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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.