A **proportion **is a set of 2 fractions that equal each other. This article focuses on how to solve proportions.

### Real World Uses of Proportions

- Modifying a budget for a restaurant chain that is expanding from 3 locations to 20 locations
- Creating a skyscraper from blueprints
- Calculating tips, commissions, and sales tax

### Use Proportions to Modify a Recipe

On Monday, you are cooking enough white rice to serve exactly 3 people.

The recipe calls for 2 cups of water and 1 cup of dry rice. On Sunday, you are going to serve rice to 12 people. How would the recipe change? If you’ve ever made rice, you know that this ratio — 1 part dry rice and 2 parts water — is important. Mess it up, and you’ll be scooping a gummy, hot mess on top of your guests' crawfish étouffée.

Because you are quadrupling your guest list (3 people * 4 = 12 people), you must quadruple your recipe. Cook 8 cups of water and 4 cups of dry rice. These shifts in a recipe demonstrate the heart of proportions: use a ratio to accommodate life's greater and smaller changes.

### Algebra and Proportions 1

Sure, with the right numbers, you can forgo setting up an algebraic equation to determine the amounts of dry rice and water. What happens when the numbers are not so friendly? On Thanksgiving, you'll be serving rice to 25 people. How much water do you need?

Because the ratio of 2 parts water and 1 part dry rice applies to cooking 25 servings of rice, use a proportion to determine the quantity of ingredients.

*Note*: Translating a word problem into an equation is super important. Yes, you can solve an incorrectly set up equation and find an answer. You can also mix rice and water together to create "food" to serve at Thanksgiving. Whether the answer or food is palatable depends on the equation.

Think about what you know:

- 3 servings of cooked rice = 2 cups of water; 1 cup of dry rice

25 servings of cooked rice = ? cups of water; ? cup of dry rice

- 3 servings of cooked rice/25 servings of cooked rice = 2 cups of water/
*x*cups of water

- 3/25 = 2/
*x*

**Cross multiply.** *Hint*: Write these fractions vertically to get the full understanding of cross multiplying. To cross multiply, take the first fraction's numerator and multiply it by the second fraction's denominator. Then take the second fraction's numerator and multiply it by the first fraction's denominator.

3 ** x* = 2 * 25

3*x* = 50

Divide both sides of the equation by 3 to solve for *x*.

3*x*/3 = 50/3*x* = 16.6667 cups of water

Verify that the answer is correct.

Is 3/25 = 2/16.6667?

3/25 = .12

2/16.6667= .12

Whoo hoo! The first proportion is right.

### Algebra and Proportions 2

Remember that *x* will not always be in the numerator. Sometimes the variable is in the denominator, but the process is the same.

Solve the following for *x*.

36/

x= 108/12

Cross multiply:

36 * 12 = 108 * *x*

432 = 108*x*

Divide both sides by 108 to solve for *x*.

432/108 = 108*x*/108

4 = *x*

Check and make sure the answer is right. Remember, a proportion is defined as 2 equivalent fractions:

Does 36/4 = 108/12?

36/4 = 9

108/12 = 9

It’s right!

### Answers and Solutions to Solving Proportions

1. *a*/49 = 4/35

Cross multiply:

a **35* = 4 * 49

35*a* = 196

Divide both sides of the equation by 35 to solve for *a*.

35*a*/35 = 196/35*a* = 5.6

Verify that the answer is correct.

Does 5.6/49 = 4/35?

5.6/49 = .114285714

4/35 = .114285714

2. 6/*x* = 8/32

Cross multiply:

6 * 32 = 8**x*

192 = 8*x*

Divide both sides of the equation by 8 to solve for *x*.

192/8 = 8*x*/8**24 = x**

Verify that the answer is correct.

Does 6/24 = 8/32?

6/24 = ¼

8/32 = ¼

3. 9/3 = 12/b

Cross multiply:

9 *

*b*= 12 * 3

9

*b*= 36

Divide both sides of the equation by 9 to solve for

*b*.

9

*b*/9 = 36/9

*b*= 4Verify that the answer is correct.

Does 9/3 = 12/4?

9/3 = 3

12/4 = 3

4. 5/60 =

*k*/6

Cross multiply.

5 *6 =

*k** 60

30 = 60

*k*

Divide both sides of the equation by 60 to solve for

*k*.

30/60 = 60

*k*/60

**½ =**

*k*Verify that the answer is correct.

Does 5/60 = (1/2)/ 6?

5/60 = .08333

(1/2)/ 6 = .08333.

5. 52/949 = *s*/365

Cross multiply.

52 *365 = *s* * 949

18,980 = 949*s*

Divide both sides of the equation by 949 to solve for *s*.

18,980/949 = 949s/949**20 = s**

Verify that the answer is correct.

Does 52/949 = 20/365?

52/949 = 4/73

20/365 = 4/73

6. 22.5/x = 5/100

Cross multiply.

22.5 * 100 =

*5 * x*

2250 = 5

*x*

Divide both sides of the equation by 5 to solve for

*x*.

2250/5 = 5

*x*/5

**450 =**

*x*Verify that the answer is correct.

Does 22.5/

*x*= 5/100?

22.5/450 = .05

5/100 = .05

7.

*a*/180 = 4/100

Cross multiply.

*a** 100 = 4 * 180

100

*a*= 720

Divide both sides of the equation by 100 to solve for

*a*.

100

*a*/100 = 720/100

*a*= 7.2Verify that the answer is correct.

Does 7.2/180 = 4/100?

7.2/180 = .04

4/100 = .04

Edited by Anne Marie Helmenstine, Ph.D.