5-5 Solving Proportions

Course 2

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Warm UpFind the unit rate.

1. 18 miles in 3 hours

2. 6 apples for $3.30

3. 3 cans for $0.87

4. 5 CD’s for $43

Course 2

5-4 Identifying and Writing Proportions

6 mi/h

$0.55 per apple

$0.29 per can

$8.60 per CD

Vocabulary

equivalent ratios

proportion

Course 2

5-4 Identifying and Writing Proportions

Quick activity

In your group measure the length and width of each others heads.

Length=

Width=

Course 2

5-4 Identifying and Writing Proportions

Students in Mr. Howell’s math class are measuring the width w and the length l of their heads. The ratio of l to w is 10 inches to 6 inches for Jean and 25 centimeters to 15 centimeters for Pat.

Course 2

5-4 Identifying and Writing Proportions

106

2515

These ratios can be written as the

fractions and . Since both simplify

to , they are equivalent. Equivalent

ratios are ratios that name the same comparison.

53

Course 2

5-4 Identifying and Writing Proportions

An equation stating that two ratios are equivalent is called a proportion. The equation, or proportion, below states that the ratios and are equivalent.

106

2515

106

= 2515

Read the proportion by saying “ten is to six

as twenty-five is to fifteen.”

Reading Math

106

=2515

Determine whether the ratios are proportional.

Additional Example 1A: Comparing Ratios in Simplest Forms

Course 2

5-4 Identifying and Writing Proportions

,2451

72128

72 ÷ 8

128 ÷ 8= 9

16

24 ÷ 351 ÷ 3

=8

17Simplify .24

51

Simplify . 72128

817

Since = , the ratios are not proportional.9

16

Determine whether the ratios are proportional.

Check It Out: Example 1A

Course 2

5-4 Identifying and Writing Proportions

,5463

72144

72 ÷ 72

144 ÷ 72 = 1

2

54 ÷ 963 ÷ 9

=67

Simplify .5463

Simplify . 72144

67

Since = , the ratios are not proportional.12

Determine whether the ratios are proportional.

Check It Out: Example 1B

Course 2

5-4 Identifying and Writing Proportions

, 135 75

94

9 4

135 ÷ 15 75 ÷ 15

= 9 5

Simplify .135 75

is already in simplest form. 9 4

95

Since = , the ratios are not proportional.94

Lesson Quiz:

Insert Lesson Title Here

Course 2

5-4 Identifying and Writing Proportions

In pre-school, there are 5 children for everyone teacher. In another preschool there are 20 children for every 4 teachers. Determine whether the ratios of children to teachers are proportional in both preschools.

20

4

5

1

=

Course 2

5-5 Solving Proportions

Quick checkDetermine whether the ratios are proportional.

1. 58

, 1524

2. 1215

, 1625

3. 1510

, 2016

4. 1418

, 4254

yes

no

yes

no

Learn to solve proportions by using cross products.

Course 2

5-5 Solving Proportions

Vocabulary

cross product

Insert Lesson Title Here

Course 2

5-5 Solving Proportions

Course 2

5-5 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 cross products of the ratios are equal, then the ratios form a proportion.

5 · 6 = 30

2 · 15 = 30=2

56

15

Course 2

5-5 Solving Proportions

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

CROSS PRODUCT RULE

In the proportion = , the cross products,

a · d and b · c are equal.

ab

cd

Use cross products to solve the proportion.

Additional Example 1: Solving Proportions Using Cross Products

Course 2

5-5 Solving Proportions

15 · m = 9 · 5

15m = 45

15m15

= 4515

m = 3

The cross products are equal.

Multiply.

Divide each side by 15 to isolate the variable.

= m5

915

Check It Out: Example 1

Insert Lesson Title Here

Course 2

5-5 Solving Proportions

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

Additional Example 2: Problem Solving Application

Course 2

5-5 Solving Proportions

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.

Course 2

5-5 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 2 Continued

Course 2

5-5 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 2 Continued

Course 2

5-5 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 2 Continued

Check It Out: Example 2

Course 2

5-5 Solving Proportions

John 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.

Course 2

5-5 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 2 Continued

Course 2

5-5 Solving Proportions

Solve33

610

= p25

10 · x = 6 · 25

10x = 150

10x10

= 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 2 Continued

Course 2

5-5 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 2 Continued

Lesson Quiz: Part I

Insert Lesson Title Here

Course 2

5-5 Solving Proportions

Use cross products to solve the proportion.

1. 2520

= 45t

2. x9

= 1957

3. 23

= r36

4. n10

=288

t = 36

x = 3

r = 24

n = 35

Lesson Quiz: Part II

Insert Lesson Title Here

Course 2

5-5 Solving Proportions

5. Carmen bought 3 pounds of bananas for $1.08. June paid $ 1.80 for her purchase of bananas. If they paid the same price per pound, how many pounds did June buy?5 pounds