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Chapter 4

Ratios and Proportions

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Section 4.3

Understanding and Solving Proportions

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Write a Proportion

A proportion is a mathematical statement showing that two ratios are equal.

A proportion is written as an equation with a ratio on each side of the equal sign.

Remember to include the units when writing a rate. Also, keep in mind that the order of the units is important. Be sure that like units for each rate are in the same position.

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Example

Write each sentence as a proportion.

a. 2 is to 7 as 4 is to 14

b. 12 eggs is to 3 chickens as 4 eggs is to 1 chicken

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Solution Strategy

2

7=

414

12 eggs

3 chickens=

4 eggs1 chicken

Write each sentence as a proportion.

a. 2 is to 7 as 4 is to 14

b. 12 eggs is to 3 chickens as 4 eggs is to 1 chicken

like units

like units

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Determine Whether Two Ratios Are Proportional

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Example

Determine whether the ratios are proportional.

If they are, write a corresponding proportion.

18

30=? 610

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Solution Strategy

Determine whether the ratios are proportional.

If they are, write a corresponding proportion.

18

30=? 610

18

30=

610

180

180

The ratios are proportional.

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Solve a Proportion

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Example

Solve for the unknown quantity. Verify your answer.

x

24=34

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Solution Strategy

Solve for the unknown quantity. Verify your answer.

x

24=34

72

4x

4x = 72

x = 18

18

24=34

24 • 3 = 72

18 • 4 = 72

It checks!

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Solving an Application Problem

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Example Apply your knowledge

A cake recipe requires 5 eggs for every 4 cups of flour. If a large cake requires 12 cups of flour, how many eggs should be used?

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Solution Strategy

x = number of eggs for large cake

5 eggs

4 cups of flour=

x eggs12 cups of flour

A cake recipe requires 5 eggs for every 4 cups of flour. If a large cake requires 12 cups of flour, how many eggs should be used?

like units

like units4x = 5 • 12

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Solution Strategy

x = 15

4x

4=604

4x = 5 • 12

4x = 60

The large cake requires 15 eggs.

5

4=1512

4 • 15 = 60

5 • 12 = 60

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Solving a Geometry Application Problem

One common application of proportions is solving problems involving similar geometric figures. Similar geometric figures are geometric figures with the same shape in which the ratios of the lengths of their corresponding sides are equal. Because these ratios are equal, they are proportional.

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Solving a Geometry Application Problem

The similar rectangles yield this proportion:7

14=1224

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Solving a Geometry Application Problem

The similar triangles yield these proportions:

1

2=36

1

2=

510

3

6=

510

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Example Using shadow proportions to find “difficult to measure” lengths

Kyle is 5.6 feet tall. Late one afternoon while visiting the Grove Isle Lighthouse, he noticed that his shadow was 8.5 feet long. At the same time, the lighthouse cast a shadow 119 feet long. What is the height of the lighthouse? (See the figure on the next slide.)

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Example Using shadow proportions to find “difficult to measure” lengths

8.5 ft shadow119 ft shadow

5.6 ft

x

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Solution Strategy

x = height of the lighthouse

5.6

8.5=

x119

Kyle is 5.6 feet tall. Late one afternoon while visiting the Grove Isle Lighthouse, he noticed that his shadow was 8.5 feet long. At the same time, the lighthouse cast a shadow 119 feet long. What is the height of the lighthouse?

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Solution Strategy

8.5x = 5.6 • 119

5.6

8.5=

x119

8.5x = 666.4

x =666.48.5

x = 78.4

5.6

8.5=78.4119

8.5 • 78.4 = 666.4

5.6 • 119 = 666.4

The height of the lighthouse is 78.4 feet.