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Studying for a test? Prepare with these 10 lessons on Ratios, rates, proportions.

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# Worked example: Solving proportions

Video transcript

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.