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### Unit 4: Lesson 6

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. •   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?
• Is there any easier way to do it? Like a way without using common core? Because this is way to confusing! •   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 = ÷2n = 45``

I hope this helped!
• Did anyone else hear breathing in between while he was talking? :( • i don't even know what he said during those whole 7 minutes • Is there anytime the answer is 0? • 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. • 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? • 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.
• i am pretty confused on how to solve problems like 7/3 = 4/t • I'm going to ask for clarification of my understanding of Sal's method at . 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. • Err... Did Sal show 5 different ways of solving one problem? • 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 :)