Writing & solving proportions
A recipe for oatmeal cookies calls for 2 cups of flour for every 3 cups of oatmeal. How much flour is needed for a big batch of cookies that uses 9 cups of oatmeal? So let's think about what they're saying. They're saying 2 cups of flour. So 2 cups of flour for every 3 cups of oatmeal. And so they're saying, how much flour is needed for a big batch of cookies that uses 9 cups of oatmeal? Now we're going to go to a situation where we are using 9 cups of oatmeal. Let me write it this way-- 9 cups of oatmeal. And I'll show you a couple of different ways to think about it. And whatever works for you, that works. So one way to think about it, so we're wondering. We're going to say, look, we know if we have 3 cups of oatmeal, we should use 2 cups of flour. But what we don't know is if we have 9 cups of oatmeal, how many cups of flour do we use? That's what they're asking us. But if we're going from 3 cups of oatmeal to 9 cups of oatmeal, how much more oatmeal are we using? Well, we're using three times more oatmeal, Right? We're multiplying by 3. 3 cups of oatmeal and 9 cups of oatmeal, we're using 3 times the oatmeal. Well, if we want to use flour in the same proportion, we have to use 3 times the flour. So then we're also going to multiply the flour times 3. We're going to multiply the flour times 3, so we're going to have to use 6 cups of flour. Ignore that question mark. And that answers the question. That's how much flour we need for a big batch of cookies that uses 9 cups of oatmeal. The other thing is you could set up a proportion. You could say 2 cups of flour over 3 cups of oatmeal is equal to question mark. And instead of writing question mark, I'll put a variable in there. Actually, let me put a question mark there just so you really understand it is equal to a question mark in a box number cups of flour over 9 cups of oatmeal. And so I like this first way we did it because it's really just common sense. If we're tripling the oatmeal, then we're going to have to triple the flour to make the recipe in the same proportion. Another way, once you set up an equation like this, is actually to do a little bit of algebra. Some people might call it cross-multiplying, but that cross-multiplying is still using a little bit of algebra. And I'll show you why they're really the same thing. In cross-multiplication, whenever you have a proportion set up like this, people will multiply the diagonals. So when you use cross-multiplication, you'll say that 2 times 9 must be equal to question mark times 3, must be equal to whatever is in this question mark, the number of cups of flour times 3. Or we get 18 is equal to whatever our question mark was times 3. So the number of cups of flour we need to use times 3 needs to be equal to 18. What times 3 is equal 18? You might be able to do that in your head. That is 6. Or you could divide both sides by 3, and you will get 6. So we get question mark in a box needs to be equal to 6 cups of flour. Same answer we got through kind of common sense. Now, you might be wondering, hey, this cross-multiplying doesn't make any intuitive sense. Why does that work? If I have something set up like this proportion set up, why does it work that if I take the denominator here and multiply it by the numerator there that that needs to be equal to the numerator here times the denominator there? And that comes from straight up algebra. And to do that, I'm just going to rewrite this part as x just to simplify the writing a little bit. So we have 2/3 is equal to-- instead of that question mark, I'll write x over 9. And in algebra, all you're saying is that this quantity over here is equal to this quantity over here. So if you do anything to what's on the left, if you want it to still be equal, if the thing on the right still needs to be equal, you have to do the same thing to it. Now, what we want to do is we want to simplify this so all we have on the right-hand side is an x. So what can we multiply this by so that we're just left with an x? So that we've solved for x? Well, if we multiply this times 9, the 9's are going to cancel out. So let's multiply the right by 9. But of course, if we multiply the right by 9, we have to still multiply the left by 9. Otherwise they still wouldn't be equal. If they were equal before being multiplied by 9, for them to still be equal, you have to multiply 9 times both sides. On the right-hand side, the 9's cancel out, so you're just left with an x. On the left-hand side, you have 9 times 2/3, or 9/1 times 2/3. Or this is equal to 18/3. And we know that 18/3 is the same thing as 6. So these are all legitimate ways to do it. I wanted you to understand that what I'm doing right here is algebra. That's actually the reasoning why cross-multiplication works. But for a really simple problem like this, you could really just use common sense. If you're increasing the cups of oatmeal by a factor of 3, then increase the cups of flour by a factor of 3.