# Algebra: Ratio Word Problems

In these lessons, we will learn how to solve ratio word problems that have two-term ratios or three-term ratios.

Ratio problems are word problems that use ratios to relate the different items in the question.

The main things to be aware about for ratio problems are:

1. Change the quantities to the same unit if necessary.
2. Write the items in the ratio as a fraction.
3. Make sure that you have the same items in the numerator and denominator.

### Ratio Problems: Two-Term Ratios

Example 1:
In a bag of red and green sweets, the ratio of red sweets to green sweets is 3:4. If the bag contains 120 green sweets, how many red sweets are there?

Solution:
Step 1: Assign variables:
Let x = number of red sweets.

Write the items in the ratio as a fraction. Step 2: Solve the equation.
Cross Multiply
3 × 120 = 4 × x
360 = 4x

Isolate variable x Answer: There are 90 red sweets.

Example 2:
John has 30 marbles, 18 of which are red and 12 of which are blue. Jane has 20 marbles, all of them either red or blue. If the ratio of the red marbles to the blue marbles is the same for both John and Jane, then John has how many more blue marbles than Jane?

Solution:
Step 1: Sentence: Jane has 20 marbles, all of them either red or blue.
Assign variables:
Let x = number of blue marbles for Jane
20 – x = number red marbles for Jane

We get the ratio from John
John has 30 marbles, 18 of which are red and 12 of which are blue. We use the same ratio for Jane. Step 2: Solve the equation

Cross Multiply
3 × x = 2 × (20 – x)
3x = 40 – 2x

Isolate variable x John has 12 blue marbles. So, he has 12 – 8 = 4 more blue marbles than Jane.

Answer: John has 4 more blue marbles than Jane.

#### How To Solve Word Problems Using Proportions?

This is another word problem that involves ratio or proportion.

Example:
A recipe uses 5 cups of flour for every 2 cups of sugar. If I want to make a recipe using 8 cups of flour. How much sugar should I use?

#### How To Solve Proportion Word Problems?

When solving proportion word problems remember to have like units in the numerator and denominator of each ratio in the proportion.

Examples:

1. Biologist tagged 900 rabbits in Bryer Lake National Park. At a later date, they found 6 tagged rabbits in a sample of 2000. Estimate the total number of rabbits in Bryer Lake National Park.
2. Mel fills his gas tank up with 6 gallons of premium unleaded gas for a cost of \$26.58. How much would it costs to fill an 18 gallon tank? 3 If 4 US dollars can be exchanged for 1.75 Euros, how many Euros can be obtained for 144 US dollars?

### Ratio problems: Three-term Ratios

Example 1:
A special cereal mixture contains rice, wheat and corn in the ratio of 2:3:5. If a bag of the mixture contains 3 pounds of rice, how much corn does it contain?

Solution:
Step 1: Assign variables:
Let x = amount of corn

Write the items in the ratio as a fraction. Step 2: Solve the equation
Cross Multiply
2 × x = 3 × 5
2x = 15

Isolate variable x Answer: The mixture contains 7.5 pounds of corn.

Example 2:
Clothing store A sells T-shirts in only three colors: red, blue and green. The colors are in the ratio of 3 to 4 to 5. If the store has 20 blue T-shirts, how many T-shirts does it have altogether?

Solution:
Step 1: Assign variables:
Let x = number of red shirts
and y = number of green shirts

Write the items in the ratios as fractions. Step 2: Solve the equation
Cross Multiply
3 × 20 = x × 4
60 = 4x
x = 15

5 × 20 = y × 4
100 = 4y
y = 25

The total number of shirts would be 15 + 25 + 20 = 60

#### Algebra And Ratios With Three Terms

Let’s study how algebra can help us think about ratios with more than two terms.

Example:
There are a total of 42 computers. Each computer runs one of three operating systems: OSX, Windows, Linux. The ratio of the computers running OSX, Windows, Linux is 2:5:7. Find the number of computers that are running each of the operating systems.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. 