Ratios and Proportions

Objective

• I will solve proportions using reciprocals and cross products.

• I will use proportions to solve real-life problems.

Vocabulary1. Proportion - an equation in which one

ratio is set equal to another ratio

(Written in the form of , which is read as “a is to b, as c is to d”)

a

bc

d

Using the Reciprocal PropertySolve the proportion .SOLUTION

4

y8

3

y

43

8

y 4 •3

8

y 12

8

y 3

2

4

y8

3

Write original proportion

Use reciprocal property

Multiply each side by 4

Simplify

Using the Reciprocal Property

Solve the proportion .

CHECK

Substitute

4

y8

3

4

3/24 •

2

38

3

Using the Cross Product Property

Solve the proportion .

x

416

x

x

416

xx • x 4 •16

x 2 64x 8

Write original proportion

Use cross product property

Simplify

Take square root of both sides

The solutions are x=-8 and x=8. Check both answers.

Checking SolutionsSolve the proportion.

x 2 16x 4

x 42

2(x 2 16)(x 4)(x 4)

2x 2 32x 2 16

x 2 16x 4

Use cross product property

Use distributive property

Isolate variable term

Take square root of both sides

Checking SolutionsCheck each solution.

x = 4: x = -4

x 2 16x 4

x 42

42 164 4

4 42

0

80

2

00

x 2 16x 4

x 42

( 4)2 16 4 4

4 42

0

0

82

The solution x = -4 is extraneous because the check results in a false statement. The correct solution is x = 4.

Proportions in Real-LifeArchaeologists excavated three pits containing the

clay army. To estimate the number of warriors in Pit 1 shown below, an archaeologist might excavate three sites. The sites at the ends together contain 450 warriors. The site in the central region contains 282 warriors. This 10-meter-wide site is thought to be representative of the 200-meter central region. Estimate the number of warriors in the central region. Then estimate the total number of warriors in Pit 1.

Proportions in Real-Life

Central Region 200m

282 warriors

240 warriors

10m

5m 5m

210 warriors

62m

SOLUTION Let n represent the number of warriors in the 200-meter central region. You can find the value of n by solving a proportion.

Number of warriors found Number of meters excavated

=

Total number of warriors Total number of meters

282 = 10

n 200

Proportions in Real-Life

The solution is n = 5640, indicating that there are about 5640 warriors in the central region. With the 450 warriors at the ends, that makes a total of about 6090 warriors in Pit 1.

Practice

1.16

412

x

2.5

8x

9

3.x 3x

x

x 6