CHAPTER 4 Ratio, Proportion, and Percent

# 4.3 Solve Proportions and their Applications

Learning Objectives

By the end of this section, you will be able to:

• Use the definition of proportion
• Solve proportions
• Solve applications using proportions
• Write percent equations as proportions
• Translate and solve percent proportions

# Use the Definition of Proportion

When two ratios or rates are equal, the equation relating them is called a proportion.

Proportion

A proportion is an equation of the form , where .

The proportion states two ratios or rates are equal. The proportion is read is to , as is to

The equation is a proportion because the two fractions are equal. The proportion is read is to as is to

If we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.

EXAMPLE 1

Write each sentence as a proportion:

1. is to as is to .
2. hits in at bats is the same as hits in at-bats.
3. for ounces is equivalent to for ounces.
Solution
 a) 3 is to 7 as 15 is to 35. Write as a proportion.
 b) 5 hits in 8 at-bats is the same as 30 hits in 48 at-bats. Write each fraction to compare hits to at-bats. Write as a proportion.
 c) $1.50 for 6 ounces is equivalent to$2.25 for 9 ounces. Write each fraction to compare dollars to ounces. Write as a proportion.

TRY IT 1.1

Write each sentence as a proportion:

1. is to as is to .
2. hits in at-bats is the same as hits in at-bats.
3. for ounces is equivalent to for ounces.
Show answer

TRY IT 1.2

Write each sentence as a proportion:

1. is to as is to .
2. adults for children is the same as adults for children.
3. for ounces is equivalent to for ounces.
Show answer

Look at the proportions and . From our work with equivalent fractions we know these equations are true. But how do we know if an equation is a proportion with equivalent fractions if it contains fractions with larger numbers?

To determine if a proportion is true, we find the cross products of each proportion. To find the cross products, we multiply each denominator with the opposite numerator (diagonally across the equal sign). The results are called a cross products because of the cross formed. The cross products of a proportion are equal.

Cross Products of a Proportion

For any proportion of the form , where , its cross products are equal.

Cross products can be used to test whether a proportion is true. To test whether an equation makes a proportion, we find the cross products. If they are the equal, we have a proportion.

EXAMPLE 2

Determine whether each equation is a proportion:

Solution

To determine if the equation is a proportion, we find the cross products. If they are equal, the equation is a proportion.

 a) Find the cross products.

Since the cross products are not equal, , the equation is not a proportion.

 b) Find the cross products.

Since the cross products are equal, , the equation is a proportion.

TRY IT 2.1

Determine whether each equation is a proportion:

Show answer
1. no
2. yes

TRY IT 2.2

Determine whether each equation is a proportion:

Show answer
1. no
2. no

# Solve Proportions

To solve a proportion containing a variable, we remember that the proportion is an equation. All of the techniques we have used so far to solve equations still apply. In the next example, we will solve a proportion by multiplying by the Least Common Denominator (LCD) using the Multiplication Property of Equality.

EXAMPLE 3

Solve: .

Solution
 To isolate , multiply both sides by the LCD, 63. Simplify. Divide the common factors. Check: To check our answer, we substitute into the original proportion. Show common factors. Simplify.

TRY IT 3.1

Solve the proportion: .

Show answer

77

TRY IT 3.2

Solve the proportion: .

Show answer

104

When the variable is in a denominator, we’ll use the fact that the cross products of a proportion are equal to solve the proportions.

We can find the cross products of the proportion and then set them equal. Then we solve the resulting equation using our familiar techniques.

EXAMPLE 4

Solve: .

Solution

Notice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.

 Find the cross products and set them equal. Simplify. Divide both sides by 9. Simplify. Check your answer: Substitute a = 64 Show common factors. Simplify.

Another method to solve this would be to multiply both sides by the LCD, . Try it and verify that you get the same solution.

TRY IT 4.1

Solve the proportion: .

Show answer

65

TRY IT 4.2

Solve the proportion: .

Show answer

24

EXAMPLE 5

Solve: .

Solution
 Find the cross products and set them equal. Simplify. Divide both sides by 52. Simplify. Check: Substitute y = −7 Show common factors. Simplify.

TRY IT 5.1

Solve the proportion: .

Show answer

−7

TRY IT 5.2

Solve the proportion: .

Show answer

−9

# Solve Applications Using Proportions

The strategy for solving applications that we have used earlier in this chapter, also works for proportions, since proportions are equations. When we set up the proportion, we must make sure the units are correct—the units in the numerators match and the units in the denominators match.

EXAMPLE 6

When pediatricians prescribe acetaminophen to children, they prescribe millilitre s (ml) of acetaminophen for every pounds of the child’s weight. If Zoe weighs pounds, how many millilitre s of acetaminophen will her doctor prescribe?

Solution
 Identify what you are asked to find. How many ml of acetaminophen the doctor will prescribe Choose a variable to represent it. Let ml of acetaminophen. Write a sentence that gives the information to find it. If 5 ml is prescribed for every 25 pounds, how much will be prescribed for 80 pounds? Translate into a proportion. Substitute given values—be careful of the units. Multiply both sides by 80. Multiply and show common factors. Simplify. Check if the answer is reasonable. Yes. Since 80 is about 3 times 25, the medicine should be about 3 times 5. Write a complete sentence. The pediatrician would prescribe 16 ml of acetaminophen to Zoe.

You could also solve this proportion by setting the cross products equal.

TRY IT 6.1

Pediatricians prescribe millilitre s (ml) of acetaminophen for every pounds of a child’s weight. How many millilitre s of acetaminophen will the doctor prescribe for Emilia, who weighs pounds?

Show answer

12 ml

TRY IT 6.2

For every kilogram (kg) of a child’s weight, pediatricians prescribe milligrams (mg) of a fever reducer. If Isabella weighs kg, how many milligrams of the fever reducer will the pediatrician prescribe?

Show answer

180 mg

EXAMPLE 7

One brand of microwave popcorn has calories per serving. A whole bag of this popcorn has servings. How many calories are in a whole bag of this microwave popcorn?

Solution
 Identify what you are asked to find. How many calories are in a whole bag of microwave popcorn? Choose a variable to represent it. Let number of calories. Write a sentence that gives the information to find it. If there are 120 calories per serving, how many calories are in a whole bag with 3.5 servings? Translate into a proportion. Substitute given values. Multiply both sides by 3.5. Multiply. Check if the answer is reasonable. Yes. Since 3.5 is between 3 and 4, the total calories should be between 360 (3⋅120) and 480 (4⋅120). Write a complete sentence. The whole bag of microwave popcorn has 420 calories.

TRY IT 7.1

Marissa loves the Caramel Macchiato at the coffee shop. The oz. medium size has calories. How many calories will she get if she drinks the large oz. size?

Show answer

300

TRY IT 7.2

Yaneli loves Starburst candies, but wants to keep her snacks to calories. If the candies have calories for pieces, how many pieces can she have in her snack?

Show answer

5

EXAMPLE 8

Josiah went to Mexico for spring break and changed dollars into Mexican pesos. At that time, the exchange rate had U.S. is equal to Mexican pesos. How many Mexican pesos did he get for his trip?

Solution

# Attributions

This chapter has been adapted from “Solve Proportions and their Applications” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

## License

Introductory Algebra Copyright © 2021 by Izabela Mazur is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.