Video transcript
We're asked to solvethe proportion. We have 8 36ths isequal to 10 over what. Or the ratio of 8/36 is equalto the ratio of 10 to what. And there's a bunch ofdifferent ways to solve this. And I'll explorereally all of them, or a good selection of them. So one way to think aboutit is, these two need to be equivalent ratios, orreally, equivalent fractions. So whatever happenedto the numerator also has to happen tothe denominator. So what do we have tomultiply 8 by to get 10? Well you couldmultiply 8 times 10/8. It will definitely give you 10. So we're multiplyingby 10/8 over here. Or another way to write 10/8,10/8 is the same thing as 5/4. So we're multiplying by 5/4to get to 10, from 8 to 10. Well, if we did thatto the numerator, in order to have anequivalent fraction, you have to do the samething to the denominator. You have to multiply it. You have to multiplyit times 5/4. And so we could say this n, thisthing that we just solved for, this n is going to be equalto 36 times 5 divided by 4. Or you could saythat this is going to be equal to 36times 5 divided by 4. And now, 36 divided by4, we know what that is. We could divide both thenumerator and the denominator by 4. You divide the numeratorby 4, you get 9. Divide the denominatorby 4 you get 1. You get 45. So that's one wayto think about it. 8/36 is equal to 10/45. Another way tothink about it is, what do we have to multiply8 by to get its denominator. How much larger is thedenominator 36 than 8? Well let's just divide 36/8. So 36/8 is the samething as-- so we can simplify, dividing thenumerator and the denominator by 4. That's the greatestcommon divisor. That's the same thing as 9/2. So if you multiplythe numerator by 9/2, you get the denominator. So we're multiplying by 9/2 toget the denominator over here. Well, then we have to dothe same thing over here. If 36 is 9/2 times8, let me write this. 8 times 9/2 is equal to 36. Right? That's how we go from thenumerator to the denominator. Then to figure out whatthe denominator here is, if we wantthe same fraction, we have to multiplyby 9/2 again. So then we'll get 10 times9/2 is going to be equal to n, is going to be equalto this denominator. And so this is the samething as saying 10 times 9/2. Divide the numerator andthe denominator by 2, you get 5/1, which is 45. So 45 is equal to n. Once again, we got the sameway, completely legitimate way, to solve it. Now sometimes when yousee proportion like this, sometimes people say, ohyou can cross-multiply. And you can cross-multiply. And I'll teach youhow to do that. And that's sometimesa quick way to do it. But I don't like teachingit the first time you look at proportions, becauseit's really just something mechanical. You really don't understandwhat you're doing. And it really comes out ofa little bit of algebra. And I'll show youthe algebra as well. But if you don'tunderstand it, or if it doesn't make as much senseto you at this point, don't worry too much about it. So we have 8/36is equal to 10/n. When you cross-multiply,you're saying that the numerator here, timesthe denominator over here, is going to be equalto, so 8 times n, is going to be equal tothe denominator over here, let me just different color,the denominator over here, times the numerator over here. This is what it meansto cross-multiply. So this is going to beequal to 36 times 10. Let me do this in aneutral color now. You could say that8n is equal to 360. And so you're saying 8times what is equal to 360. Or to figure outwhat that times what is, you divide 360 divided by 8. So we could divide, and this isa little bit of algebra here, we're dividing both sidesof the equation by 8. And we're getting n isequal to 360 divided by 8. You could do thatwithout thinking in strict algebraic terms. You could say 8times what is 360. Well 8 times 360/8. If I write 8 times questionmark is equal to 360, well, question mark coulddefinitely be 360/8. If I multiply these out, thisguy and that guy cancel out, and it's definitely 360. And that's why it's 360/8. But now we wantto actually divide this to actually get our rightanswer, or a simplified answer. 8 goes into 360, 8 goes into36 4 times, 4 times 8 is 32. You have a remainder of 4. Bring down the 0. 8 goes into 40 5 times. 5 times 8 is 40. And then you have no remainder. And you're done. Once again, we gotn is equal to 45. Now the last thingI'm going to show you involves a littlebit of algebra. If any of the ways beforethis worked, that's fine. And where this issitting in the playlist, you're not expectedto know the algebra. But I want to showyou the algebra just because I wanted to show youthat this cross-multiplication isn't some magic,that using algebra, we will get thisexact same thing. But you could stopwatching this, if you'll find thispart confusing. So let's rewrite our proportion,8/36 is equal to 10/n. And we want to solve for n. Well the easiest way to solvefor n is maybe multiply both-- this thing on the left is equalto this thing on the right. So we can multiply themboth by the same thing. And the equalitywill still hold. So we could multiplyboth of them by n. On the right-hand side,the n's cancel out. On the left-hand side, we have8/36 times n is equal to 10. Now if we want to solve for n,we could literally multiply. If we want just ann here, we would want to multiplythis side times 36-- I'll do that in adifferent color-- we'd want to multiply thisside times 36 times 8, because if you multiplythese guys out, you get 1. And you just have an n. But since we're doing itto the left-hand side, we also have to do it to theright-hand side, so times 36/8. These guys cancel outand we're left with n is equal to 10times 36 is 360/8. And notice, we're gettingthe exact same value that we got withcross-multiplying. And withcross-multiplying, you're actually doing two steps. Actually, you're doingan extra step here. You're multiplying both sidesby n, so that you had your 8n. And then you're multiplyingboth sides by 36, so that you get your36 on both sides. And you get this value here. But at the end, whenyou simplify it, you'll get theexact same answer. So those are all differentways to solve this proportion. Probably the most obviousway, or the easiest way to do it in your head,was either just looking at what you have tomultiply the numerator by and then doing the samething to the denominator, or maybe bycross-multiplication.
