Proportion Word Problems

There are lots of situations that can create proportion word problems. We will illustrate these situations with some examples.

In this lesson, you will learn

  • how to set up a proportion with both ratios written in the same order
  • how to replace the known values and use a letter for the unknown
  • how to cross multiply and solve, with problems about mixing, shadows, paint and time

How to Set Up a Proportion Word Problem

Problem #1

Mix 3 liters of water with 4 lemons to make lemonade. How many liters of water are mixed with 8 lemons?

Set up the ratios, but make sure that the two ratios are written in the same order.

Let x be the number of liters of water. All the following can be used to solve this problem:

1. 34 = x8

2. 43 = 8x

3. 3x = 48

4. x3 = 84

Rule

It is very important to notice that if the ratio on the left is a ratio of number of liters of water to number of lemons, you have to do the same ratio on the right before you set them equal.

Take a closer look, and you'll see that the first proportion maintains the original ratio of liters of water to lemons. In contrast, the second proportion reverses the ratio, focusing on the number of lemons per liter of water as shown below.

Number of lemonsNumber of liters of water = Number of lemonsNumber of liters of water

When solving proportion word problems, make sure it is set up correctly. Once you set up your proportion correctly, all you have to do is to replace values that you know and use an x or any other variable for the value you don't know.

Let us solve the second proportion. I already showed you how to solve a proportion. If you do not remember, go to solving proportions.

43 = 8x

Cross product is usually used to solve proportion word problems. If you do a cross product, you will get: 4 × x = 3 × 8

4 × x = 24. Since 4 × 6 = 24, x = 6

6 liters should be mixed with 8 lemons.

Common mistake

Writing the two ratios in a different order, like 34 = 8x. The left ratio is water over lemons, but the right ratio is lemons over water. Cross multiplying gives 3 × x = 32, so x = 10.666…, which is wrong. Before you cross multiply, say both ratios out loud: “water over lemons equals water over lemons.”

Mix 2 cups of juice with 5 cups of water. How many cups of juice are mixed with 20 cups of water?

Show answer

8 cups. 25 = x20, so 5 × x = 2 × 20 = 40 and x = 8.

More Interesting Proportion Word Problems: Shadows

Problem #2

A boy who is 3 feet tall can cast a shadow on the ground that is 7 feet long. How tall is a man who can cast a shadow that is 14 feet long?

A man and a boy standing in the sun. The boy’s shadow is short and the man’s shadow is twice as long.
Worked example: How tall is the man?
  1. Set up the proportion by doing ratios of height to length of shadow: Height of BoyLength of Boy's Shadow = Height of ManLength of Man's Shadow
  2. Replace the known values and use H for the unknown height of the man: 37 = H14
  3. Cross multiply: 3 × 14 = 7 × H, so 42 = 7 × H.
  4. Since 7 × 6 = 42, H = 6.

The man is 6 feet tall.

A 5-foot girl casts a 4-foot shadow. At the same time, a tree casts a 20-foot shadow. How tall is the tree?

Show answer

25 feet. 54 = H20, so 4 × H = 5 × 20 = 100 and H = 25.

Proportion Word Problem with Paint

Problem #3

3 gallons of paint cover 900 square feet. How many gallons will cover 300 square feet?

1. 3900 = x300

2. 9003 = 300x

3. 3x = 900300

We will solve the first one:

3900 = x300

Doing a cross product will give you 3 × 300 = x × 900

900 = x × 900. Thus x = 1 because 1 × 900 = 900

1 gallon of paint will cover 300 square feet.

Firefighter Math and Proportion

Problem #4

A fire truck can hold 3,000 gallons of water. A firefighter can deliver 160 gallons of water every 2 minutes.

  1. How much water will be delivered in 10 minutes?
  2. How long will it take for the firefighter to empty the tank?

1. How much water will be delivered in 10 minutes?

Set up a proportion by doing ratios of number of gallons to time it takes: Number of GallonsTime it Takes = Number of GallonsTime it Takes

Replace the known values and use G to represent the number of gallons in 10 minutes: 1602 = G10

Cross multiply: 160 × 10 = 2 × G

1,600 = 2 × G

Since 2 × 800 = 1,600, G = 800

800 gallons of water will be delivered in 10 minutes.

2. How long will it take for the firefighter to empty the tank?

Set up a proportion by doing ratios of number of gallons to time it takes, and use T to represent the time it takes to deliver 3,000 gallons: 1602 = 3,000T

Cross multiply: 160 × T = 2 × 3,000

160 × T = 6,000

Divide 6,000 by 160 to get T. 6,000 divided by 160 = 37.5

T = 37.5 minutes, or 37 minutes and 30 seconds.

Remember

The unknown can be on the top or on the bottom of a ratio. In part 1, G is on top. In part 2, T is on the bottom. Cross multiplying works the same way in both cases.

A printer prints 30 pages every 2 minutes. How long does it take to print 150 pages?

Show answer

10 minutes. 302 = 150T, so 30 × T = 2 × 150 = 300 and T = 10.

I welcome any questions about these proportion word problems if you have any.

Check this site if you want to solve more proportion word problems.

Summary

  • A proportion word problem compares two ratios of the same two quantities, such as liters of water to lemons.
  • Write both ratios in the same order. If the left ratio is water over lemons, the right ratio must also be water over lemons.
  • Replace the values you know, and use x or another letter for the value you don't know.
  • Cross multiply, then divide to find the unknown: from 34 = x8, 4 × x = 24 and x = 6.
  • Check that the answer makes sense: more lemons need more water, and a longer shadow belongs to a taller person.

Practice: Solve 5 Proportion Word Problems

Now solve some proportion word problems yourself. Problems 1 and 2 walk you through every step of the method above. Problem 3 has fewer steps. In problems 4 and 5, you follow the key steps and find every answer yourself. You can skip a problem at any time, and each time you play you get new problems.

Problem 1 of 5Score: 0

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