holt ca course 1 5-4 solving proportions ns1.3 use proportions to solve problems (e.g., determine...

Holt CA Course 1

5-4 Solving Proportions

NS1.3 Use proportions to solve problems (e.g., determine the value of N if = , find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. Also covered: AF2.2, AF2.3

California Standards

4 7

N 21

Holt CA Course 1

5-4 Solving Proportions

Vocabulary

cross productThe product of the numerator in one ratio and the denominator in the other when comparing two ratios.

Holt CA Course 1

5-4 Solving Proportions

For two ratios, the product of the numerator in one ratio and the denominator in the other is a cross product.

If the two ratios form a proportion, then the cross products are equal.

5 · 6 = 30

2 · 15 = 30=2

56

15

Holt CA Course 1

5-4 Solving Proportions

You can use the cross product rule to solve proportions with variables.

Holt CA Course 1

5-4 Solving Proportions

Use cross products to solve the proportion.

Example 1: Solving Proportions Using Cross Products

15 · m = 9 · 5

15m = 45

15m15

= 4515

m = 3

The cross products are equal.

Multiply.

Divide each side by 15.

= m5

915

Holt CA Course 1

5-4 Solving Proportions

It is important to set up proportions correctly. Each ratio must compare corresponding quantities in the same order.

16 mi4 hr

= 8 mix hr

16 mi8 mi

= 4 hrx hr

Holt CA Course 1

5-4 Solving Proportions

The graph shows the time and distance Greg walked on his way to school. At this rate, how long would it take Greg to walk 7.5 miles?

Example 2: Sports Application

The labeled point on the graph shows that Greg walked 0.75 miles in 0.25 hr. Let t represent the time in hours it will take Greg to walk 7.5 miles. 0

0.1 0.2 0.3 0.4 0.5

0.25

0.5

0.75

Dis

tan

ce (

mi)

Walking Rate

Time (hr)

(0.25, 0.75)

Holt CA Course 1

5-4 Solving Proportions

The graph shows the time and distance Greg walked on his way to school. At this rate, how long would it take Greg to walk 7.5 miles?

Example 2 Continued

0.75 · t = 0.25 · 7.5

0.75t = 1.875

0.75t0.75

= 1.8750.75

t = 2.5 hr

DistanceTime

Divide each side by 0.75.

= 7.5 mi t hr

0.75 mi0.25 hr

Set up a proportion in which each ratio compares distance to the time needed to walk that distance.

The cross products are equal.

Multiply.

It will take Greg 2.5 hours to walk 7.5 miles.

Holt CA Course 1

5-4 Solving ProportionsExample 3: Problem Solving Application

If 3 volumes of Jennifer’s encyclopedia takes up 4 inches of space on her shelf, how much space will she need for all 26 volumes?

11 Understand the Problem

Rewrite the question as a statement.

• Find the space needed for 26 volumes of the encyclopedia.

List the important information:

• 3 volumes of the encyclopedia take up 4 inches of space.

Holt CA Course 1

5-4 Solving Proportions

22 Make a Plan

Set up a proportion using the given information. Let x represent the inches of space needed.

3 volumes4 inches

= 26 volumesx

volumesinches

Additional Example 3 Continued

Holt CA Course 1

5-4 Solving Proportions

Solve33

34

= 26x

3 · x = 4 · 26

3x = 104

3x3

= 1043

x = 34 23

She needs 34 23

inches for all 26 volumes.

Write the proportion.

The cross products are equal.

Multiply.

Divide each side by 3 to isolate the variable.

Additional Example 3 Continued

Holt CA Course 1

5-4 Solving Proportions

Look Back44

34

= 2634 2

3

4 · 26 = 104

3 · 34 23

= 104

The cross products are equal, so 34 23

is theanswer.

Additional Example 3 Continued

Holt CA Course 1

5-4 Solving Proportions

Check It Out! Example 1

Use cross products to solve the proportion.

7 · m = 6 · 14

7m = 84

7m7

= 847

m = 12

The cross products are equal.

Multiply.

Divide each side by 7 to isolate the variable.

67

= m14

Holt CA Course 1

5-4 Solving Proportions

The graph shows the time and distance Paige ran in one marathon. At this rate, how long would it take Paige to run 8.5 miles?

Check It Out! Example 2

The labeled point on the graph shows that a Paige ran 2.5 miles in 0.15 hr. Let t represent the time in hours it will take Paige to run 8.5 miles.

0 0.1 0.2 0.3 0.4 0.5

1.0

2.0

3.0

Dis

tan

ce (

mi)

Running Rate

Time (hr)

(0.15, 2.5)

Holt CA Course 1

5-4 Solving ProportionsCheck It Out! Example 2 Continued

2.5 · t = .15 · 8.5

2.5t = 1.275

2.5t2.5

= 1.2752.5

t = .51 hr

DistanceTime

Divide each side by 2.5.

= 8.5 mi t hr

2.5 mi.15 hr

Set up a proportion in which each ratio compares distance to the time needed to run that distance.

The cross products are equal.

Multiply.

The graph shows the time and distance Paige ran in one marathon. At this rate, how long would it take Paige to run 8.5 miles?

Holt CA Course 1

5-4 Solving Proportions

Check It Out! Example 3John filled his new radiator with 6 pints of coolant, which is the 10 inch mark. How many pints of coolant would be needed to fill the radiator to the 25 inch level?

11 Understand the Problem

Rewrite the question as a statement.• Find the number of pints of coolant required

to raise the level to the 25 inch level.

List the important information:• 6 pints is the 10 inch mark.

Holt CA Course 1

5-4 Solving Proportions

22 Make a Plan

Set up a proportion using the given information. Let p represent the pints of coolant.

6 pints10 inches

= p25 inches

pints inches

Check It Out! Example 3 Continued

Holt CA Course 1

5-4 Solving Proportions

Solve33

610

= p25

10 · p = 6 · 25

10p = 150

10p10

= 15010

p = 15

15 pints of coolant will fill the radiator to the 25 inch level.

Write the proportion.

The cross products are equal.

Multiply.

Divide each side by 10 to isolate the variable.

Check It Out! Example 3 Continued

Holt CA Course 1

5-4 Solving Proportions

Look Back44

610

= 1525

10 · 15 = 150

6 · 25 = 150

The cross products are equal, so 15 is the answer.

Check It Out! Example 3 Continued