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

## 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

## Modifying 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 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: using 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, however? 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

Freeze- verify that the answer is correct.

Is 3/25 = 2/16.6667?

3/25 = .12

2/16.6667= .12

Whoo hoo! The answer 16.6667 cups of water is correct.

## Ratio and Proportions Word Problem 1: The Brownie Recipe

Damian is making brownies to serve at the family picnic. If the recipe calls for 2 ½ cups of cocoa to serve 4 people, how many cups will he need if there will be 60 people at the picnic? **37.5 cups**

What do you know?

2 ½ cups = 4 people

? cups = 60 people

2 ½ cups/*x* cups = 4 people/60 people

2 ½/*x* = 4/60

Cross Multiply.

2 ½ * 60 = 4 * *x*

150 = 4*x*

Divide both sides by 4 to solve for *x*.

150/4 = 4*x*/4

37.5 = *x*

37.5 cups

Use common sense to verify that the answer is correct.

The initial recipe serves 4 people and is modified to serve 60 people. Of course, the new recipe has to serve 15 times more people. Therefore, the amount of cocoa has to be multiplied by 15. Is 2 ½ * 15 = 37.5? Yes.

## Ratio and Proportions Word Problem 2: Growing Little Piglets

A piglet can gain 3 pounds in 36 hours. If this rate continues, the pig will reach 18 pounds in** 216 **hours.

What do you know?

3 pounds = 36 hours

18 pounds = ? hours

3 pounds/18 pounds = 36 hours/ ? hours

3/18 = 36/*x*

Cross Multiply.

3 * *x* = 36 * 18

3*x* = 648

Divide both sides by 3 to solve for *x*.

3*x*/3 = 648/3*x* = 216

216 hours

Use common sense to verify that the answer is correct.

A piglet can gain 3 pounds in 36 hours, which is a rate of 1 pound for every 12 hours. That means that for every pound a piglet gains, 12 hours will pass. Therefore 18 *12, or 216 pounds, is the correct answer.

## Ratio and Proportions Word Problem 3: The Hungry Rabbit

Denise’s rabbit can eat 70 pounds of food in 80 days. How long will it take the rabbit to eat 87.5 pounds? **100 days**

What do you know?

70 pounds = 80 days

87.5 pounds = ? days

70 pounds/87.5 pounds = 80 days/*x *days

70/87.5 = 80/*x*

Cross Multiply.

70 * *x* = 80 * 87.5

70*x* = 7000

Divide both sides by 70 to solve for *x*.

70*x*/70 = 7000/70*x* = 100

Use Algebra to verify the answer.

Is 70/87.5 = 80/100?

70/87.5 = .8

80/100 = .8

## Ratio and Proportions Word Problem 4: The Long Road Trip

Jessica drives 130 miles every two hours. If this rate continues, how long will it take her to drive 1,000 miles? **15.38 hours**

What do you know?

130 miles = 2 hours

1,000 miles = ? hours

130 miles/1,000 miles = 2 hours/? hours

130/1000 = 2/*x*

Cross Multiply.

130 * *x* = 2 * 1000

130*x* = 2000

Divide both sides of the equation by 130 to solve for *x*.

130*x*/130 = 2000/130*x* = 15.38 hours

Use Algebra to verify the answer.

Does 130/1000 = 2/15.38?

130/1000 = .13

2/15.38 is approximately .13