Warm Up Find two ratios that are equivalent to each given ratio. 3535 1. 45 30 3. 90 60 3232, 10 12 2. 20 24 5656, 8989 4. 24 27 16 18, 9 15 6 10, Possible.

Warm UpFind two ratios that are equivalent to each given ratio.

35

1.

4530

3. 9060

32

,

1012

2. 2024

56

,

89

4. 2427

1618

,

915

610

,Possible answers:

Vocabulary

Proportion

Proportional

Cross products

Direct-proportional relationship

Constant of proportionality

An equation that states that two ratios are

equivalent is called a proportion. For example,

the equation, or proportion, states that

the ratios and are equivalent.

Ratios that are equivalent are said to be

proportional, or in proportion.

46

23

=

46

23

Proportion

To find cross products, you multiply the numerator of one

ratio by the denominator or another, then multiply the second

numerator by the first denominator.

In the proportion , the products a ∙ d and b ∙ c

are called cross products.

a∙ d = b ∙ c Cross Products

One way to find whether two ratios are equivalent is to find their cross products. If the cross products are equal, the proportions are equivalent.

Tell whether the ratios are proportional.

410

615

Since the cross products are equal, the ratios are proportional.

=?

Class Example

Find the cross products.

60 = 60

410

615

=?

6 10 = 4 15 ?

Tell whether the ratios are proportional.

Individual Practice

Since the cross products are equal, the ratios are proportional.

Find the cross products.

24

510

=?

20 = 20

24

510

=?

5 4 = 2 10 ?

A mixture for a certain brand of tea should be 3 parts tea to 1 part sugar. If you combine 4 tablespoons of sugar with 12 tablespoons of tea, will the mixture be correct?

Class Example

3 parts tea 1 part sugar

=? 12 tablespoons tea4 tablespoons sugar

The ratios are equal. The mixture will be correct.

Set up equal ratios.

Find the cross products.

12 = 12

124

31

=?

3 4 = 1 12 ?

The ratio of the length of the actual height of a person to the length of the shadow cast by the person is 1:3. At the same time, a lighthouse casts a shadow that is 36 meters long. What is the height of the lighthouse?

Write a ratio comparing height of a person to shadow length.

Set up the proportion. Let x represent the lighthouse height.

Partner Practice

13

height of personlength of shadow

12 = x

13

= x36

The height of the lighthouse should be 12 meters.

Find the cross products. 1 36 = 3 x

36 = 3x 3 3

Solve for x by dividing both sides of the equation by 3

What does this mean for this situation?

The constant of proportionality is the value that relates the two amounts in a direct-proportional relationship.

These ratios are directly proportional. What are the constants of proportionality?

35

1. 1012

2. 2024

915

In a direct-proportional relationship, as one amount increases, another amount increases at the same rate.

3 2

A mixture of fuel for a certain small engine should be 4 parts gasoline to 1 part oil. If you combine 5 quarts of oil with 15 quarts of gasoline, will the mixture be correct?

4 parts gasoline1 part oil

=? 15 quarts gasoline5 quarts oil

The ratios are not equal. The mixture will not be correct.

Set up equal ratios.

Find the cross products.

Partner Practice

20 15

155

41

=?

4 5 = 1 15 ?

For most cats, the ratio of the length of their head to their total body length is 1:5. If a cat is 20 inches in length, what should the total length of their head be?

Write a ratio comparing head length to total length.

Set up the proportion. Let x represent the length of the cat's head.

Individual Practice

15

head lengthtotal length

Since x is divided by 20, multiply both sides of the equation by 20.

4 = x

15

= x20

(20) = (20)15

x20

The length of the cat's head should be 4 inches.

There's another way to solve fractions with a variable in the numerator...