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## Algebra basics

### Course: Algebra basics > Unit 3

Lesson 7: Writing & solving proportions# Worked example: Solving proportions

Learn the reasoning behind solving proportions. We'll put some algebra to work to get our answers, too. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- In the first example on how to find the proportion, Sal said to multiply 8x10/8 to get 10. He said he made the calculation on his head, but I wonder what steps should I follow to get that answer? The 10/8 looks obvious after he gave it away but if he hadn't I don't think I could have find it on my own. At least I don't know how to do it at the moment.(76 votes)
- Well, let's see...we're basically asking 8 times what = 10, right? So, in algebra terms, 8 x n = 10 or 8n = 10. Divide both sides by 8 to get the n by itself and you get n = 10/8. Does that make sense?(68 votes)

- Is there any easier way to do it? Like a way without using common core? Because this is way to confusing!(33 votes)
- In that first example that Sal gave you, try checking for fractions to simplify first. You can see that the first fraction
`8/36`

simplifies to`2/9`

.`2 10`

-- = --

9 n

Now, we need to ask ourselves this question: "2 times what equals 10?" And to answer that question, 2 times**5**equals 10. So, we now know to multiply 5 on the denominator of the first fraction to find**n**. 9 times 5 equals 45, so`n = 45`

.

There is another way: it's to cross-multiply and then solve the equation. But you won't learn about solving equations until much later in Pre-Algebra: https://www.khanacademy.org/math/pre-algebra/pre-algebra-equations-expressions/pre-algebra-intro-equations/v/variables-expressions-and-equations

Let's use Sal's example again:`2 10`

-- = --

9 n

Try drawing an 'X' with your fingers on the proportion above. That's how we'll cross-multiply it. The equation will ultimately look like this:`2 ⋅ n = 10 ⋅ 9`

. Simplify the equation and you get:`2n = 90`

. To solve it, you just divide 2 on both sides:`2n = 90`

÷2 = ÷2

n = 45

I hope this helped!(74 votes)

- Did anyone else hear breathing in between while he was talking? :((19 votes)
- i did :( i was genuinely worried but then i realised its probably because he's the one narrating almost every khan academy video so i do think its a case of tired-of-talking(13 votes)

- i don't even know what he said during those whole 7 minutes(19 votes)
- same :/ normally all his videos help me but this one kinda was a lil you know confusing. to help tho i'd just use the community's help and look at the comments :)(12 votes)

- Is there anytime the answer is 0?(14 votes)
- maby if you had 0/0 = y/0(2 votes)

- I was doing the "Solving Proportions" and the question was like "12/7 = k/8", and whatever I did I kept getting 13.7142857143, but it was wrong. I don't understand.(8 votes)
- Your answer is technically correct, but I think they want you to leave it as a fraction (96/7) or a mixed number (13 5/7)(10 votes)

- Err... Did Sal show 5 different ways of solving one problem?(7 votes)
- Yes he told many ways and some of them were confusing. The simplest way I think is Cross multiplication. Here's how to do it.

Lets take a number 3/12=4/X

1. First cross multiply the digits i.e, multiply 12 x 4 and 3 x X

2. 12 x 4 = 48 and 3 x X = 3X

3.Next, According to question 3/12=4/X so, 48 will be equal to 3X

4.48=3X

5. Now I need to find the value for X So in order to isolate X I need to divide 3X by 3 (In order to find the value for X)

6. 48= [4X/3] If I divide one side by 3 then I must also divide the other side by 3 (That is the rule of the Equation).

So, we get [48/3]=[3X/3]

7. Next lets solve it 48/3 is 16 and 3X/3 is X

8. Now we get 16=X

9. Finally, the X value is 16

10. So X=16

Hope this helps. Feel free to Comment me if you have any questions :)(6 votes)

- The video is a bit confusing, and I'm struggling to transfer this to solving the questions for "Solving Proportions". For example in the question:

4/z = 12/5

I understand that you begin by multiplying by z.

z * 4/z = 12/5*z

--> 4 = 12/5*z

After this, the solution set asks you to multiply both sides by 5/12, the opposite fraction of the right side. Why is that?

And how does multiplying a fraction with its opposite give you one?(6 votes)- This concept is based upon the Inverse Property of Multiplication that says:

Any number multiplied by its reciprocal = 1

For example: 12/5 * 5/12 = 60/60 = 1

If you find it easier, you can do cross multiplication. This is where you multiply along each diagonal of the proportion.

4/z = 12/5

12(z) = 4(5)

12z = 20

Then, divide by 12: z = 20/12 = 5/3

Hope this helps.(10 votes)

- i am pretty confused on how to solve problems like 7/3 = 4/t

PLEASE HELP!(7 votes)- All you have to do is cross multiply.

7/3 * 4/t = 12/7t

12/7=1 1/3(3 votes)

- I'm going to ask for clarification of my understanding of Sal's method at0:45. I believe I understand how it works, but it actually greatly confused me, so maybe someone else reading this can get something from it.

What confused me is that while it makes sense that 8 * 10 / 8 = 10 because that's how the reciprocal works, I didn't get why 36 * 8 / 10 gave the correct answer.

Thinking over it, I'm pretty sure this only works because if 10/8 raises 8 to 10, it follows that multiplying the denominator by that same fraction would give you a proportional increase that is the same as 8 -> 10. IE, you're increasing both numbers, 8 and 36, by the same multiplier. It would be the same as multiplying both numbers by 2, or 3. You'd still have a proportional fraction after multiplying, as long as both numbers are multiplied by the same value.

And obviously, choosing the multiplier to use, it makes sense to go with the multiplier which raises 8 to 10, because raising 36 by that multiplier reveals what number n must be. This makes sense to me now, but initially it was incredibly confusing (I kept thinking it should have been 36 * n / 36).

GL to anyone else as confused by this as I was/am.(9 votes)

## 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.