Worked example: Solving proportions

We're asked to solve the proportion. We have 8 36ths is equal to 10 over what. Or the ratio of 8/36 is equal to the ratio of 10 to what. And there's a bunch of different ways to solve this. And I'll explore really all of them, or a good selection of them. So one way to think about it is, these two need to be equivalent ratios, or really, equivalent fractions. So whatever happened to the numerator also has to happen to the denominator. So what do we have to multiply 8 by to get 10? Well you could multiply 8 times 10/8. It will definitely give you 10. So we're multiplying by 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 5/4 to get to 10, from 8 to 10. Well, if we did that to the numerator, in order to have an equivalent fraction, you have to do the same thing to the denominator. You have to multiply it. You have to multiply it times 5/4. And so we could say this n, this thing that we just solved for, this n is going to be equal to 36 times 5 divided by 4. Or you could say that this is going to be equal to 36 times 5 divided by 4. And now, 36 divided by 4, we know what that is. We could divide both the numerator and the denominator by 4. You divide the numerator by 4, you get 9. Divide the denominator by 4 you get 1. You get 45. So that's one way to think about it. 8/36 is equal to 10/45. Another way to think about it is, what do we have to multiply 8 by to get its denominator. How much larger is the denominator 36 than 8? Well let's just divide 36/8. So 36/8 is the same thing as-- so we can simplify, dividing the numerator and the denominator by 4. That's the greatest common divisor. That's the same thing as 9/2. So if you multiply the numerator by 9/2, you get the denominator. So we're multiplying by 9/2 to get the denominator over here. Well, then we have to do the same thing over here. If 36 is 9/2 times 8, let me write this. 8 times 9/2 is equal to 36. Right? That's how we go from the numerator to the denominator. Then to figure out what the denominator here is, if we want the same fraction, we have to multiply by 9/2 again. So then we'll get 10 times 9/2 is going to be equal to n, is going to be equal to this denominator. And so this is the same thing as saying 10 times 9/2. Divide the numerator and the denominator by 2, you get 5/1, which is 45. So 45 is equal to n. Once again, we got the same way, completely legitimate way, to solve it. Now sometimes when you see proportion like this, sometimes people say, oh you can cross-multiply. And you can cross-multiply. And I'll teach you how to do that. And that's sometimes a quick way to do it. But I don't like teaching it the first time you look at proportions, because it's really just something mechanical. You really don't understand what you're doing. And it really comes out of a little bit of algebra. And I'll show you the algebra as well. But if you don't understand it, or if it doesn't make as much sense to you at this point, don't worry too much about it. So we have 8/36 is equal to 10/n. When you cross-multiply, you're saying that the numerator here, times the denominator over here, is going to be equal to, so 8 times n, is going to be equal to the denominator over here, let me just different color, the denominator over here, times the numerator over here. This is what it means to cross-multiply. So this is going to be equal to 36 times 10. Let me do this in a neutral color now. You could say that 8n is equal to 360. And so you're saying 8 times what is equal to 360. Or to figure out what that times what is, you divide 360 divided by 8. So we could divide, and this is a little bit of algebra here, we're dividing both sides of the equation by 8. And we're getting n is equal to 360 divided by 8. You could do that without thinking in strict algebraic terms. You could say 8 times what is 360. Well 8 times 360/8. If I write 8 times question mark is equal to 360, well, question mark could definitely be 360/8. If I multiply these out, this guy and that guy cancel out, and it's definitely 360. And that's why it's 360/8. But now we want to actually divide this to actually get our right answer, or a simplified answer. 8 goes into 360, 8 goes into 36 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 got n is equal to 45. Now the last thing I'm going to show you involves a little bit of algebra. If any of the ways before this worked, that's fine. And where this is sitting in the playlist, you're not expected to know the algebra. But I want to show you the algebra just because I wanted to show you that this cross-multiplication isn't some magic, that using algebra, we will get this exact same thing. But you could stop watching this, if you'll find this part 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 solve for n is maybe multiply both-- this thing on the left is equal to this thing on the right. So we can multiply them both by the same thing. And the equality will still hold. So we could multiply both of them by n. On the right-hand side, the n's cancel out. On the left-hand side, we have 8/36 times n is equal to 10. Now if we want to solve for n, we could literally multiply. If we want just an n here, we would want to multiply this side times 36-- I'll do that in a different color-- we'd want to multiply this side times 36 times 8, because if you multiply these guys out, you get 1. And you just have an n. But since we're doing it to the left-hand side, we also have to do it to the right-hand side, so times 36/8. These guys cancel out and we're left with n is equal to 10 times 36 is 360/8. And notice, we're getting the exact same value that we got with cross-multiplying. And with cross-multiplying, you're actually doing two steps. Actually, you're doing an extra step here. You're multiplying both sides by n, so that you had your 8n. And then you're multiplying both sides by 36, so that you get your 36 on both sides. And you get this value here. But at the end, when you simplify it, you'll get the exact same answer. So those are all different ways to solve this proportion. Probably the most obvious way, or the easiest way to do it in your head, was either just looking at what you have to multiply the numerator by and then doing the same thing to the denominator, or maybe by cross-multiplication.