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Evaluating Algebraic Expressions

5-3 Proportions

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California StandardsCalifornia Standards

Lesson Presentation

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5-3 Proportions

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:


5-3 Proportions

AF4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation.

California Standards


5-3 Proportions

Vocabulary

proportion

cross products


5-3 Proportions

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


5-3 Proportions

Proportion

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

are called cross products.

cd

ab =

a∙ d = b ∙ c Cross Products

One way to find whether two ratios are equivalent is to find their cross products.

cd

ab =


5-3 Proportions


5-3 Proportions

Tell whether the ratios are proportional.

410

615

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

=?

Additional Example 1A: Using Cross Products to Identify Proportions

Find the cross products.

60 = 60

410

615

=?

6 10 = 4 15 ?


5-3 Proportions

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.


Additional Example 1B: Using Cross Products to Identify Proportions

20 15

155

41

=?

4 5 = 1 15 ?


5-3 Proportions


Check It Out! Example 1A

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


24

510

=?

20 = 20

24

510

=?

5 4 = 2 10 ?


5-3 Proportions

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?

Check It Out! Example 1B

3 parts tea 1 part sugar

=? 12 tablespoons tea4 tablespoons sugar

The ratios are equal. The mixture will be correct.

Set up equal ratios.


12 = 12

124

31

=?

3 4 = 1 12 ?


5-3 Proportions

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 should the length of its shadow be?

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

Set up the proportion. Let x represent the shadow length.

Additional Example 2: Using Properties of Equality to Solve Proportions

13

height of personlength of shadow

Since x is divided by 36, multiply both sides of the equation by 36.

12 = x

13

= x36

(36) = (36)13

x36

The length of the lighthouse’s shadow should be 12 meters.


5-3 Proportions

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.

Check It Out! Example 2

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.


5-3 Proportions

Allyson weighs 55 pounds and sits on a seesaw 4 feet away from it center. If Marco sits on the seesaw 5 feet away from the center and the seesaw is balanced, how much does Marco weigh?

55 ∙ 4 = 5w Find the cross products.

Divide both sides by 5.5w5

2205

=

Additional Example 3: Using Cross Products to Solve Proportions

=weight 1length 2

weight 2length 1

555

= w4

44 = w Simplify.

Set up a proportion using the information. Let w represent Marco’s weight.

Marco weighs 44 lb.


5-3 Proportions

Austin weighs 32 pounds and sits on a seesaw 6 feet away from it center. If Kaylee sits on the seesaw 4 feet away from the center and the seesaw is balanced, how much does Kaylee weigh?

32 ∙ 6 = 4w Find the cross products.

Divide both sides by 4.4w4

1924

=


=weight 1length 2

weight 2length 1

324

= w6

48 = w Simplify.

Set up a proportion using the information. Let w represent Kaylee’s weight.

Kaylee weighs 48 lbs.


5-3 Proportions

30 ∙ 225 = 45x Find the cross products.

Divide both sides by 45.45x45

675045

=

3045

= x225

150 = x Simplify.

It will take 150 minutes to complete the job. Nate has already spent 30 minutes, so it will take him 150 – 30 = 120 more minutes to finish the job.

Nate has 225 envelopes to prepare for mailing. He takes 30 minutes to prepare 45 envelopes. If he continues at the same rate, how many more minutes until he has completed the job?

Additional Example 4: Business Application

Set up the proportion.

Let x represent the number of minutes it takes to complete the job.


5-3 Proportions

21 ∙ 160 = 24m Find the cross products.

Divide both sides by 24.24m24

336024

=

2124

= m160

140 = m Simplify.

It will take 140 minutes to complete the job. Nemo has already spent 21 minutes, so it will take him 140 – 21 = 119 more minutes to finish the job.

Nemo has to make 160 muffins for the bake sale. He takes 21 minutes to make 24 muffins. If he continues at the same rate, how many more minutes until he has completed the job?


Set up the proportion.

Let m represent the number of minutes it takes to complete the job.


5-3 ProportionsLesson Quiz: Part I


4842 =? 16

141. 40

15 =? 34

2.

3. The ratio of violins to violas in an orchestra is 5:3. t The orchestra has 9 viola players. How many t violinists are in the orchestra?

yes no

15

0.5 ft

4. Two weights are balanced on a fulcrum. If a 6 lb weight is positioned 1.5 ft from the fulcrum, at what distance from the fulcrum must an 18 lb weight be placed to keep the weights balanced?


5-3 Proportions

Lesson Quiz: Part II

5. An elevator travels 342 feet as it goes from the lobby of an office building to the top floor. It takes 7 seconds to travel the first 133 feet. If the elevator travels at the same rate, how much longer does it take to reach the top floor?11 s

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