equal to 0.20. Choice A right away is our answer. And if we want to actually multiply this out, what we could do is we could say 0.40 40 * 050. This is equal to, and for this one, there's a few different ways we could handle this. We could turn this back into fraction form and multiply. So, if I have 40 out of 100 times 50 out of 100. To multiply here, I'm doing 4 * 5. So, I'm multiplying these two. 4 * 5 is 20. And then I just tack on the trailing zeros here. So, we have two trailing zeros that were not accounted for. So, I have 2,00. And when I multiply 100 times 100, I do 1 times 1 is one. And then notice that we have two four trailing zeros. So, that's 1 2 3 4. And now when I simplify this, I could cross off common zeros. So, I have one pair of common zeros, another pair, and a final pair like this. And this works out to 2/10. And 2/10 is the same thing as 0.2, 2, which I could also express as 0.20. Okay, so choice A definitely checks out. Question 10. Which letter on the number line best identifies the location of NE5? Well, when we're dealing with negative numbers, everything goes backwards. If I were to count going forward here, we have zero and then we would have 1 2 3 4 5 and this would continue this way. But when we go backwards, we have 0, then we have -1, then -2, -3, and then we have -4 is already labeled, then -5, and then -6. So if we're looking for the location here of five, which letter on the number line best identifies the location of five? That's going to be over here at point W. Okay, so choice A is our answer. Question 11. A school orders 46 boxes of pencils. Each box contains 125 pencils. How many pencils does the school order in total? So for this one, if it's not obvious right away that we have to multiply, what you could do is you could draw this out. So we have a box of pencils and this box of pencils has 125 pencils in it. And then the second box here would have another 125 pencils. And I could draw this out, but drawing out 46 boxes is going to take a long time. So, what we have here is we're just going to say that there's 46 of these. And by the time we get to the end, the 46th box has 125 pencils. So, we could sit there and just add these numbers up like this. But which operation takes care of repeated addition? That would be multiplication. So, we're going to be doing 125 times 46. And for this one here, if I multiply by the six first, this is in the onees place. I'm going to do 6 * 5 is 30 and I carry the three. And then I do 6 * 2 is 12. 12 + 3 is 15. So I write my five and then I carry the 1. And then we have 6 * 1 is 6 and 6 + 1 is 7. And then once I complete this here, I go on to the next row here and I multiply by the four in the 10's place. But I'm just going to give us a little bit of space here. I'm going to move this up. So now we're going to multiply through by the four. And let's go ahead and do that. But first, we have to put a placeholder. If we're multiplying by a number from the 10's place, we have to put a placeholder here in the ones place. We put a zero. And we start multiplying, but once again from the 10's place. So we have four * 5 and we're going to get 20. So I put a zero here and I put a two up top. I could cross out these numbers from before because now I'm multiplying by a new number. So the old numbers that we carried we could just get rid of. So we have once again 4 * 5 is 20. I put a zero and I carry the two. And then I do 4 * 2 is 8. And then 8 + 2 is 10. So I put a zero and I carry the one. And now finally 4 * 1 is 4. And four + 1 is five. And then all we have to do here is just add all this stuff up. So for the final product here, we're going to have 0 plus 0 is 0. 5 + 0 is 5. 7 + 0 is 7. And the five just carries down. So the final answer here should be 5,750 pencils. And this matches up with choice C. Question 12. A farmer harvest 3,276 apples and wants to pack them equally into 12 crates. How many apples will be in each crate? So for this one, the key phrase is that the farmer is packing the apples equally into 12 crates. So this tells us to divide. Okay, so we're going to be taking the total here. So there are 3,276 apples and we're going to be making groups of 12. So we're going to divide here by 12. So we're going to have 3,276 divided by 12. And for this one, we're doing long division. So we go piece by piece. So I say 12 goes into 3. 3 is not divisible by 12. We have to go for a bigger number here. So we go to 32. 32 we can divide by 12. We're going to get something more than zero. So we say 32 / 12. And the thought process for this is we have to know our times tables to be able to do this without a calculator. So 12 * 1 is equal to 12. You have 12 * 2 is equal to 24. If I did 12 * 3, that's 36. And that would send me too far. I could only go up to 32. So I have to say two up here. Okay, 32 / 12 is 2. But when I multiply, I have 2 * 12 is 24 and I subtract. And when you do 32 - 24, you're going to get 8. So we have a remainder of 8. And then the process continues. I bring down the next number and we're going to do 87 / 12. So if we continue, I could list all the times tables here. But let's say I go up to I skip a bit. I have 12 * 5 is 60. And then next I would have 12 * 6. I could just add 12 to this. 60 + 12 is 72. And if I go one more, 12 * 7 gives us 84. If I go another one, 12 * 8, that's going to bring me to 96, which is too far. So I could put a seven next. So, I could say 87 / 12 is 7. And when I do 7 * 12, I get 84. And I subtract. And I have a remainder of 3. And then I bring down the next number here. So, we're going to bring down the six. And finally, for this one, 36 / 12 will divide evenly by 3. Okay? And then we do 3 * 12 is 36. We subtract 36. We get a remainder of zero. So, this divides evenly. So each crate is going to have 273 apples. And choice B is our answer. Question 13. Lena bought two notebooks at $3.85 each, three pens at $1.45 each, and one folder for $2.30. She paid with a $20 bill. How much change should she receive? So for this one, let's go piece by piece. She bought notebooks, and the notebooks were $3.85 each, and she bought two of them. So, what we could do is we could take $3.85 and we could add it to itself. Okay? So, I'm going to do $3.85 plus $3.85. And this is for notebooks. So, we have 5 + 5 is 10. We put a zero and carry the one. And then we're going to have 8 + 8 is 16. 16 + 1 is 17. We carry the one. The decimal could just drop straight down like this. And now we have 3 + 3 is 6. 6 + 1 is 7. So, the total cost of notebooks is $7.70. And now we'll focus on pens next. So, the pens are $1.45 each. And she's buying three of these. So, I could add $1.45 to itself three times. Or I could just multiply by three. So, we'll do it with the multiplication. 3 * 5 is 15 and we carry the 1. And then we have 3 * 4 is 12. 12 + 1 is 13 and we carry the 1. And then we have 3 * 1 is 3 and 3 + 1 is four. So if we look at the total bill here, let's think about where the decimal goes. Just think about it. The pens are $1.45 each. So we could even use estimation for this just to see how reasonable our answer is. So let's think about is $435 a reasonable price for the three pens. Well, no, because let's say I even rounded this up like way up to $2. Even though this would round closer to a dollar, let's say the pens were only $2 each. So, if the pens were $2 each and Lena bought three of them, that would be $6. So, $435 is way off. So, the decimal should go here. Okay, it should be $4.35. We could even think about why wouldn't the decimal go over here? Because $43.50 50 cents is also very unreasonable. And if we put the decimal here, is the pen going to be around 44 cents if we round to the nearest 100th? No. Three pens would not be cheaper than the cost of one pen. So that's why here $4.35 is the way. Okay, that's what we're going to go with. And now for the rest here, we have one folder. So the folder costs $2.30. Okay, so one folder is just $2.30. So now what we do is we just add these three items up. So we're going to have 7.70 plus 4.35 plus 2.30. We add this up and this is going to tell us the total bill. Okay, so the total bill is going to be 0 + 5 + 0 is 5. Then we have 7 + 3 is 10 and then 10 + 3 gives us 13. We carry the one. And then we have 1 plus 7 is 8. And then I could do, if I want to do this in a certain order, I could do 1 plus 7 is 8. And then 8 + 2 makes 10. And then I could add in the plus4. And if I do 10 + 4, I get 14. So the total bill here is $14.35. So what we need to do is we see if Lena pays with a $20 bill, how much change should she receive? So we have to subtract Okay, we're going to be subtracting from 20. So, we'll write this out now. We have $20. She's paying with a 20 and they're going to take $14.35. So, for this one, we borrow. So, we're going to borrow from here and make this a one. We move it over. And then we just keep borrowing again. So, we're going to make this into a nine. Put the one here. We borrow from here and make this a nine. Put the one here. And then we have 10 - 5 is 5. 9 - 3 is going to give us six. I could drop down the decimal. And then I have 9 minus 4 is five. And the 1 min - 1 is zero. But I don't actually write it because you wouldn't want to put a zero in front of a nonzero digit to the left of the decimal. Okay? If I write 05.65, that just doesn't make sense. We just leave it like this. So the change is going to be $5.65. And this matches up with choice D. A bottle holds 1.25 L of juice. If Jackson buys four bottles, how many liters of juice does he have? So for this one, we could even just draw this out. So we have four bottles here. So Jackson buys four bottles and each bottle holds 1.25 L of juice. So I'll just write this in here. We have 1.25 all the way through like this for the four bottles. So how many lers of juice does he have in total here? All we have to do is multiply 1.25 25 by 4 because I'm really just doing 1.25 plus itself four times. But when we have repeated addition, that's the same thing as multiplication. So, we could just multiply by 4. So, we have 4 * 5 is 20 and we carry the two. And then we have 4 * 2 is 8. 8 + 2 is 10. We carry the 1. And then 4 * 1 is 4 + 1 is 5. But then here, notice that I ignored two decimal places. So, I just have to pay these two decimal places back. And 5.00 is going to be our answer. Choice C. Question 15. A 3.6 m rope is cut into four equal pieces. How long is each piece? So, for this one, we are cutting the rope into four equal pieces. So, this tells us to divide. So, what we're doing here, we're doing 3.6 / 4. Okay? So, we're going to do 3.6 / 4. And what we have next is I could write this as 3.6 over 4. So I could write this in fraction form. And now I'm just going to do something a little bit different. What I'm going to do here is I'm going to multiply the top and bottom of this fraction by 10. And what that's going to do when I multiply this decimal by 10 here, the decimal is going to move one space to the right. And that's going to make 36 in the numerator. Okay? So I'm doing 3.6* 6 * 10 in the numerator over 4 * 10 in the denominator. So we do 3.6 * 10 that gives us 36 over 4 * 10 is 40. And now what I'll do is I'll reduce this fraction. I'll divide the top and bottom by 4. So I divide top and bottom by 4. 36 / 4 is 9. 40id 4 is 10. And I get 9/10 which is the same thing as 0.9. So our solution here is going to be 0.9 m. But now I want to look at one more thing before we move on. Let's say I look at 36 / 4. 36 / 4 is equal to 9. If I do 360 / 4, 360 / 4, this would give me 90. Okay? So we would get 90 here. And now I'm just going to throw in a decimal like this. We have 0.0. So notice when I do 360 / 4, I get 90. 36 / 4, the decimal moves one space to the left. So if I go down now to 3.6 / 4, 3.6 / 4, the decimal would move one over and we would get 0.9. Okay? And if I really want to go one extra step here, I could say if we did 0.36 / 4, then the decimal would move one more to the left and we'd have 0.09. We'd have 9 hundreds. Okay? So just this is a good skill to have in math where you compare the division to something familiar and you look for a pattern here. But either way we're getting choice B. Hank is building a pig pen for his pig Hadley. He has a 7 foot long wooden beam and wants to cut it into pieces that are half a foot long. How many pieces will Hank have when he's finished cutting? So for this one, let's draw out a 7 foot long wooden beam. Okay. Okay. Well, this won't actually be 7 ft, but you get the idea. This is just to visualize what's happening here. So, here's the 7 foot beam. And what I'm going to do is I'm actually going to count to seven here. So, we have one 2 3 four five six. And that'll be the seventh piece here. Okay? So, we have 1 2 3 4 5 6 7. Not drawn exactly to scale here where each piece is exactly equal, but this is just to visualize what's happening. Okay? Okay. So, we have once again 1 2 3 4 5 6 7 like this. And if he's cutting each piece a half a foot long. So, he's cutting once again each piece of wood from the entire 7 foot piece is being cut half a foot long. That means the first cut would be around here. Okay? See this is half a foot. And then he would cut over here. He would cut over here. He would cut over here. So, you see I'm just cutting this at the halfway mark. So for something like this, we want to use division because this question would be way more difficult to do visually if this piece of wood was, let's say, 700 ft long. I'm not going to count to 700 and make that many chops. That would take forever. But for a 7ft piece, this visual is really good to see what's happening. Okay, so we cut in half here. We cut at the half foot mark here. And we keep going. But notice between any one piece, we're going to get two pieces of wood out of that. So the idea is we're taking seven and we're multiplying it by two. So that's how I know it's going to be 14. If we had to, we could just count these. See how we have 14 pieces here. So once again, we could just draw out a model here for this 7ft piece of wood. And then we could just count how many half-t blocks we have and we have 14 of them. But the fast way to do this without drawing it out would be to take seven and divide it by 1/2. Okay, this is what 7 / a half means visually and the technique for doing this we use keep change flip. We keep the first number, we change the operation from division to multiplication and then we flip the fraction. Okay, so this is a technique when you are dividing fractions. So if you want to add this somewhere to your notes, just remember keep change flip. You do this when you are dividing fractions. So now we just simplify this here. This is equal to 7 * 2 / 1 is 2 and then 7 * 2 is 14. We get the exact same answer. Question 17. What is the decimal form of 3/4s? Now for this one, where my head goes is I just think three quarters. So just pretend we're talking about money here. And if we have three quarters, that's 25 cents each for each quarter. So when we have three quarters, we have 75 cents. So that's 0.75. And that's how I know right away just thinking of money that this is choice C. But another way to see this would be to take 3/4s and to turn this into something base 10. So multiplying four to get to 10 would be difficult. We'd have to multiply by 2 1/2. But if we multiply 4 by 25, that would turn this denominator into 100. But if I multiply the bottom by 25, I have to do the same thing on top. So I'd have 3 * 25 is 75 over 4 * 25 is 100 and 75 hundreds is the same thing as 0.75. So either way we're getting choice C. Question 18. What is the value of the expression shown below and we are subtracting two mixed numbers. And what makes this question a bit tricky is that the fraction over here is less than the fraction over here. So we are going to have to borrow. Now, how do I know that this fraction is smaller? Well, I always just think in terms of pizza. So, if I have a pizza pie cut into four slices, if I eat one out of the four slices, that's how much pizza I'm eating. But if I had the same size pizza pie, but this time it's cut into six slices. So, I have something, let's say, like this. And let me just draw that a little bit neater. So, we have going to cut it here, here, and then let's say here, like this. If I eat five out of six slices, I've eaten a lot more pizza on this pie than I did on that first pie. I ate almost the whole thing. So that's how I know that 1/4 is less than 5 over 6. So what that means is when we do the subtraction here, we are going to have to borrow in order to do this subtraction. Okay, so this is for the method here where we keep the mixed numbers as mixed numbers, but we borrow so that we could actually do this subtraction. So the second idea that we need in order to do this is we need to be able to find a common denominator for four and six. So we have to think of the least common multiple of four and six. Well, if I start counting by fours, I get four and then 4 + 4 is 8 plus another four is 12. Then 16 and so on. If I count by sixes, I get 6 12 18 and so on. The first number that these two lists have in common is 12. So that's the least common multiple of four and six. So now on the side I'll find the common fraction here for 1/4. So I'm going to multiply this one by 3 over 3. And we're going to have 1 * 3 is 3 over 4 * 3 is 12. And for the second fraction 56, I'm going to multiply top and bottom by 2. And that's going to give us 10 12. Okay. So the first thing I'm going to do is I'm going to rewrite this subtraction here as 11 and 3 12ths minus 6 and 10 12ths. So I want those common denominators. But once again, notice that three is less than 10. So when I do the subtraction here, I have to fix this over here. So then the concept that I'm going to use is I could borrow from this whole number here. So what I'm going to do is just know 11 and 312ths is the same thing as 11 + 3 12ths. Okay? These are the same thing. And when you write mixed numbers, you basically just get rid of that plus sign and you squeeze the numbers together. So, what I could do from here, so I'm just move all this stuff over is I'm going to borrow from the 11. So, what I'm going to do is I'm going to break one off. I'm going to call 11 10 + 1. So, I'm just breaking this into two pieces. And then I write my plus 3 12ths. And then from here, what I'm going to do is I'm going to call this 10 + instead of one, I'm going to call that 12 over 12. And I'll just keep moving this over here because we're going to simplify this at the end. So notice once again I'm doing 10 + 12 and then + 32. And now what I'll do is I'll combine these fractions. This is equal to 10 + and then I get 12 + 3 over 12. So this becomes 10 + 12 + 3 is 15 and I get 15 over 12. And as a mixed number I could call this 10 and 15 12ths. Okay. Okay, so we're going to rewrite 11 and 3 12ths as 10 and 15 over 12. If you could do all of this stuff in your head, like you just know to borrow from here to turn this into a 10 and to add 12 here to make 15, then great. But this is the concept. So over here, I'm just outlining the concept of what's actually happening. Okay, so the concept behind why borrowing works for mixed numbers is all outlined here. So now we'll just make that substitution. 11 and 3 12ths we are going to replace with 10 and 15 12ths. So we could say that this stuff over here is equal to we have 10 and 15 12ths minus 6 and 10 12ths. So now we could just go piece by piece. If I do 10 - 6, I subtract the whole numbers here. Then I'm going to get 10 - 6. This part is equal to it's going to work out to four. And then for the other part here we are subtracting these two fractions. 15 / 12 minus 10 / 12 is going to give us 15 - 10 is 5 and then over 12. So now I could just throw in this is the end of my mixed number here. So we're looking for the answer 4 and 5 12ths which is going to give us choice A. Now before we move on I just want to look at another option and we'll call this option two. So for this method here, we're going to turn each mixed number into an improper fraction and then subtract. So let's say I start with 11 and 1/4. We're going to do 4 * 11 plus and then we just add this numerator 1. And we're over the original denominator here is 4. And then we have minus for the next mixed number. We're going to turn this into an improper fraction. We're doing 6 * 6. And then plus, we're just adding this numerator 5. and we're dividing all of this by that denominator 6. So then from here we just simplify. We have 4 * 11 is 44 and 44 + 1 is 45. So we get 45 over 4 and then we have minus and from here we have 6 * 6 is 36. 36 + 5 is 41 and we're over six. So we still have to find common denominators. So that method of finding the least common multiple before is still going to be helpful here. So what we could do is we could multiply that first fraction by 3 over3 because we want that common denominator of 12. And then here we would multiply by 2 over2. So then in the next line what we're going to have we're going to have 3 * 45 is going to give us 135. And let me just change colors here. So we're going to have for this next part we'll have 135 over 3 * 4 is 12. and then minus 41 * 2 gives us 82 over 6 * 2 gives us 12. And then from here we just have to do this subtraction. So I'll do that over here. 135 - 82 we have 5 - 2 is 3. And then 13 - 8 is 5. So we get 53 over 12. But then from here we just have to do this division. We have to turn this improper fraction into a mixed number. So I'll go ahead and do that over here. So let's say, let me just move this out of the way, but let's say we have the division. We're doing 53 / 12. So 53 / 12. And we could say 12 goes into 53 four times. 4 * 12 is 48. We subtract and 53 - 48 is going to give us 5. So that's our remainder. But we write our remainder over the original divisor. So we get 4 and 5 12ths. So either way here both methods are going to get us choice A. Question 19. Which fraction is equal to 0.6 in simplest form. So we have to know that 0.6 means 610. So when you hear 610 that means 6 over 10. And now all we have to do is write this in simplest form. So we have to reduce. So we're going to divide the numerator and denominator by two. And this fraction will reduce to 3 fifths. Okay? Okay, we're doing 6 / 2 is 3 and 10 / 2 is 5. So 3 fths choice C is our answer. Question 20. Which list contains only prime numbers? So for this question, we have to know the definition of prime numbers. A prime number has only two factors. So we'll say that prime numbers have only two factors. And the factors I'll just write over here. The factors are one and the original number. Okay, so that means that once again our prime number only has two factors. Now I just want to clear up some confusion that pops up around the agreed upon definition for primes is that the first prime number is two. So we'd have two 3 5 7 I wouldn't say nine. Nine is not prime because nine is divisible by three. So nine has more factors than just one and the original number itself. So we'd have to skip nine and then go to 11, 13, and so on. But one thing I really want to mention is the number one. So just know this forever. One is not a prime number. Okay? And I want to point this out because if this pops up as sort of a sneaky question, one is not prime because one does not have two factors. The only factor of one is one. So one does not have two factors. It has one factor. So one is unique. It's like its own entity here. It is the identity element of multiplication. So we do not consider one to be a part of the primes. The prime numbers start at two. So now we'll go through the answer choices. And which list contains only prime numbers? So 29 looks good. 29 is prime. 31 is prime. 37 is prime. 41 is prime. Right away this answer choice is looking good. So we could say here that this one we could we could chop out. And now let's say we look at choice B because of course we should see why the other answer choices are no good. So the culprit here in choice B is 27. 27 is no good because 27 is equal to 9 * 3. And notice that these are numbers that are not originally 27 or 1. So 27 has more than two factors. So that's why B is out. Choice C is out because of a few answers or a few numbers in the list. 21 and 33. 21 is equal to 7* 3 and 33 is equal to 3 * 11. So C is wrong twice. And now choice D. Why is choice D no good? So this list is a little bit sneaky because this number is prime. This number is prime. And so is this one. But 51 is a sneaky composite number. Okay, composite numbers are numbers that are not prime. They are divisible by numbers other than one in themselves. So 51 is actually equal to 3 * 17. Now a little divisibility trick that you could use when you take the number 51, you could say, let's just take a look at this. If we have 51, I'm going to look at the sum of the digits. So I want to do 5 + 1 = 6. And 6 is divisible by 3. Okay, so this number is divisible by three. And what that means is that this number in the beginning is also divisible by three. So 51 is also divisible by three. So this is the sum of the digits trick. If you take any number and you add up the digits and you get a sum, if that sum is divisible by three, the original number is divisible by three. And that also works for nine. If the sum is divisible by 9, then the original number is also divisible by 9. Okay, so that tells us 51 here is not a prime number. So D is out. Choice A is definitely our answer. Question 21. A water bottle contains 2 L of water. Mia drinks 3/4 L in the morning, 2s L at lunch, and 1/2 L during soccer practice. How much water is left in the bottle? So for this question, the first thing that jumps out is we have two 3/4s, two- fifths, and 1/2. And we have a lot of fractions here. So what we're going to have to do is we're going to have to find common denominators because we're going to be subtracting these fractions from two. So that's where my head goes right away. So if I look at these denominators and I write them in order from least to greatest, we have denominators of two, four, and five. And if I start listing out the multiples of two, I'm going to have to list out quite a bunch. Although this list you could just make in your head, but I will write out a few of the numbers in the list for the multiples of two. All right. And I'll stop at 24. And then for four, we'd have 4 8 12 16 and we go to 20 24 28 and so on. And if I count by fives, I have 5, 10, 15, 20, and I'll go to 25, 30, and so on like this. But what I'm looking for is the least common multiple. So the least common multiple, the number that shows up in all three lists first is 20. So this is what I want to work with in this question. So now I look at all of these numbers. So I have two and I could write two as 2 over one. And I'm going to multiply the one on bottom by 20 because 20 is the number I want. But if I multiply the bottom by 20, I have to multiply the top by 20. And this is going to give us 40 over 20. Okay? So I'm just rewriting two as 40 over 20. And then next I'll focus on the 3/4s. So 34s, if I want a denominator of 20, I have to multiply the bottom by 5 because 4 * 5 is 20. And then I do 5 on top. So that gives us 3 * 5 is 15 over 4 * 5 is 20. And then next I could focus on the two- fifths. So for two fifths, I could do 4 over 4 because 5 * 4 is 20 on bottom and then 2 * 4 is 8 on top. And now I'll just make a little bit of space here so we could fit the last one. Okay, so we'll just move this stuff over and move this one over a little bit too. So now for the last one here, we will look at the fraction 1/2. If I do 1/2, I'm going to multiply by 10 over 10 because once again, I want a denominator of 20. And on top, I have 1 * 10 is 10. So now I'm ready to work through this question. So once again, Mia has a water bottle that holds 2 L of water. Okay, it has 2 L of water in it. And what's happening here is Mia is drinking the water. So that's going to be subtraction here. We're going to be subtracting water out. And now we have all the pieces we need. So let's start off here with the total amount of water is 2 L right now. That's at the start of the question. And I'm not going to write two. I'm going to write the 40 over 20. So 40 over 20. And then minus we have Mia drinks 3/4 lers in the morning. So this is the morning amount that's being subtracted. We're going to subtract not 3/4s. I'm going to write it as 15 over 20 because now what we do is we could just work out. We're doing 40 minus 15, which I could write I I'll write over here on the side. So I borrow 10 minus 5 is five. 3 minus one is two. So this tells us after the morning, Mia has 25 over 20 lers of water left. And then after the morning, it's the lunch. It's lunchtime for Mia and she drinks two-fifths of a liter. So the two- fifths, notice we wrote with a denominator of 20 and that worked out to 8 over 20. So we're going to subtract now 8 over 20 and this represents how much water Mia drinks at lunch. Okay, so now we're going to subtract the water from lunch and this is going to work out to. So we're doing now 25 minus 8 is going to give us 17 and we're over 20. And now finally the last one we're subtracting is the half liter that Mia drank during soccer practice. So minus and this is from soccer practice. So now we just subtract out the 10 over 20 and this is going to give us our final answer. We have 17 - 10 is 7 over 20. And this brings us right to the correct answer. Choice B. Question 22. Which inequality correctly compares the two fractions? So for this one I'm going to write out 3/8 and 5 12ths. And first I'll show a trick. We're going to multiply, but we're going to multiply up and across. So, I'm going to do 8 * 5 is 40. And then I'm going to do 12 * 3 and that's 36. And I'll write the product above. And now I compare those two numbers. 40 is greater than 36, which tells us the greater fraction is 51 12ths. So, if I want to write out the inequality, I would say 3/8 is less than 52. And remember the idea behind the inequality symbol. Like I said before, you could think of the inequality symbol as Pac-Man. And Pac-Man wants to eat the bigger number. So since this is the bigger number, his mouth is going to be facing this way. So we're going to get 3/8 is less than 51 12ths. Now the next the next way we could do this is we could rewrite these fractions to have common denominators. So if I write out 38 and 5 12ths, I could look for the least common multiple of 8 and 12. And the least common multiple of 8 and 12 is going to be 24. If we count by 8s, we have 8 16 24 32 and so on. And if I count by twelves, I get 12 24 36 and so on. So the least common multiple is 24. So I'm going to make 24 on bottom by doing times three. So then I have to multiply by three on top. And then I got 3 * 3 is 9 over 8 * 3 is 24. And now to turn this fraction into a fraction with denominator 24, I'm going to multiply the bottom by two. So I have to multiply the top by 2. And now 5 * 2 is 10 over 12 * 2 is 24. And now it's very obvious that this fraction is greater because we have the same denominator. So now the fraction with a bigger numerator is bigger. So 5 12ths is definitely going to be greater than 3/8. So we could say 3/8 is less than 512ths. So choice A is definitely correct. Question 23. A recipe for one loaf of banana bread uses 1.6 cups of flour. How many cups of flour are needed to make five loaves? So for this one, we could just draw out five loaves like this. I'm just I'm just going to draw out rectangles like this. And I'll make five of them. And for each of these loaves, we need 1.6 cups of flour. So this one's going to take 1.6. This one will take 1.6. 6 and so on. So, how many cups of flour are needed to make five loaves? Well, for this, what we could do is just multiply 1.6 by 5. So, I could do 1.6 * 5. And now, the thing to think about here is if I were to do 16 * 5, 16 * 5, I have 5 * 6 is 30, and I carry the three. And then I have 5 * 1 is 5 + 3 is 8, and I get 80. So I'm going to say here 16 * 5 is 80. If I were to do 160 * 5, that would be 800. So if I do 1.6 * 5, see how here I'm trending backwards. I have 160, then 16, then 1.6, then this is going to shrink down to eight. Okay? So here we're going to have eight cups, and this is going to match up with choice A. Question 24. The line plot shows the lengths of 15 jump ropes used in gym class. and we want to know what is the total combined length in feet of all 15 jump ropes. So for this line plot question, what we could do is we could just go one at a time. We're going to be adding up all these x's, but we have to make sure that we label them correctly based on where they are along this number line. So these first two x's here are in the seven and 1/2 foot section over here. So when I add these up to find the total combined length, first I'm doing seven and a half plus 7 and 1/2. And now if this is math that you just can't do comfortably in your head, what you could do is you could break up these mixed numbers into a sum. I could say 7 and 1/2 is 7 plus a half. So if I add 7 and 1/2 + 7 and 1/2, I'm doing 7 + 1/2 plus 7 + 1/2. And then the concept is we could add the whole numbers together and we could add the fractions together and we'll write the results over here. So I'll put the sum of the whole numbers here and the sum of the fractions over here. So in the next line we're going to have 7 + 7 is 14 and then 1/2 + 1/2 is going to give us 1. So then when I do 14 + 1 this gives us 15 and this represents the length of the jump ropes that are 7 and 1/2 ft long. Okay. So in total here, if I look at these two, they're going to be 15 together. And now I'll go to the next part here. So we have these jump ropes here in the 8t section, there's four of them. So I only have to do 8 * 4, which is equal to 32. So when we have just whole numbers here, it's a bit easier to work with. So I'm just going to put a 32 up here like this to represent the length of the 8ft jump ropes in total. And then next, we could focus on the 8 and 1/2t ones. So, this one is definitely a little bit trickier. There are a few ways we could handle it. I mean, for this, we could just go ahead and do 8 1/2 + 8 1/2 + 8 1/2 or we could do 8 1/2* 3. So, I'm going to write out 8 and 1/2* 3, but I'll write 3 * 8 1/2. But I'm going to break 8 1/2 up as I'm going to write three times and then we're going to have 8 plus a half. Okay? Just remember in math parentheses means multiply. So I'm doing 3 * 8 1/2. So what I'm doing is I'm multiplying the whole number by 3 and then I'm multiplying the fraction part by 3 and then adding. So what we're doing is we're using the distributive property and this is going to help us find the total length of the jump ropes that are 8 1/2 ft long. So I do 3 * 8 is 24. And when I do 3 * a half, 3 * a half I'll do off to the side is the same thing as doing 3 over 1 * 1 over2. And that gives us 3 * 1 is 3 over 1 * 2 is 2. And I could also say that 3 over two is equal to 1 and a2. If I have to, I could just do that over here off to the side. So I'll do 2 going into 3. 3 / 2 is 1. 1 * 2 is 2. I subtract and I have a remainder of 1. And my original divisor is 2. So I could say once again I have 3 * 8 is 24 plus 3 * a half is 3 over2 which is equal to 1 and a2. So when I do 24 + 1 and 1/2 that's 25 1/2 and that's going to go up here in the purple section. So 25 1/2. And now we can move on to the 9 ft jump ropes. And this one is a little bit easier to work with because we're just going to do 9 * 3 is equal to 27. Okay. So 27 is going to go up here. And now for the last one here, we just have a 9 and a half foot jump rope. There's only one, so I could just write nine and a half feet. So now they want the total combined length. So now I just add all of these numbers up. But one thing I'm going to do off to the side here, I want to add these numbers separately. I want to add 25 1/2 and 9 and 1/2 because those mixed numbers could be a little bit annoying to deal with. But they they both have a fraction of 1/2 trailing, which means when we add them together, we're going to get a whole number. So we're doing 25 1/2 plus 9 and a half. And this I could think of as 25 plus a half + 9 plus a half. And then I could switch the order here. Addition is commutative. The order doesn't matter. So I could choose to do 25 + 9 plus a half plus a half. And now from here I have 25 + 9 is 34 and then I have 1/2 + 1/2 is 1. So that part of the sum is going to work out to 35. So now I'll just write out everything in one space here. So let's put that all over here like this. And I'll just make this a little bit neater. So we're going to section this off. So we're going to have 15 plus 32. And then we have 27. And now the two green ones that I highlighted combine to 35. So I'm going to put a 35 here and add all this up. So we have five. Uh these I could actually to show a little trick when you're doing long addition. You could do 5 + 5 first is 10. And that's going to make it easier to close this out. So I have 10 and then plus 7 is 17 + 2 is 19. I carry the 1. And then I have 1 + 1 is 2. 2 + 3 is 5. 5 + 2 is going to give us 7. and 7 + 3 is going to give us 10. So I write my 10 and then our answer is going to be 109 choice B. Question 25. Four students participated in a classroom election. The bar graph shows how many votes each student received. And we want to know what fraction of all the votes went to Millie. So for this question, the big idea is part over whole. Okay, so let's think part over whole. That's going to tell us what fraction of all the votes went to Millie. So the part here represents Millie. So this is the number on top here is going to be the number of Milliey's votes. Okay? So the number of Milliey's votes goes on top and on bottom we're going to have the total number of votes. Okay? So this is going to be over the total number of votes. So let's go ahead and find each of those pieces. So, if we look at this bar graph here, the way to read a bar graph is that you just go to the height of the bar and you go all the way across like this. So, if we follow this all the way across, this tells us that Millie got 20 votes. All right, let me just make that a little bit neater. So, we go all the way across and we can see Millie got 20 votes. So, I'll write a 20 above Millie. And if we look at Lily, Lily and Frank got the same amount of votes. They're at the same height and they're at a height of 10. So, I'm going to put a 10 here and a 10 here. And then Hank, if we look at Hank's votes, Hank got 15 votes. So the total here, we'll find the total first. The total number of votes. And maybe I'll write it off to the side. The total is equal to 20 + 10 + 10 + 15. So this tells us if we add, we have 0 plus 0 plus 0 + 5 is 5. And then 2 plus this many ones, we have 2 + 1 is 3 + 1 is 4 + 1 is 5. There are 55 votes in total. Okay. Okay, so this is the total once again the total number of votes is here and it's 55. But now Milliey's votes is just 20. Okay, is here. So Millie got 20 out of 55 votes. And then we look and we see that our answer doesn't show up and that's because we have to reduce our fraction. So we could divide the numerator and denominator by 5. And we do 20 / 5 is 4 over 55 / 5 is 11. So 4 11ths choice A is our answer. Question 26. Ava is baking muffins for a school fundraiser. She uses six cups of flour, one cup of sugar, two pints of milk, and one quart of oil. How many quarts of ingredients does Ava use in total? So for this question, we have to know that one pint equals two cups. We also have to know that one quart is equal to two pints. And this question is not making use of gallons, but as a side note, so I'll write this over here. So as a side note, we should know that one gallon equals four quarts. Okay, so in this question here, we're not using this, but this may pop up on, you know, a fifth grade test where you have to make use of this information. But for now, we're just going to be using these two. So what I want to do is I'm going to turn everything into cups. And from here, I'm going to turn the total amount of cups into quarts because if I turn everything into quarts, I'm going to have a lot of decimals and fractions. And I want to just work with whole numbers because I believe that'll make everything much easier. So, we already have six cups of flour, one cup of sugar, but now we have pints and quarts over here. So, these need to be converted. So, let's look at the two pints of milk. So two pints I could say is equal to and I'm going to replace one pint with two cups. So that tells us we're going to have two times and remember a pint is equal to two cups. So that's 2 * 2 cups which gives us four cups. Okay. And this I could even just draw out. I could say one pint. So let's say that this is a pint over here. One pint has two cups. And if I draw out another pint, so here's my second pint. I have another two cups. And in total, I have two plus two or four cups. And then now for quarts, I have to turn quarts into pints. And then pints I have to turn into cups. So this one is a little bit more involved. Okay? So we have one quart of oil. And one quart is equal to two pints. And think about what we just said before. Two pints in total gives us four cups. So I could say that one quart is going to give us four cups. And now I just have to find the total of everything. So we're just going to add all this stuff up. So if we look at the measurements here, we have six cups of flour. So first I'll just write down we have six cups of flour. So we have six cups of flour. We have one cup of sugar, but I'll just write cups. So everything says cups. And then we have the two pints was equal to four cups. And then for the last one, one quart was also equal to four cups. So now we just add all this up. So we'll do 6 + 1 + 4 + 4. And I'm just making groups of 10 in my head. When I do this addition, I want to do 6 + 4 first is 10. And then we have 10 + 1 plus 4 is 15. So we have 15 cups. But now we have to go from cups back to quarts. So what we're going to do is we're going to have to see the relationship between quarts and cups. And we have it over here. One quart we could say is equal to four cups. So if I want to go from cups to quarts, what I could do is I could divide the number of cups by four. So I'm going to do 15 / four. And the reason I'm doing this is once again for every quart I have I have four cups. So I'm just going to make groups of four here. So I'm going to divide 15 by four. So we have 15 divided by 4 gives us three. 3 * 4 is 12. And we subtract. We have a remainder of three. So the three remainder goes here. And we divided by four originally. So we have 3 and 3/4s quarts. And that is going to correspond to choice C. Question 27. A group of three friends has half of a pizza to share equally. How much pizza does each person get? So we have a group of three friends and they are sharing the pizza equally. So this tells us we're going to divide. So we're going to take the half of a pizza and we're going to divide that by three. And what I'm going to do is I'm going to rename three. I'm going to rename that three over one. So that way we could use keep change flip. So anytime we divide by a fraction or if we're dividing a fraction by a whole number, we could use the keep change flip technique. So we keep the first part here. We keep 1/2. We change the operation from division to multiplication. And we flip the fraction at the end to 1 over 3 in this case. And now we could do 1* 1 over 2 * 3 on bottom. And this gives us 1 over 6. So our solution here is choice A. But now let's think about the concept behind this question. So we have this pizza and there's only half a pizza left. So let's just say that this pizza over here, like this piece is eaten. It's gone. So the friends, there's three of them are going to share the remaining part equally. So if we break this into three equal pieces, well, that means we're going to be cutting it around here, and we're going to be cutting it around here. So this would be cutting it into three equal pieces. And let me just adjust this one a little bit more so it's a little bit more equal looking. So that's much better. But now imagine the whole pie was there because this represents what each person will get. So let's say one person is going to get this slice. The other friend, the second friend is going to get this one and the third friend is going to get this one. But let's think about what fraction is being modeled here. If I extend this line through, I pretend the pizza is whole again. Well, if I extend it all the way through, you could see that we have six slices and one of the slices here being eaten represents 16th of the original pie. Okay, so that's why choice A is our answer. But once again, we could just use this division of fractions technique over here to get 1 over six. Question 28. The table represents a relationship between X and Y. And we want to know which statement about the relationship between X and Y is true. So if we look at choice A, we have there is an additive pattern because each Y-value increases by 10. So if we look, the Y values are definitely increasing by 10. So, choice A is good. No, choice A is a very dangerous bear trap. Notice the question is saying which statement about the relationship between X and Y is true. And that's not referring to the relationship between X and Y. That's only talking about the pattern in the Y column. Okay, that's not considering the X values. So, that's why A is no good. So, now let's look at the remaining choices. There's a multiplicative pattern because each y-value is eight more than the x value. Well, yes, it's true that 10 is eight more than is eight more than two. But notice 20 is not eight more than four and multiplicative wouldn't mean eight more. It would be eight times. But even 2 * 8 would give us 16, not 10. So b is out. This answer choice is no good. If we look at the remaining choices, there's a multiplicative pattern because each y-value is five times the x value. This one is going to be it because if we look, let's say for each pair here, if I take the two starting here and I do 2 * 5, that's going to give me 10, which gives me the y value right next to it. Okay? So, we see we get y= 10, which is right here. And now, let's see if that pattern holds for the others. So, if I take the four and I multiply that one by five, that gives us 20. and that matches the y-value that's written right next to it. Okay, so once again, it's about the relationship between x and y. So this one is holding strong and I'll just do it for the others. So let's say we take six. If we do 6 * 5, that gives us 30 and that gives us the y value right next to this x value. And then if we do the last one here, we have 8. And if we do 8 * 5, 8 * 5 is 40. So the last entry also checks out. So this one is definitely good. Now let's see why choice D is no good. There's a multiplicative pattern because each X value is 1/2 of the Y value plus one. But that one doesn't even work. Let's say I take the first entry here. If we look at let's say these two and we throw this rule through, we look each x value is half of the y-value plus one. If I take half of the y-value, so let's say this y-value 10, and I take half of that, I get five. And then if I go plus one, that's not going to give me two. Okay? Even if I went minus one, that's still not going to give me two. So choice D is is off. Choice C is definitely correct. Question 29. What is the value of the expression shown below? And for this one, we have parentheses, we have subtraction, we have multiplication. So we're going to use order of operations. We're going to use PEMDAZ. and PEMDAZ. I like to remember it as please excuse my dear aunt Sally. But what it really stands for is do parentheses, exponents, multiplication, division, addition, subtraction. Just know for these questions though that multiplication and division and then addition, subtraction, these have the same rank. So if those are the only operations left, we could just go from left to right. So the first one I see though is the parenthesis. So the first thing I'm drawn to is we do have these brackets which also act like parenthesis but within the brackets we have a set of parentheses. So that's what I want to do first. So I'm going to write everything else we're going to have we'll have 2 * 9 minus and then in the parentheses here we are doing 4 1/2 - 2 1/2. And for this this is a lot like doing 45 minus 25 which is going to give us 20. But now I just throw in the decimals like this and that's going to give us 2.0 zero or it's going to give us two. So inside the parentheses here, I'm going to have 4 and 1/2 - 2 and 1/2 is 2. And now I'm multiplying by two on the outside. Okay, so I could just say times two on the outside. And then I'll throw in the the bracket at the end. Okay, so just know these symbols here are interchangeable. If this didn't make sense what I just did here, if I had something, you know, just a side note, not part of the question, but if I had something like, let's say, 5* 3, I could say this is equal to 15. I could also say parentheses 5, parenthesis 3 is equal to 15. So these mean the same thing. And that's the form I'm going with here. And now we'll just close this out. So let's see the next stage here. We are still inside the parenthesis. So I'm still focusing on this piece here. And for this now we're just going to simplify the next thing in the order of operations. See how we have subtraction and we have multiplication. But multiplication comes before subtraction. So we're going to do the multiplication here first. So we're going to do next we're going to have 2 * 9 minus and we do 2 * 2 is 4. And now we close this. And we're once again using order of operations, but we have multiplication, subtraction, but we have these brackets which act as parentheses. So we're going to do 9 - 4 first. And this is going to give us next we'll have 2 times. And now I'll switch the brackets to parenthesis. But 9 - 4 gives us 5. And now I just do 2 * 5 is equal to 10. And this is going to be our solution to question 29. Question 30. The diagram below shows how some shapes are related. And we want to know which shape could go in the empty box to complete the diagram. So we have a quadrilateral here at the top and then it's branching out like this. So everything in these boxes should be quadrilaterals. So a trapezoid is a quadrilateral because it has four sides. Okay? So just know quad means four and lateral means sides. So that means we're looking for shapes that have four sides. And a trapezoid is definitely a shape that fits that description. Okay? Okay, a trapezoid has one pair of parallel sides like this and it is a quadrilateral. A square counts as a quadrilateral because a square has four sides as well. Okay, so this is the trapezoid. This is just an example over here. And if I draw a square off to the side, a square has four sides. All four sides are equal in length and a square has four right angles as well. Okay, so a square is a quadrilateral. So when we look through the answer choices, we have to find a four-sided figure. triangle is no good because a triangle has three sides. A rhombus is the answer because a rhombus, some people like to think of it as a slanted square, but a rhombus is a parallelogram where all four sides are equal. Okay, so here I'll just draw that a little bit neater, but a rhombus is once again it is a parallelogram where all four sides are equal. So a rhombus is definitely a quadrilateral. So this is going to be our answer. And now let's see why are the other answer choices no good. So a pentagon is no good because a pentagon has five sides. So if I draw one out like this. So here's a pentagon. It has five sides. And a hexagon is no good because a hexagon has six sides. So if I draw one out, a hexagon would look something like this. Six sides. It is not in the quadrilateral family. So rhombus choice B is our answer. Question 31. A school playground is shaped like a rectangle that is 85 ft long and 60 ft wide. And we want to know what is the perimeter of the playground. So we could draw this one out here. So I'm going to draw out a rectangle. And for this we have the longer side is 85 ft and the other side is 60 ft. So it's 85 ft long and 60 ft wide. But what we should know about rectangles is that rectangles have four right angles and they also have opposite sides equal. So if this side over here is 85 ft, then this one over here is also 85 ft. And if this side is 60 ft, then this side over here is also 60 ft. And our goal is to find the perimeter. So when we find the perimeter here, the perimeter is equal to the sum of all the side lengths going all the way around. So when we find it, we could just go piece by piece. So I'm going to add these one at a time. So I'm going to go 85 first. So I'm going to write that first and then plus. And I'm just going to trace the path as I write it. So we have 85 plus next I'll write 60. And then we have plus after this I'm going to go 85. So we have plus 85. And then plus one more. We have four sides here. We have to add up that fourth side which is 60 ft. Okay. So now we just add all the stuff up. I'll write it vertically over here. So I have 85. I'll write the two 85s next to each other. And then the two 60s. And I have 5 + 5 is 10. So I have a zero. I carry the one. And then I have I'll do 8 + 8 is 16. And 16 + 1 is 17. Then I have 17 + 6 is 23. 23 + 6 is 29. Okay. So this is going to work out to in total 290 ft. Okay. So 290 ft. Choice A is our answer. Now, one thing I want to point out about the wrong answers. If we were to multiply these two sides here, we would be finding area. Area is equal to length time width. And if I worked out and I did 85 * 60. So, this one I would do 0 * 5 is 0. 0 * 8 is 0. I put a placeholder. And then I do 6 * 5 is 30. Carry the 3. 6 * 8 is 48. 48 + 3 is going to give us 51. And this gives us 5,100. But one thing to be mindful of is choice D is a very dangerous bear trap. This is talking about once again area. And the giveaway is the units. Because if I do 85 ft* 85 ft, that gives us 85 square ft. And the idea behind area versus perimeter area would talk about if I were to cut this up into let's say uh I make 85 lines like this and they're all going one foot this way. So I just keep doing that until I get to the end and then I make 60 lines going this way like this. What that answer down here represents the 5100 and I'll just say dot dot dot the whole way down is that represents the number of little squares that would be cut into this big rectangle. Okay, but that's if we're finding area, not perimeter. Perimeter is a distance concept, so it should just be feet. Okay, like this. So, our answer is going to be choice A. Question 32. The figure below is made up of eight small cubes. And we want to know which best shows the side view of the figure. So, if we look over here, the side view is this part over here. And I will highlight it. So, that's this piece. So, pretend that I rotate this shape in such a way that we're looking directly at this. Notice we would be looking at just these two squares like this. Okay. So this would be the side highlighted. If I rotate the shape so it's facing directly at the screen like this. And then this would correspond to this shape over here. So we could say that choice C is our answer. Now why are the other answer choices no good? Well this one, choice A would be a front view. Okay. If we were looking for the front view of the figure, that would be the the uh eight squares like this, the 4x two. So that's why this one's no good. The top view would give us choice B. So that's why B is no good. And then D is just wrong for, you know, a lot of reasons. There's five squares, but we have eight small cubes, so that means, you know, we're going across four over one up two. This one doesn't match the figure in any way. So D is out. It's definitely going to be choice C. So now we're moving on to the part two short answer questions and for question 33 we have consider the ordered pairs below and we have these three points and we want to know what patterns do you see between the numbers in each pair explain your thinking. So for this one let's first think our way through this. So if we look at the first point we have 4a 8 and let's think about how do we get from 4 to 8. Well we could do 4 * 2 is equal to 8. We could also do 4 + 4 equals 8. But how do we know which one to go with? Well, whatever one we go with has to work for all of the pairs. So, it has to work for 48, 59, and 610. So, let's say I were thinking here like, all right, we're going to go with four * 2. The pattern is time 2. The problem with that is when we go to use the x 2 pattern for the next one, if we do for 59, if we do 5 * 2, we get 10, which is not equal to 9. So the x two pattern is out. So let's try the plus4 pattern. So if we do 5 + 4, notice that does give us 9. So the plus4 pattern works for the first one, the second one, and let's see if it works for the third one. Okay. So if we were to use that pattern for the 6 comma 10, we do 6 + 4 that gives us 10. Okay. So this is the thought process. But now we have to explain our thinking. So what patterns do you see between the numbers? Say that there is an additive pattern. We have to add four to the first number to get the second number. Now one last thing before we move on. I just want to point out that when you have to write a sentence, when you have to explain, it really helps to gather math evidence to do some investigating because I think of the phrase a picture is worth a thousand words, but so is all of your math work. If you just do the research here or you look for the pattern and you gain or you gather some evidence here and you're like, "Ooh, the pattern is plus4." Then all you have to do is just write about the evidence that you gathered. Okay? So I like to do this part first and that helps me build nice sentences here and this would get us full credit for question 33. Question 34. Find the prime factorization of 72 and then use your prime factorization to help you find all the factors of 72. So we start out with 72. And what I'm going to do is I'm going to make a factor tree. So I'm going to write down let's say 8 * 9 is 72. So I write these two factors. And when we're finding the prime factorization, what we want to do is we want to keep breaking down the factors until we get all primes. So next I'll write 8 is equal to 2 * 4. And two is a prime number. So I could just circle it like this. And then four I'll break down to 2 * 2. And I could circle these twos because 2 is a prime number. And that takes care of the 8. So now I'll break down 9. 9 is equal to 3 * 3. 3 is prime. So I could circle it. And now this is going to help us find the prime factorization. So the prime factorization we could say that 72 is equal to 2 * 2 * 2. And then times we have 3 * 3. So we break this down. And now I'm going to write it in exponent form. So 2 * 2 * 2 is 2 to the 3 power. And then we have times. 3 * 3 gives us 3 to the 2 power. So this is the prime factorization of 72. And from here, this is going to help us find all the factors of 72. Now, one thing I want to show off to the side sort of as a bonus is a trick. When you want to count how many factors you have in total, what you could do is you could focus on the powers in your prime factorization and you add one to each power. So notice I have a three and a two. So, I'm going to do 3 + 1 and then I'm going to multiply that by 2 + 1. And what that's going to give me, I have 3 + 1 is 4 and 2 + 1 is 3. And this tells us that there are 12 factors of 72. Okay, so 72 has 12 factors. So that way when we start looking for the factors, we know how many we have to get to. We have to get to 12 factors in order to be done. So let me start writing these out. Let's say I start with 1 * 72 and then 72 is even. So I could say 2 * 36 also works. And now another divisibility trick here is once again we could add the digits 7 + 2 is 9 and 9 is is divisible by 3. But if we use the prime factorization to help us notice that there's already a three in the prime factorization. So I know 72 is going to be divisible by three. So, what I could do is I could just sort of group off over here everything but one of the threes. So, if we do 2 * 2 * 2 * 3 and then we just do 3 at the end, this will tell us what 3 needs to multiply by to get to 72. So, this would give us we would have 8 * 3 * 3 because when I do 2 * 2 * 2, I get 8 and then time 3. So in the parentheses here we would have 24 * 3 is equal to 72. Okay. So this is how we could use the prime factorization to help us. So we have 3 * 24 gives us 72. If I go to four next four I could just do the division here or I could use the prime factorization again to help us. So let's say I use the prime factorization to help. So for that one, what I could do is I could write 2 * 2. That's equal to four. And then the rest would be I would have 2 * 3 * 3. And this would give us 4 * 2 * 3 is 6. And 6 * 3 is 18. So 4 * 18 would be another pair of factors here. And I might think like, okay, am I done yet? Well, notice I only have 1 2 3 4 5 6 7 8. I need 12 factors in total. So I have to keep going here. So next I have 6 * 12 is 72 and I also have 8 * 9. So we have the prime factorization over here. But now I'm going to say the factors of 72 are and I'll just write out all the numbers in this list and I will throw them inside curly brackets like this. Question 35. We're going to plot and label point A at 47 and point B at 92 on the xycoordinate grid below. And we have to be sure to plot each point in the correct spot and label each point clearly with its letter. So here, just know when you're plotting a point x, y, what you should be thinking of is that you go left or right first and then up or down. Okay? So the x tells you how far to go left or right. But for this question here, we're dealing with all positive numbers. So, we're only going to the right. And the yvalue tells you how much to go up or down. So, let's apply that here. If we look, notice that for point A, we have an x value of four and a y value of seven. So, we're going to go first to the right. And this is always starting at the origin. So, we go here from the point 0 0. We go to the right 1 2 3 4. And then we go up seven. So we got 1 2 3 4 5 6 7 and then we plot this point and this is going to be the location of A. And notice that it lines up with seven on the y- axis. So this one is definitely good and we're lined up with four on the x- axis. So now next we'll do point b and this is located at 92. So I could count one at a time or I could just go hyper speed here right from the origin to x= 9 and then I have to go over nine. So I'm going nine to the right and then we go up two. So we're going over nine and then up one, two. And this would be the location of point B. Question 36. The perimeter of a square is one/10enth of a unit. What is the length in units of each side of the square? And we have to show our work. So for this question, we have to think really carefully about what's happening here. So we have some square. So let me just draw this a little bit neater. We have a square. And with this square, we know the perimeter. And the perimeter is the distance going all the way around. So the perimeter of the square, we're going all the way around like this. Now, some important ideas about a square is that the side length of a square is the same all the way around. And I could say that the perimeter is equal to 4 * s where s is the length of one side. So this is the length of one side of the square. So if we think about this very carefully like let's say we think about an easy square first where fractions are not involved. Let's say I had a square where each side was let's say equal to three. Okay so three units. Then the perimeter I would just do 4 * 3 is equal to 12. But now let's think about what's the relationship between the side length. So we have the side length and we have the perimeter. Well, the side length is three and the perimeter is 12. So to go from the side length to the perimeter, that's going to be a time four pattern. Okay? So when I'm going forward like this, I'm going x times 4. When I go backwards from the perimeter to the side length, I divide by four to go backwards. Okay? So that's exactly what we want to do here. If we know the perimeter is equal to one/10enth of a unit, we're going to do one/10enth / 4. Okay? Okay, so that's what's going to happen here. And now we're going to use that technique, keep, change, flip. And let's think of four as 4 over 1. Now we do 1/10th times because we're changing the operation from division to multiplication. And we flip the fraction at the end to 1 4. So we're going to get 1 * 1 / 10 * 4. And that's going to give us 1 over 40. So now we'll just write the answer out. We'll say that the side length is equal to 1 over 40 and we'll just say it's 140th of a unit. Okay, so this is the length of the side of the square. Question 37. In the number 523 and 347,000, how does the value of the digit 3 to the left of the decimal point compare to the value of the digit three to the right of the decimal point? And we have to explain our answer. So let's look at both of those threes. We have a three over here and this three is in the onees place. And if we look at the other three over here to the right of the decimal point, this one is in the 10th's place. So now what I want to consider is let's say we look at 3 * 10. 3 * 10 is equal to 30. If I do 30 * 10, that gives us 300. And I'll do one more this way. We have 300 * 10 next would give us 3,000. So every time we multiply by 10, that moves the three along the place value line like this. Okay? It moves it from right to left. And if I go backwards this way, like let's say I look at 0.3 * 10, that would bring us to three. And if I do one more, let's say 0.03 * 10, that would bring us to 0.3. Okay? But once again, when we multiply by 10, that decimal moves one space to the right. And we could see the three is moving along the place value line going from right to left. So now the line I want to focus on here is this one 0.3 * 10 is equal to 3. Because when we look 3/10th what we're going to compare here we are comparing or we are going to compare three 1's. So we're comparing just three and we're comparing 3/10th. And just know 3/10 is the same thing as 0.3. So when we look at the line that I boxed out that I boxed off over here, we have 0.3 * 10 is equal to 3. So how do these digits compare? The three to the left is 10 times more than the three to the right. Okay? Because if we do 10 * 3/10, we get three. Okay? So we could say when we write our sentence out that the three on the left is 10 times more than the three on the right. So, now we're moving on to part three, the extended response. And for question 38, we have 16 students in the bear club are planning a trip to visit a wildlife sanctuary to learn about black bears. The ticket price is $40 per student, and the cost of transportation and food for the whole group is $800. To raise money, the students design and sell bear themed t-shirts, making an $18 profit on each shirt sold. If each student sells the same number of shirts, how many shirts must each student sell so the group has enough money to cover the total cost of the trip? And we have to show our work. So, we could start by gathering data. We're told that the ticket price is $40 per student and the cost of transportation and food for the whole group is $800. So, notice they also said that there's 16 students in the bear club. So, we can start finding out the cost of the trip. Okay, so the cost is going to be equal to we have 16 * 40 because once again the ticket price is $40 per student. So we'll go ahead and do that. We have 16 and we multiply that by 40 and we're going to have 0 * 6 is 0. 0 * 1 is 0. And then we put a placeholder here and multiply 4 * 6 is 24. Carry the two and then 4 * 1 is 4 + 2 is 6. So we work this out, we add this up, and we're going to get 640. But this is the cost of just tickets. Okay, so we'll say the cost of tickets. We also have to consider here that the cost of transportation and food for the whole group is $800. Okay, so let me just move this over here so it's lined up nice. This is the cost of tickets that we found over here. So now we could just say that the total cost, so the total cost is equal to we're going to take the 640 and we're going to add the $800. So the total cost for the trip is going to be we have 4 + 0 is 4. 6 + 8 is going to give us 14. So the total cost of the trip is $1,440. And now let's think about this. We have that the students design and sell bare themed t-shirts. They're going to be making an $18 profit on each shirt. And we're told that each student sells the same number of shirts. Okay, so what that tells us, we're going to divide the 16 students into the total cost here. So this will tell us how much money each student has to raise. Okay, so each student, I'll write that over here. Each student has to raise and we'll go ahead and do the division. Each student has to raise,440 divided by 16. Okay? Because each one once again is going to sell the same number of shirts. So let's go ahead and do this division. We could say that 16 goes into 144. And for this one, what I'm thinking of, the mental math that I'm doing here to make this division easier is I think 16* 10 is 160. I just throw on an extra zero. So if I do 16 * 9, I just subtract 16 from 160. And if I go - 16 like this, I'll go ahead and work this out. So we borrow 10 - 6 is 4. 5 - 1 is 4. And we get 144 exactly. Okay. So I'm going to put a nine here. And when we multiply, we're subtracting 144. We bring down the zero. We could bring down this last zero. And then 16 goes into 0 0 times. And then 0 * 16 is 0. And when we subtract, we have no remainder here. So each student has to raise $90. Okay? So this is representing once again how much each student has to raise. But if we think about this, each student is going to make for each t-shirt sold, there's going to be an $18 profit on each shirt sold. So then what we do is we have to look for how many shirts need to be sold by each student to raise $90. So each student has to raise this much. So we're going to take the 90 here. So we're taking this 90 and we're going to divide this by 18. Okay? So we divide by 18 because once again each t-shirt sold makes an $18 profit and each student has to make $90. So now we say 18 goes into 90. This one could be a little bit tricky to think of, but we'll go ahead and think about this. Let's say over here. So if I do 18 * 10, that's equal to 180. If I go halfway, 18 * 5, then I just divide this by two and I get exactly 90. Okay, so 18 * 5 is 90. Divides evenly. There's no remainder. So what this tells us, how many shirts must each student sell? So we'll just say that at the end here, each student must sell five t-shirts. Question 39. Jordan's doctor recommends that Jordan drink 64 flu ounces of water each day. Jordan has a water bottle that holds 1 and 1/4 pints of water when filled. Today, he filled the bottle three times and drank all of the water each time. Jordan claims that he drank the full amount of water recommended by his doctor. Explain why Jordan's claim is not true. So, for this one, let's look at the target amount. Jordan needs to drink 64 flu ounces of water each day. And if we think about his water bottle, his water bottle holds 1 and 1/4 pints. Okay, so here's his water bottle and it holds this much water. And we're told next that he filled the bottle three times and drank all of the water each time. So right away, what I'm doing here is I'm finding out how many cups of water did Jordan drink because the conversion we're going to use is one pint is equal to two cups. And then we could go from cups to ounces. One cup is equal to 8 flu ounces. But I'll just say 8 ounces like this. So now we could just start the conversion process. So we're going to say 1 and 1/4 pints is equal to 2 * 1 and 1/4 cups. Okay. So when we want to go from pints to cups, we just double the number in front. And off to the side, what I'll do is I'll turn 1 and 1/4 into an improper fraction. So I'm going to do 4 * 1 plus the numerator on top. So plus one over the original denominator is 4. And this gives us 5 over 4. So what I'm going to do over here is we're going to do 2 * 5 over 4 cups. And now this I'll say is equal to I'll call this 2 over 1. 2 * 5 is 10 over 1 * 4 is 4. So we have 10 over four cups like this. And there's a few ways we could go forward from here. One thing we could do is reduce. Divide the top and bottom by two. And this is going to give us 10 / two is 5 over 4 / two is two. And we have cups. Okay. So this is telling us how many cups of water are in Jordan's water bottle when full. So then from here, the thing to think about is how many ounces is here. So one cup equals 8 ounces. So if he has 5 over two cups, we could multiply by eight to get the number of ounces because remember for every one cup, we have eight ounces. So when we're in cups and we want to go to ounces, we could just multiply the number by 8 and change the unit to ounces. Okay, so let's go ahead and do that over here. We're going to say that Jordan has once again 5 over 2 * 8 ounces. So this is the amount in one water bottle. So now this gives us if we just call this 8 over 1 5 * 8 is 40 over 2 * 1 is 2 and we have ounces and now we just reduce this this reduces to 40 / 2 is 20. Okay so this is the amount in one full water bottle. Okay so this is the amount in Jordan's water bottle. And now let's think about how to work with this. So from here, what we're going to do with this information is just go back to the question. And for word problems like this, sometimes you have to reread it a few times. So let's read this part again. We have Jordan filled the bottle three times and drank all of the water each time. So we could just say here that Jordan drank. So we'll say Jordan drank and since he filled the bottle three times, he drank three times 20 ounces. Okay? So this is how much water Jordan drank. So, he drank 60 ounces. Okay. And what this tells us, if we look at the original goal, Jordan's doctor recommends that he drink 64 flu ounces of water. So, explain why Jordan's claim is not true. So, we could say that Jordan's claim is not true because Jordan drank 60 ounces, which is less than 64 ounces. Question 40, our last question. The figure below shows two rectangular prisms stacked to form one solid figure. What is the volume in cubic centimeters of the solid figure? And we have to show our work and explain how we found our answer. So the formula we need to know here is volume equals length* width* height. This is the formula for the volume of a rectangular prism. And what we're going to do is we're going to look at these shapes one at a time. So, I'm going to look at this one over here in blue. So, we're going to say here the total volume. So, I'm just going to write that over here. The total volume equals and we're going to add the volumes of both shapes together. So, we'll start with the blue one here. Notice that we have a length of 2.5 cm. So, we have 2.5. And just know that we can just write the units at the end. We're going to say cubic centimeters at the very very end. I don't have to write it all throughout the question. So we have 2.5 is the length. The width over here we could say is 10 cm. So we're going to multiply by 10. And then we're going to multiply by the height of this blue rectangular prism which is labeled 4 cm. Okay? So if this piece over here is 4 centimeters, then this one over here we could also say is 4 cm. So either way we're multiplying by four. And then we have plus we're going to add the volume of the other rectangular prism which I'll shade over here in purple. So we're going to use that formula again. Length time width time height. So we'll say in this case here that the length is equal to 2 cm. We have two and then we're going to multiply by the width which is 10. And then the height, the vertical distance here is equal to 3 cm. And now we just have to work this out and add everything up. Just remember order of operations. We're going to do the multiplication first before we add these values together. So, we're going to say the total volume equals, and now we could just multiply. So, when we work this out, I'll just do this off to the side. 2.5 * 10 is equal to 25. Remember, when you multiply by 10, the decimal just moves one space to the right. And then if we do 25 * 4, this is equal to 100. Okay? So, the blue stuff multiplies to 100. And then we have plus, we're going to multiply the purple stuff. And if we do 2 * 10 first, so I'll do that over here. 2 * 10 is equal to 20. And then I take the product of these two and multiply that by 3. So 20 * 3 is equal to 60. So I'll just do plus 60 over here. So the total volume is equal to 160 and the units here are cubic centimeters. Okay. So now I'll write the units in at the end. So we have 160 cubic cm. And now what we're going to do from here, we just have to explain how we found our answer. Well, let's think about what we did. We found the volume of the first rectangular prism. We found the volume of the second rectangular prism. And we added those two volumes together.