Proportion word problem: cookies

a recipe for a recipe for oatmeal cookies calls for two cups of flour for every three cups of oatmeal how much flour is needed for a big batch of cookie that uses 9 cups of oatmeal so let's think about what they're saying they're saying two cups of flour so two cups of flour two cups of flour flour for every three cups of oatmeal for every three cups of oatmeal for every three cups of oatmeal so for every three cups of oatmeal for every three 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 so now we're going to a situation now we're going to go to the situation where we are using nine cups of oatmeal let me write it this way 9 cups of oatmeal 9 and I'll show you a couple of different ways to think about it and whatever works for you that's that that works so first of all the 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 they 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 we're multiplying by 3 3 cups of oatmeal 2 9 cups of oatmeal we're using three times the oatmeal well if we want to use flour in the same proportion we have to use three times the flour so then we're also gonna have to multiply the flour times three we're going to multiply the flour times three so we're going to have to use 6 cups of flour we're going to have to use six cups of ignore that question mark 6 cups of flour another way you could think about it that answers your 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 you could say 2 cups 2 cups of flour over 3 cups of oatmeal 3 cups of oatmeal is equal to question mark and I'll say instead of writing question mark I'll put a variable in there I'll say is equal to a question actually let me put a question mark there just so you really understand is equal to a question mark in a box number cups of flour cups of flour cups of flour over 9 cups of oatmeal 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 the oatmeal then we're going to have to triple the flour to Matt 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 represent will multiply the diagonals so that when you multiply it we use cross multiplication you'll say that 2 times 9 so 2 times 9 must be equal to question mark times 3 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 18 is equal to whatever our question mark was whatever our question mark was times 3 so the number of Klout cups of flours we need to use times three needs to be equal to 18 what times 3 is equal to 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 you know why does that work if I have something set up like this proportion set up why does it work that if I 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 is X just to simplify the writing a little bit so we have 2 over three is equal to instead of that question mark I'll write x over nine 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 the 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'd have to do the same thing to it and 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 nine the nines are going to cancel out so let's multiply the right by nine but of course if we multiply the right by nine we have to still multiply the left by nine otherwise they still wouldn't be equal if they were equal before being multiplied by nine for them to still be equal you have to multiply nine times both sides on the right hand side the 9s cancel out so you're just left with an X on the left hand side you have nine times two-thirds or nine over one times two-thirds or this is equal to eighteen over three and we know that 18 over three is the same thing as six so these are all legitimate ways to do it I want 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 three that increase the cups of flour by a factor of three