Do NowFind 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

Hwk: p 45 & 46

EQ: How do I find equivalent ratios and to identify proportions and solve proportions by using cross products?

Course 2

5-4 Identifying and Writing Proportions

M7N1.b Compare and order rational numbers, including repeating decimals; M7N1.d Solve problems using rationalnumbers;

M7A2.a Given a problem, define a variable, write an equation, solve the equation, and interpret the solution;M7A2.b Use the addition and multiplication properties of equality to solve one- and two-step linear equations

Vocabulary

equivalent ratiosproportion

Course 2

5-4 Identifying and Writing Proportions

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. 10

62515

106

= 2515

Read the proportion by saying “ten is to six

as twenty-five is to fifteen.”

Reading Math

106

= 2515

Course 2

5-4 Identifying and Writing Proportions

If two ratios are equivalent, they are said to be proportional to each other, or in proportion.

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 ÷ 8128 ÷ 8

= 916

24 ÷ 351 ÷ 3

= 817

Simplify .2451

Simplify . 72128

817

Since = , the ratios are not proportional.9

16

Determine whether the ratios are proportional.

Additional Example 1B: Comparing Ratios in Simplest Forms

Course 2

5-4 Identifying and Writing Proportions

,150 105

9063

90 ÷ 9 63 ÷ 9

= 107

150 ÷ 15105 ÷ 15

= 10 7

107

Since = , the ratios are proportional.107

Determine whether the ratios are proportional.

Check It Out: Example 1A

Course 2

5-4 Identifying and Writing Proportions

,5463

72144

72 ÷ 72144 ÷ 72

= 1 2

54 ÷ 963 ÷ 9

= 67

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

95

Since = , the ratios are not proportional.94

Directions for making 12 servings of rice call for 3 cups of rice and 6 cups of water. For 40 servings, the directions call for 10 cups of rice and 19 cups of water. Determine whether the ratios of rice to water are proportional for both servings of rice.

Additional Example 2: Comparing Ratios Using a Common Denominator

Course 2

5-4 Identifying and Writing Proportions

Write the ratios of rice to water for 12 servings and for 40 servings.

Ratio of rice to water, 12 servings:36

Ratio of rice to water, 40 servings: 1019

36

= 3 · 196 · 19

= 57114

1019

= 10 · 619 · 6

= 60114

57114

60114Since = , the two ratios are not proportional.

Servings of Rice

Cups of Rice

Cups of Water

12 3 6

40 10 19

Use the data in the table to determine whether the ratios of beans to water are proportional for both servings of beans.

Check It Out: Example 2

Course 2

5-4 Identifying and Writing Proportions

Write the ratios of beans to water for 8 servings and for 35 servings.

Ratio of beans to water, 8 servings:43

Ratio of beans to water, 35 servings: 13 9

43

= 4 · 93 · 9

= 3627

13 9

= 13 · 3 9 · 3

= 3927

3627

3927

Servings of Beans Cups of Beans Cups of Water

8 4 3

35 13 9

Since = , the two ratios are not proportional.

Course 2

5-4 Identifying and Writing Proportions

You can find an equivalent ratio by multiplying or dividing the numerator and the denominator of a ratio by the same number.

Additional Example 3: Finding Equivalent Ratios and Writing Proportions

Find a ratio equivalent to each ratio. Then use the ratios to find a proportion. Possible Answers:

Insert Lesson Title Here

Course 2

5-4 Identifying and Writing Proportions

A. 3535

= 3 · 25 · 2

610

=

35

= 610

B. 28162816

= 28 ÷ 416 ÷ 4

2816

= 74

= 74

Check It Out: Example 3

Find a ratio equivalent to each ratio. Then use the ratios to find a proportion. Possible Answers:

Insert Lesson Title Here

Course 2

5-4 Identifying and Writing Proportions

A. 2323

= 2 · 33 · 3

69

=

23

= 69

B. 16121612

= 16 ÷ 412 ÷ 4

1612

= 43

= 43

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

= 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

67

= m14

Course 2

5-5 Solving Proportions

When setting up a proportion to solve a problem, use a variable to represent the number you want to find. In proportions that include different units of measurement, either the units in the numerators must be the same and the units in the denominators must be the same or the units within each ratio must be the same.

16 mi4 hr

= 8 mix hr

16 mi8 mi

= 4 hrx hr

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?

Course 2

5-5 Solving Proportions

34

= 26x

3 · x = 4 · 26

3x = 104

3x3

= 1043

x = 34 23

She needs 34 23

inches for all 26 volumes.

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?

Quiz

Insert Lesson Title Here

Course 2

5-4 Identifying and Writing Proportions

Determine whether the ratios are proportional.

1. 930 ,

1240

2. 1221 ,

1015

Find a ratio equivalent to each ratio. Then use the ratios to write a proportion.

3. 38

4. 37

Quiz

Insert Lesson Title Here

Course 2

5-4 Identifying and Writing Proportions

5. 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

=

Lesson Quiz: Part I

Insert Lesson Title Here

Course 2

5-5 Solving Proportions

Use cross products to solve the proportion.

6. 2520

= 45t

7. x9

= 1957

8. 23

= r36

9. n10

=288

Lesson Quiz: Part II

Insert Lesson Title Here

Course 2

5-5 Solving Proportions

10.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?