CorrectionKey=A Proportions and Percent MODULE 5

Proportions and Percent

? ESSENTIAL QUESTION How can you use proportions and percent to solve real-world problems?

5 MODULE

LESSON 5.1

Percent Increase and Decrease

7.RP.3

LESSON 5.2

Rewriting Percent Expressions

7.RP.3, 7.EE.2, 7.EE.3

LESSON 5.3

Applications of Percent

7.RP.3, 7.EE.3

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Real-World Video

A store may have a sale with deep discounts on some items. They can still make a profit because they first markup the wholesale price by as much as 400%, then markdown the retail price.

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137

Are YOU Ready?

Complete these exercises to review skills you will need for this module.

Percents and Decimals

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EXAMPLE 147% = 100% + 47%

=

_1_00_ 100

+

_4_7_ 100

= 1 + 0.47

Write the percent as the sum of 1 whole and a percent remainder. Write the percents as fractions.

Write the fractions as decimals.

= 1.47

Simplify.

Write each percent as a decimal.

1. 22%

2. 75%

3. 6%

4. 189%

Write each decimal as a percent.

5. 0.59

6. 0.98

7. 0.02

8. 1.33

Find the Percent of a Number

EXAMPLE

30% of 45 = ?

30% = 0.30

45 ?_ 0.3 13.5

Write the percent as a decimal. Multiply.

Find the percent of each number.

9. 50% of 64

10. 7% of 30

12. 32% of 62

13. 120% of 4

11. 15% of 160 14. 6% of 1,000

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138 Unit 2

Reading Start-Up

Visualize Vocabulary

Use the words to complete the triangle. Write the review word that fits the description in each section of the triangle.

a statement that two ratios are equivalent

compares a number to 100

Vocabulary

Review Words

proportion (proporci?n) percent (porcentaje)

rate (tasa) ratio (raz?n)

unit rate (tasa unitaria)

Preview Words

percent decrease (porcentaje de disminuci?n) percent increase (porcentaje de aumento) principal (capital) simple interest (inter?s simple)

a comparison of two numbers by division

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Understand Vocabulary

Complete the sentences using the preview words.

1. A fixed percent of the principal is

.

2. The original amount of money deposited or borrowed is the

.

3. A

is the amount of increase divided by the original amount.

Active Reading

Tri-Fold Before beginning the module, create a tri-fold to help you learn the concepts and vocabulary in this module. Fold the paper into three sections. Label the columns "What I Know," "What I Need to Know," and "What I Learned." Complete the first two columns before you read. After studying the module, complete the third.

Module 5 139

GETTING READY FOR

Proportions and Percent

Understanding the Standards and the vocabulary terms in the Standards will help you know exactly what you are expected to learn in this module.

7.RP.3

Use proportional relationships to solve multistep ratio and percent problems.

Key Vocabulary

proportion (proporci?n) An equation that states that two ratios are equivalent.

ratio (raz?n) A comparison of two quantities by division.

percent (porcentaje) A ratio that compares a part to the whole using 100.

What It Means to You

You will use proportions to solve problems involving ratio and percent.

EXAMPLE 7.RP.3

Find the amount of sales tax if the sales tax rate is 5% and the cost of the item is $40.

5%

=

__5 _ 100

=

_1_ 20

Multiply

_1_ 20

times

the

cost

to

find

the

sales

tax.

_1_ 20

?

40

=

2

The sales tax is $2.

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7.EE.2

Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.

Key Vocabulary

expression (expresi?n) A mathematical phrase containing variables, constants and operation symbols.

What It Means to You

You will find helpful ways to rewrite an expression in an equivalent form.

EXAMPLE 7.EE.2

A store advertises that all bicycle helmets will be sold at 10% off the regular price. Find two expressions that represent the value of the sale price p for the helmets that are on sale. Sale price = original price minus 10% of the price

= p - 0.10p

Equivalently,

p - 0.10p = p(1 - 0.10) = 0.90p

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140 Unit 2

5.1 LESSON Percent Increase and Decrease

7.RP.3

Use proportional relationships to solve multistep ratio and percent problems.

? ESSENTIAL QUESTION How do you use percents to describe change?

Finding Percent Increase

Percents can be used to describe how an amount changes.

Percent

Change

=

A__m_o_u_n_t_o_f _C_h_a_n_g_e Original Amount

The change may be an increase or a decrease. Percent increase describes how much a quantity increases in comparison to the original amount.

EXAMPLE 1

7.RP.3

Amber got a raise, and her hourly wage increased from $8 to $9.50. What is the percent increase?

STEP 1 Find the amount of change.

Amount of Change = Greater Value - Lesser Value

= 9.50 - 8.00 Substitute values.

= 1.50

Subtract.

STEP 2 Find the percent increase. Round to the nearest percent.

Percent

Change

=

A__m_o_u_n_t_o_f _C_h_a_n_g_e Original Amount

=

_1_.5_0_ 8.00

= 0.1875

Substitute values. Divide.

19%

Write as a percent and round.

Reflect

1. What does a 100% increase mean?

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YOUR TURN 2. The price of a pair of shoes increases from $52 to $64. What is the percent increase to the nearest percent?

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Lesson 5.1 141

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My Notes

Finding Percent Decrease

When the change in the amount decreases, you can use a similar approach to find percent decrease. Percent decrease describes how much a quantity decreases in comparison to the original amount.

EXAMPLE 2

7.RP.3

David moved from a house that is 89 miles away from his workplace to a house that is 51 miles away from his workplace. What is the percent decrease in the distance from his home to his workplace?

STEP 1

Find the amount of change.

Amount of Change = Greater Value - Lesser Value

= 89 - 51 Substitute values.

= 38

Subtract.

STEP 2 Find the percent decrease. Round to the nearest percent.

Percent

Change

=

A__m_o_u_n_t_o_f _C_h_a_n_g_e Original Amount

=

_3_8 89

0.427

Substitute values. Divide.

= 43%

Write as a percent and round.

Reflect

Math Talk

Mathematical Practices

3. Critique Reasoning David considered moving even closer to his workplace. He claims that if he had done so, the percent of decrease would have been more than 100%. Is David correct? Explain your reasoning.

How is finding percent decrease the same as finding

percent increase? How is it different?

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142 Unit 2

YOUR TURN 4. The number of students in a chess club decreased from 18 to 12. What is the percent decrease? Round to the nearest percent. 5. Officer Brimberry wrote 16 tickets for traffic violations last week, but only 10 tickets this week. What is the percent decrease?

Using Percent of Change

Given an original amount and a percent increase or decrease, you can use the percent of change to find the new amount.

EXAMPLE 3

7.RP.3

The grizzly bear population in Yellowstone National Park in 1970 was about 270. Over the next 35 years, it increased by about 115%. What was the population in 2005?

STEP 1

Find the amount of change. 1.15 ? 270 = 310.5 Find 115% of 270. Write 115% as a decimal.

311 Round to the nearest whole number.

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STEP 2

Find the new amount.

New Amount = Original Amount + Amount of Change

= 270 + 311 Substitute values.

= 581

Add.

The population in 2005 was about 581 grizzly bears.

Add the amount of change because the population increased.

Reflect

6. Why will the percent of change always be represented by a positive number?

7. Draw Conclusions If an amount of $100 in a savings account increases by 10%, then increases by 10% again, is that the same as increasing by 20%? Explain.

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YOUR TURN

A TV has an original price of $499. Find the new price after the given percent of change.

8. 10% increase

9. 30% decrease

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Lesson 5.1 143

Guided Practice

Find each percent increase. Round to the nearest percent. (Example 1)

1. From $5 to $8

2. From 20 students to 30 students

3. From 86 books to 150 books

4. From $3.49 to $3.89

5. From 13 friends to 14 friends

6. From 5 miles to 16 miles

7. Nathan usually drinks 36 ounces of water per day. He read that he should drink 64 ounces of water per day. If he starts drinking 64 ounces, what is the percent increase? Round to the nearest percent. (Example 1)

Find each percent decrease. Round to the nearest percent. (Example 2)

8. From $80 to $64

9. From 95 ?F to 68 ?F

10. From 90 points to 45 points

11. From 145 pounds to 132 pounds

12. From 64 photos to 21 photos

13. From 16 bagels to 0 bagels

14. Over the summer, Jackie played video games 3 hours per day. When school began in the fall, she was only allowed to play video games for half an hour per day. What is the percent decrease? Round to the nearest percent. (Example 2)

Find the new amount given the original amount and the percent of change. (Example 3)

15. $9; 10% increase

16. 48 cookies; 25% decrease

17. 340 pages; 20% decrease

18. 28 members; 50% increase

19. $29,000; 4% decrease

20. 810 songs; 130% increase

21. Adam currently runs about 20 miles per week, and he wants to increase his weekly mileage by 30%. How many miles will Adam run per week? (Example 3)

? ESSENTIAL QUESTION CHECK-IN

22. What process do you use to find the percent change of a quantity?

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144 Unit 2

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