# Proportions and Percent

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Real-World Video

### ESSENTIAL QUESTION

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## 5

### Percent

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### solve real-world problems?

Math On the Spot

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.

LESSON 5.1

### Decrease

7.RP.3 LESSON 5.2

### Expressions

7.RP.3, 7.EE.2, 7.EE.3 LESSON 5.3

### Percent

7.RP.3, 7.EE.3

You can use percent and proportions to find the amount by which real-world quantities have increased or decreased.

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### Are

Personal Math Trainer Online Practice and Help my.hrw.com Completetheseexercisestoreviewskillsyou will need

forthismodule.

Percents and Decimals EXAMPLE 147% = 100% + 47%

= 100___ 100 + ___ 10047

= 1 + 0.47 = 1.47

1. 22% 2. 75% 3. 6% 4. 189%

Writeeachdecimalasa 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 4 5 ×0.3 _ 13.5

Findthe percentof each number.

9. 50% of64 10. 7% of30 11. 15% of160 12. 32% of62 13. 120% of 4 14. 6% of1,000

Write the percent as a decimal. Multiply.

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

Write the percents as fractions. Write the fractions as decimals. Simplify. 32 59% 0.22 19.84 2.1 98% 0.75 24 2% 0.06 133% 1.89 60 4.8 Unit 2 138 © H ough to n Miff l in H ar cour t Pu b l ishing Compan y Math Trainer Online Assessment and Intervention Personal my.hrw.com 1 2 3 Response to Intervention 1 2 3 Response to Intervention Professional Development

### PROFESSIONAL DEVELOPMENT VIDEO

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Author Juli Dixon models successful teaching practices as she explores percent problems in an actual seventh-grade classroom.

Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills.

Intervention Enrichment

Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.

Online and Print Resources

Skills Intervention worksheets • Skill 30 Percents and

Decimals

• Skill 46 Find the Percent of a Number

Differentiated Instruction • Challenge worksheets

PRE-AP

Extend the Math PRE-AP

Lesson Activities in TE

### Real-World Video Viewing Guide

After students have watched the video, discuss the following: What is a discount?

How is the amount of the discount calculated? Multiply the original price by the discount expressed as a fraction or a decimal.

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### Start-Up

Tri-FoldBeforebeginningthemodule, create atri-foldtohelpyou learntheconceptsand

vocabularyinthismodule. Foldthepaperinto threesections. Label thecolumns “What I Know,” “What I Needto Know,” and “What I Learned.”

Visualize Vocabulary

Usethewordstocompletethetriangle. Writethereview wordthat fitsthedescription in eachsection of thetriangle.

Understand Vocabulary

Completethesentences using the previewwords.

1.Afixedpercentoftheprincipal is .

2.Theoriginal amountofmoneydepositedorborrowedisthe .

3.A istheamountofincreasedividedbytheoriginal amount. astatementthat

tworatiosareequivalent

Vocabulary Review Words ✔ proportion (proporción) ✔ percent (porcentaje) rate (tasa) ✔ ratio (razón) unit rate (tasa unitaria) Preview Words percent decrease

(porcen-taje de disminución) percent increase

(porcen-taje de aumento) principal (capital) simple interest (interés

simple)

comparesanumberto 100

acomparisonoftwonumbersbydivision

proportion percent ratio simpleinterest principal percentincrease 139 Module 5 © H ough to n Miff l in H ar cour t Pu b l ishing Compan y

Have students complete the activities on this page by working alone or with others.

### Strategies for English Learners

Each lesson in the TE contains specific strategies to help English Learners of all levels succeed.

Emerging: Students at this level typically progress very quickly, learning to use English for immediate needs as well as beginning to understand and use academic vocabulary and other features of academic language.

Expanding: Students at this level are challenged to increase their English skills in more contexts, and learn a greater variety of vocabulary and linguistic structures, applying their growing language skills in more sophisticated ways appropriate to their age and grade level.

Bridging: Students at this level continue to learn and apply a range of high-level English language skills in a wide variety of contexts, includ-ing comprehension and production of highly technical texts.

Integrating Language Arts

Students can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary.

### EL

After Students will:

solve real-world problems using percent

### In this module

Students represent and solve problems involving proportional relationships:

solve problems involving percent increase, percent decrease, and percent of change

solve markup and markdown problems

use percents to find sales tax, tips, total cost, simple interest

Before

Students understand proportional relationships:

convert units within a measurement system solve real-world problems

involving percent

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What It Means to You

What It Means to You

You will useproportionstosolveproblemsinvolvingratio 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.

Findtheamountofsalestax ifthesalestax rateis5% andthe costoftheitemis \$40.

5% = ___ 5 100 = __ 201

Multiply __ 1

20 timesthecosttofindthesalestax.

1

__ 20 × 40 = 2

Thesalestax is \$2.

Astoreadvertisesthatall bicyclehelmetswill besoldat 10% off theregularprice. Findtwoexpressionsthatrepresentthe valueof thesalepricepforthehelmetsthatareonsale.

Saleprice = original priceminus 10% oftheprice

= p - 0.10p

Equivalently,

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

### Proportions and Percent

Useproportional relationships tosolvemultistepratioand percentproblems.

Key Vocabulary

proportion (proporción) An equation that states that two ratios are equivalent.

ratio (razón) A comparison of two quantities by division.

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

Understandthatrewritingan expressionindifferentformsina problemcontextcanshed light ontheproblemandhowthe quantitiesinitarerelated.

Key Vocabulary

expression (expresión) A mathematical phrase containing variables, constants and operation symbols.

EXAMPLE 7.RP.3 EXAMPLE 7.EE.2 Visit my.hrw.com to see all CA Common Core Standards explained. 7.EE.2 7.RP.3 Unit 2 140 © H ough to n Miff l in H ar cour t Pu b l ishing Compan y • I mage Cr edits : © H emer a T echno l ogies /Al am y I mages

### Proportions and Percent

Use the examples on the page to help students know exactly what they are expected to learn in this module.

my.hrw.com Go online to see a complete unpacking of the CA Common Core Standards.

### Standards

Content Areas

Ratios and Proportional Relationships—7.RP

Analyze proportional relationships and use them to solve real-world and mathematical problems.

Expressions and Equations—7.EE

Cluster Use properties of operations to generate equivalent expressions.

### ore Standards

Lesson

5.1

Lesson 5.2

Lesson 5.3

7.RP.3 Recognize and represent proportional relationships between quantities.

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.

7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.

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80% 50% The ﬁrst decimal square shows 30% more than the second decimal square.

### Content Objective

Students will learn to use percents to describe change.

### Language Objective

Students will show how to use percents to describe change.

### Building Background

Visualize Math Have students work with a partner to shade

in two decimal squares each representing a different percent. Then have them compare the percents.

### Learning Progressions

In this lesson, students continue to build their understanding of percents. They will use percent to describe change as percent increase and percent decrease. Some key understandings for students are the following:

Percent change (increase or decrease) is always the amount of change divided by the original amount. Percent increase describes how much a quantity

increases in comparison to the original amount. Percent decrease describes how much a quantity

decreases in comparison to the original amount. The original amount and the percent of change can be

used to determine the new amount.

The concepts of percent increase and percent decrease will be used to solve a variety of real-world problems, such as problems involving price markups and markdowns.

### Cluster Connections

This lesson provides an excellent opportunity to connect ideas in this cluster: Analyze proportional relationships and use them to solve real-world and mathematical problems. Give students the following prompt: “One week a store

decreased the price of potatoes by 25%. The next week the price was increased by 25%. The original price of 5 pounds of

potatoes was \$4.00. How does the final price compare with the original price?” Have students justify their answer.

The final price was less. 4 × 0.25 = 1.00, 4 - 1 = 3. The first week the price was \$3.

Then 3 × 0.25 = 0.75, 3 + 0.75 = 3.75. The final price was \$3.75, which is less than the original price, \$4.00.

Focus | Coherence | Rigor

### ore Standards

7.RP.3 Use proportional relationships to solve multistep ratio and percent problems.

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Math Talk

### EL

percent – The word percent may be familiar to speakers of Spanish and other Latin-based languages. The root word percent means one part out of each hundred. Point out that one cent is another word for one penny and that there are 100 pennies in a US dollar. In Spanish the meaning of the word for percent

porciento is much more evident because the word

ciento is also the word for hundred.

### Multiple Meaning Words

change – Explain to students that the word change

can have different meanings in mathematics. When working with money, change can indicate an amount of money returned from a transaction, or it can indicate the coins. In this lesson, percent change

describes the percent of increase or decrease in an amount compared to the original amount. Review with students the words increase and decrease.

### Leveled Strategies for English Learners

Emerging Have students select a problem from this lesson and demonstrate how to find the

percentage increase or the percentage decrease.

Expanding Pair students and have them select a problem from the lesson and explain it to each

other. Provide a sentence frame:

This problem is an example of percentage increase/decrease because ______________.

Bridging Have students write in their journal when they think showing a percent increase, rather

than an actual amount of increase, can be more useful.

Model for English learners how to begin their responses with a sentence frame.

Finding percentincrease and finding percentdecrease are alikebecause ______________. Finding percentincrease and finding percentdecrease are not alike

because ______________.

### alifornia ELD Standards

Emerging 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual cues to determine the meaning of unknown and multiple-meaning words on familiar topics.

Expanding 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual cues to determine the meaning of unknown and multiple-meaning words on familiar and new topics.

Bridging 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual cues to determine the meaning, including figurative and connotative meanings, of unknown and multiple-meaning words on a variety of new topics.

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### ESSENTIAL QUESTION

How do you use percents to describe change? Sample answer: Dividing the amount of the change by the original amount results in a percent increase or decrease.

Motivate the Lesson

Ask: What does it mean to have a 100% increase in something? Begin the lesson to find out.

### Explore

Motivate the Lesson

A pen costs \$1, and a jacket costs \$199; both prices increase by \$1. Have students discuss how they preceive both increases, and use this opportunity to introduce relative increases.

### EXAMPLE 1

Focus on Communication

Make sure students can express in their own words that a percent increase is always the amount of change divided by the original amount.

Questioning Strategies Mathematical Practices

How do you find the amount of change? Subtract the lesser value from the greater value.

Focus on Technology Mathematical Practices

If students are using a calculator, make sure parentheses are inputted to find (64 - 52) ÷ 52.

Check for Understanding

Ask: If you were not told a situation is a percent increase, how could you recognize that it is? The new amount is greater than the original amount.

### EXAMPLE 2

Questioning Strategies Mathematical Practices

How could you find 50% of 89? 50% of a number is half the number. 89 ÷ 2 is 44.5. Avoid Common Errors

Remind students that when changing a fraction to a decimal, the top number is divided by the bottom number. For example, __ 3889 means 38 ÷ 89 or 89 ⟌ ⎯ 38 .

Focus on Modeling Mathematical Practices

Have students show symbolically the connection between the original amount, 18, the new amount, 12, the amount of change, 6, and the percent decrease, 33%.

### Percent Increase and Decrease

Interactive Whiteboard

Interactive example available online ADDITIONAL EXAMPLE 1

The number of people signed up for a bus trip increased from 32 to 45. What is the percent increase? Round to the nearest percent. 41%

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Interactive Whiteboard

Interactive example available online ADDITIONAL EXAMPLE 2

The regular price of a scooter is \$65.50. It is on sale for \$52.40. What is the percent decrease from the regular price to the sale price of the scooter?

20%

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### dards

Thestudentisexpectedto:

Ratio and Proportional Relationships—7.RP.3

Useproportional relationshipstosolvemultistepratioand percentproblems.

Mathematical Practices

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My Notes Math Trainer Online Practice and Help Personal my.hrw.com

Math On the Spot

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### Finding Percent Decrease

Whenthechangeintheamountdecreases, you can useasimilarapproach tofindpercentdecrease.Percent decreasedescribeshowmuchaquantity decreasesincomparisontotheoriginal amount.

Davidmoved fromahousethatis 89 milesaway fromhisworkplaceto ahousethatis 51 milesaway fromhisworkplace. Whatisthe percent decreasein thedistance fromhishometohisworkplace?

EXAMPLE 2

Findtheamountofchange.

AmountofChange = Greater Value - Lesser Value

= 89 - 51 = 38

Findthepercentdecrease. Roundtothenearestpercent. PercentChange = Amount of Change______________ Original Amount

= 38__ 89

≈ 0.427 = 43%

Reflect

STEP 1

STEP 2

4. Thenumberofstudentsinachessclubdecreasedfrom 18 to 12. What is thepercentdecrease? Roundtothenearestpercent. 5. Officer Brimberrywrote 16 ticketsfortraffic violations lastweek, but

only 10 ticketsthisweek. Whatisthepercentdecrease? YOUR TURN

How is finding percent decrease the same as finding

percent increase? How is it different?

Itsthesamebecause you subtractthe

lesser valuefrom thegreater value tofindtheamount ofchangeandthen dividetheamount ofchangebythe original amountto findpercentchange.

Itsdifferentbecause theoriginal amount isgreaterthanthe newquantity.

Math TalkMathematical Practices

7.RP.3

Substitute values. Subtract.

Substitute values. Divide.

Write as a percent and round.

No; The leastdistanceDavidcould livefromhis workplaceis 0 miles, whichcorrespondstoa 100%

33% 37.5% © H ough to n Miff l in H ar cour t Pu b l ishing Compan y Unit 2 142

Howdoyou usepercentstodescribechange?

### ?

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### Finding Percent Increase

Percentscanbe usedtodescribehowanamountchanges. PercentChange = Amount of Change______________ Original Amount

Amber gotaraise, andherhourlywageincreased from \$8 to \$9.50. Whatis the percentincrease?

Findtheamountofchange.

AmountofChange = Greater Value - Lesser Value

= 9.50 - 8.00 = 1.50

Findthepercentincrease. Roundtothenearestpercent. PercentChange = Amount of Change______________ Original Amount

= ____ 18.5.000

= 0.1875

≈ 19%

Reflect

1. Whatdoesa 100% increasemean?

EXAMPL EXAMPLE 1

STEP 1

STEP 2

L E S SO N

### and Decrease

ESSENTIAL QUESTION

2. Thepriceofapairofshoesincreasesfrom \$52 to \$64. Whatisthe percentincreasetothenearestpercent? YOUR TURN

7.RP.3

7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. Substitute values. Subtract. Substitute values. Divide. Write as apercent and round.

Theamountofchangeisequal totheoriginal amount;

the valuedoubles.

23%

141

Lesson 5.1

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MathBackground

The percent of change compares the amount of change to the original amount. When there is a series of percent changes, the original amount changes with each additional percent increase or decrease. For example, the percent increase from 10 to 15, where 10 is the original amount, is a 50% increase

### (

_____ 1510-10 = __ 105 = 50%

### )

, but the percent decrease of 15 back to 10 is a 33.3% decrease

### (

_____ 1515-10 = __ 155 = 33. _3 %

### )

because 15 is now the original amount.

IntegrateMathematical Practices MP.2

This lesson provides an opportunity to address this Mathematical Practice standard. It calls for students to create and use representations to organize, record, and communicate mathemati-cal ideas. Students use verbal equations to model a relationship among the percent increase or decrease, the amount of change, and the original amount. Students use these equations

to then write numerical equations to find the percent of change.

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### EXAMPLE 3

Engage with the Whiteboard

Cover up the solution and have students read the Example a couple of times. Then invite a student to circle all the information needed to solve the problem and to draw a line through any extraneous information.

Questioning Strategies Mathematical Practices

Why is 1.15 used as a factor for 115%? 1.15 is the decimal equivalent for 115%. To multiply by a percent, the percent must be represented by either a decimal or a fraction.

How do you know whether to add or subtract the amount of change? Since the population increased, the amount of change is added to the original amount.

Focus on Critical Thinking Mathematical Practices

Be sure students understand how to change any percent to a decimal. Remind them that percents less than 100% will equal decimals less than 1.

Avoid Common Errors

When solving Exercises 8 and 9, students may skip Step 2 as shown in Example 3. Remind them that the new amount for a percent increase is found by adding the original amount to the amount of change. The new amount for a percent decrease is found by subtracting the amount of change from the original amount.

### Elaborate

Summarizethe Lesson

Ask: How would you explain percent increase and percent decrease? Percent increase is a ratio of the amount of the increase to the original amount expressed as a percent. Percent decrease is a ratio of the amount of the decrease to the original amount expressed as a percent.

### GUIDED PRACTICE

Engage with the Whiteboard

In the space under each of Exercises 1–6, invite volunteers to write the original amount (OM), the amount of change (AC), and the ratio they will simplify to find the percent increase or decrease (PI or PD). So, for Exercise 1 students would write OM = 5; AC = 8 - 5 or 3; PI = __ 35 .

Avoid Common Errors

Exercise 6 Remind students to use number sense to check their answers for reasonableness. 16 is more than 3 × 5, so the percent will be more than 200%.

Exercise 14 Students might question whether the 3 hours be changed to minutes or the

half hour be treated as a fraction or a decimal. Either approach will yield the correct answer. However, the math is much easier if the calculation is performed using hours as the unit of measure.

Interactive Whiteboard

Interactive example available online ADDITIONAL EXAMPLE 3

A shoe sales associate earned \$300 in August. In September she earned 8% more than she did in August. How much did she earn in September? \$324

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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)

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

CHECK-IN ESSENTIAL QUESTION

### ?

60% 74% 8% 20% 78% \$9.90 272 pages \$27,840 26 miles

Divide the amount of change in the quantity by the original amount, then express the answer as a percent.

36 cookies 42 members 1,863 songs 50% 67% 28% 9% 100% 50% 11% 220% 83% Unit 2 144 © Hough ton Mifflin Har cour t P ublishing Compan y

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### 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.

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?

Find the amount of change. 1.15 × 270 = 310.5

≈ 311 Find the new amount.

New Amount = Original Amount + Amount of Change

= 270 + 311

= 581

The population in 2005 was about 581 grizzly bears.

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.

EXAMPL EXAMPLE 3

STEP 1

STEP 2

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

8. 10% increase 9. 30% decrease

7.RP.3

Find 115% of 270. Write 115% as a decimal. Round to the nearest whole number.

Add the amount of change because the population increased.

No. An increase of 10% gives a balance of \$110. Another 10% increase would give a balance of \$121. One increase of 20% would give a balance of \$120.

\$548.90 \$349.30

Sample answer: The amount of change is equal to the greater value minus the lesser value, which is always positive. 143 Lesson 5.1 © Hough ton M ifflin Har cour t P ublishing Compan y • Image Cr edits: ©C orbis

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Cooperative Learning

Have students work in pairs to solve percent increase and decrease problems. Start with a problem, and have each person complete one step in the process. Have students exchange roles so each person has a chance to complete each step at least once. This helps emphasize that finding percent increase or decrease is a multi-step process.

Critical Thinking

Ask students to think about percent increase and decrease in the context of integers. For example, if a bank account increases from

-\$100 to \$100, can you use the formula to calculate percent increase? Does the answer make sense? The formula gives a percent increase of -200% in this context. This percent doesn’t make much intuitive sense, so percent increase may not be a useful tool for

understanding increases from negative to positive.

Differentiated Instruction includes:

Success for English Learners EL

Reteach

Challenge PRE-AP

### CorrectionKey=B

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Math Trainer Online Assessment and Intervention Personal Online homework assignment available my.hrw.com

### T PRACTICE

Concepts & Skills Practice

Example 1

Finding Percent Increase

Exercises 1–7, 23, 25

Example 2

Finding Percent Decrease

Exercises 8–14, 24, 25

Example3

Using Percent of Change

Exercises 15–21, 26

Additional Resources Differentiated Instruction includes: Leveled Practice Worksheets

Lesson Quiz available online

5.1 LESSON QUIZ

Find each percent increase or decrease to the nearest percent. 1. from 14 books to 40 books

2. from 72 points to 50 points

Find the new amount given the original amount and the percent of change.

3. \$12; 20% increase

4. 36 grams; 45% decrease

5. If 48 eggs are used in the cafeteria today but the number expected to be used tomorrow is 30% less than that, how many eggs are expected to be used tomorrow?

6. Priscilla currently reads 10 pages in her book each night. She wants to increase the number of pages by 30%. How many pages will Priscilla read each night after the increase?

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Focus | Coherence | Rigor

7.RP.3

Exercise Depth of Knowledge (D.O.K.) Mathematical Practices

23 2 Skills/Concepts MP.2 Reasoning

24 2 Skills/Concepts MP.2 Reasoning

25 2 Skills/Concepts MP.4 Modeling

26 4 Extended Thinking MP.7 Using Structure

27 3 Strategic Thinking MP.7 Using Structure

28 3 Strategic Thinking MP.3 Logic

29 3 Strategic Thinking MP.4 Modeling

7.RP.3 Answers 1. 186% increase 2. 31% decrease 3. \$14.40 4. 19.8 g 5. 34 6. 13

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Work Area 26. Percent error calculations are used to determine how close to the true

values, or how accurate, experimental values really are. The formula is similar to finding percent of change.

Percent Error = |__________________________ Experimental Value Actual Value- Actual Value| × 100

In chemistry class, Charlie records the volume of a liquid as 13.3 milliliters. The actual volume is 13.6 milliliters. What is his percent error? Round to the nearest percent.

27. Look for a Pattern Leroi and Sylvia both put \$100 in a savings account. Leroi decides he will put in an additional \$10 each week. Sylvia decides to put in an additional 10% of the amount in the account each week.

a. Who has more money after the first additional deposit? Explain.

b. Who has more money after the second additional deposit? Explain.

c. How do you think the amounts in the two accounts will compare after a month? A year?

28. Critical Thinking Suppose an amount increases by 100%, then decreases by 100%. Find the final amount. Would the situation change if the original increase was 150%? Explain your reasoning.

29. Look for a Pattern Ariel deposited \$100 into a bank account. Each Friday she will withdraw 10% of the money in the account to spend. Ariel thinks her account will be empty after 10 withdrawals. Do you agree? Explain.

FOCUS ON HIGHER ORDER THINKING

They have the same. \$100 + \$10 = \$110 and \$100 + 10%(\$100) = \$110.

Sylvia has more. Leroi has \$110 + \$10 = \$120 and Sylvia has \$110 + 10%(\$110) = \$121.

Because Sylvia will have more after the second additional deposit and she will be depositing increasing amounts, she will always have more in her account.

The final amount is always 0. A 100% decrease of any amount would leave 0.

No. Only the first withdrawal is \$10. Each withdrawal after that is less than \$10 because it is 10% of the remaining balance. There will be money left after 10 withdrawals.

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Atlantic Basin Hurricanes

Year Hurricanes Personal Math Trainer Online Practice and Help my.hrw.com

Name Class Date

### 5.1

23. Complete the table.

Item Original Price New Price Percent Change Increase or Decrease

Bike \$110 \$96 Scooter \$45 \$56 Tennis

Racket \$79 5% Increase Skis \$580 25% Decrease

24. Multiple Representations The bar graph shows the number of hurricanes in the Atlantic Basin from 2006–2011.

a. Find the amount of change and the percent of decrease in the number of hurricanes from 2008 to 2009 and from 2010 to 2011. Compare the amounts of change and percents of decrease.

b. Between which two years was the percent of change the greatest? What was the percent of change during that period?

25. Represent Real-World Problems Cheese sticks that were previously priced at “5 for \$1” are now “4 for \$1”. Find each percent of change and show your work.

a. Find the percent decrease in the number of cheese sticks you can buy for \$1.

b. Find the percent increase in the price per cheese stick.

7.RP.3

5; 5; 62.5%; 41.7%; the amount of change is the same, but the percent of change is less from 2010 to 2011.

2009 and 2010; 300% increase

Amount of change = 1; percent decrease = 1 _ 5 = 20%

\$1.00

____ 5 = \$0.20 each; \$1.00____ 4 = \$0.25 each. Amount of change = \$0.05; percent increase = 0.05 ____ 0.20 = 25%

\$82.95 ≈ 13% ≈ 24% \$435 Decrease Increase 145 Lesson 5.1 © Hough ton Mifflin Har cour t P ublishing Compan y

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### EXTEND THE MATH

PRE-AP

Activity On a grid draw a 4 × 4 square. Use the square and what you have learned about percent increase and percent decrease to determine what happens to the area of the square when the sides are increased by 50%. State by what percent the area increases. Then make a third square by decreasing the sides of the second square by 50%. State by what percent the area decreases. By what percent would you have had to change the sides of the 4 × 4 square to get the third square?

The area of the 4 × 4 square is 16 units2.

Increasing the sides by 50% makes a 6 × 6 square with an area of 36 units2. The area of the first

square is increased by 125%. Decreasing the second square’s sides by 50% makes a 3 × 3 square with an area of 9 units2. The area of the

second square is decreased by 75%. The sides of the original square could have been decreased by 25% to get the third square.

Activity available online my.hrw.com

### CorrectionKey=B

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100% 300 75 75 75 75 75 75 25% 25% 25% 25% 25%

### Content Objective

Students will learn to rewrite percent expressions to solve markup and markdown problems.

### Language Objective

Students will demonstrate and explain how to rewrite expressions to help you solve markup and

markdown problems.

### Building Background

Visualize Math Knowledge Draw the bar model on the

board. Discuss with students what the model shows and how they could use the model to find the missing information. For example, the model shows that the amount 300 is equal to 100%. Four equal parts of the second bar are equal to 100%. So, each part is 300 ÷ 4 = 75. The fifth 25% is another 75. So, the total length of the top bar is 375, which is 125% of 300.

### Learning Progressions

In this lesson, students extend their skill at using percents to solve problems by rewriting expressions for easier computation. Some key understandings for students are the following:

A percent can be written as a decimal or as a fraction. Depending on the problem, one of the two forms may provide a more efficient solution than the other. A markup is an example of a percent increase. The term

markup sometimes refers to a percent increase and sometimes to the amount of the increase. A markup of 20% on \$150 is a markup of \$30.

A markdown is an example of a percent decrease. Concepts related to percent and the use of equivalent expressions will continue to be applied in everyday life and in the study of algebra.

### Cluster Connections

This lesson provides an excellent opportunity to connect ideas in this cluster: Analyze proportional relationships and use them to solve real-world and mathematical problems. Give students the following prompt: “Elisa and Dan each are calculating the sale prices of a tablet device at two different stores. What is the sale price at each store?” Have students complete the table and ask them show a one-step calculation to find each sale price.

Store OrPirgicinae l DiPescounrcentt SalePrice Computer Deals \$280 15% \$238 Today’s Computers \$315 25% \$236.25 280 × 0.85 = 238; 315 × 0.75 = 236.25

Focus | Coherence | Rigor

### ore Standards

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.

7.RP.3 Use proportional relationships to solve multistep ratio and percent problems.

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Math Talk

### EL

increase or decrease – In this lesson, percents are used to solve markup and markdown problems. Point out to English learners the words that cue them to whether the change is increase or decrease. Words that cue an increase: markup (noun), mark up (verb), profit

Words that cue a decrease: markdown (noun), mark down (verb), loss, discount

Word that cues neither an increase nor a decrease:

break even

### Rules and Patterns

co- – Point out to English learners any prefixes, suffixes, etc. to help them figure out the meanings of words in word problems. The prefix co-, meaning with, appears in the word coefficient in this lesson.

Common words with the prefix co- include coworker, co-author, co-star, co-exist, coed. Notice that some words have a hyphen after the prefix co- and while others do not.

### Leveled Strategies for English Learners

Emerging Visual cues, like bar models, can help students at this level of English proficiency

understand an abstract idea or concept. Have students draw and label a bar model to demonstrate how to solve a markup problem.

Expanding Have pairs of students review and discuss the steps in Example 1 of the lesson before

solving one of the word problems in Independent Practice.

Bridging Pair students at this level of English proficiency to discuss and review the steps in

Example 1 of the lesson. Then have them explain the difference between how to solve a markup vs. markdown problem.

To help English learners answer the question posed in Example 1 Math Talk, give them a model to begin their answer with:

It makes sense to write the retail price as the sum of _______ because _______. A good reason for writing the retail price as the sum of _______ is _______.

### alifornia ELD Standards

Emerging 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual cues to determine the meaning of unknown and multiple-meaning words on familiar topics.

Expanding 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual cues to determine the meaning of unknown and multiple-meaning words on familiar and new topics.

Bridging 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual cues to determine the meaning, including figurative and connotative meanings, of unknown and multiple-meaning words on a variety of new topics.

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### ESSENTIAL QUESTION

How can you rewrite expressions to help you solve markup and markdown problems?

Sample answer: Markups are 1 plus a percent of the cost, and markdowns are 1 minus a percent of a price. Either can be rewritten as a single term.

Motivate the Lesson

Ask: Did you ever want to figure out the sale price of an item before you got to the check-out counter? Begin the lesson to find out how to do this.

### Explore

Multiple Representations Mathematical Practices

Explain that a certain pack of gum costs \$1 and that you have 100% of what it costs to buy that gum. Show students four quarters. Explain that you plan to sell the pack of gum to make a profit. You plan a markup of 50%. Ask how much 50% of \$1 is. Show the original cost, four quarters, in one hand and the markup, two quarters, in your other hand. Explain that the retail price is now \$1.50.

### EXAMPLE 1

Focus on Modeling Mathematical Practices

Point out to students that the part of the model in Step 1 labeled as s is equivalent to 1s and represents 100% of the original cost. The part labeled 0.42s shows 42% of the original cost, the amount being added to the original cost, while the entire model represents 142% of the original cost.

Questioning Strategies Mathematical Practices

How could you use a mathematical property to add 1s+ 0.42s? Use the Distributive Property to write 1s+ 0.42s as (1 + 0.42)s.

How could you use the expression to help you determine the retail price of a skateboard that cost the store \$50? Substitute \$50 for s in 1.42s. 1.42 × \$50 = \$71

Avoid Common Errors

Students may think that tripling a cost means a 300% markup. A cost x tripled is 3x. Breaking apart 3x into the cost plus the markup yields x+ 2x. So, you must add 200% of x or 2x to x

to get triple the cost, 3x.

Check for Understanding

Ask: Why will 1.1c work as an expression for a 10% markup, no matter what is being sold? c is a variable that can stand for any original cost. 1.1 is a constant that represents 100% plus 10%.

Interactive Whiteboard

Interactive example available online ADDITIONAL EXAMPLE 1

A shoe store buys a pair of boots from a supplier for b dollars. The store’s manager decides on a markup of 35%. Write an expression for the retail price of a pair of boots. 1.35b my.hrw.com

### dards

Thestudentisexpectedto: Expressions and Equations—7.EE.2

Understandthatrewritinganexpressionindifferent formsinaproblemcontextcanshed lightontheproblem andhow thequantitiesinitarerelated.

Ratio and Proportional Relationships—7.RP.3

Useproportional relationshipstosolvemultistepratioand percentproblems.Also 7.EE.3

Mathematical Practices

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Math On the Spot my.hrw.com Math Trainer Online Practice and Help Personal my.hrw.com Animated Math my.hrw.com p -0.2p p 0.2p

2. Rick buys remote control cars to resell. He applies a markup of 10%.

a. Write two expressions that represent the retail price of the cars.

b. If Rick buys a remote control car for \$28.00, what is his selling price?

3. An exclusive clothing boutique triples the price of the items it purchases for resale.

a. What is the boutique’s markup percent?

b. Write two expressions that represent the retail price of the clothes.

### Calculating Markdowns

An example of a percent decrease is a discount, or markdown. A price after a markdown may be called a sale price. You can also use a bar model to represent the price of an item including the markdown.

A discount store marks down all of its holiday merchandise by 20% off the regular selling price. Find the discounted price of decorations that regularly sell for \$16 and \$23.

Use a bar model.

Draw a bar for the regular price p.

Then draw a bar that shows the discount: 20% of p, or 0.2p.

The difference between these two bars represents the price minus the discount, p- 0.2p. EXAMPLE 2 STEP 1 7.EE.2, 7.RP.3, 7.EE.3 1c+ 0.1c, 1.1c 1c+ 2c; 3c \$30.80 200% Unit 2 148 © Hough ton Mifflin Har cour t P ublishing Compan y

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### ?

Math On the Spot

my.hrw.com s+0.42s 0.42s s ESSENTIAL QUESTION L E S S O N

### Calculating Markups

A markup is one kind of percent increase. You can use a bar model to represent the retail price of an item, that is, the total price including the markup.

To make a profit, stores mark up the prices on the items they sell. A sports store buys skateboards from a supplier for s dollars. What is the retail price for skateboards that the manager buys for \$35 and \$56 after a 42% markup?

Use a bar model.

Draw a bar for the cost of the skateboard s. Then draw a bar that shows the markup: 42% of s, or 0.42s.

These bars together represent the cost plus the markup, s+ 0.42s. Retail price = Original cost + Markup

= s + 0.42s

= 1s + 0.42s

= 1.42s

Use the expression to find the retail price of each skateboard.

s=\$35 Retail price = 1.42(\$35) = \$49.70

s=\$56 Retail price = 1.42(\$56) = \$79.52 Reflect

1. What If? The markup is changed to 34%; how does the expression for the retail price change?

EXAMPL EXAMPLE 1

STEP 1

STEP 2

STEP 3

How can you rewrite expressions to help you solve markup and markdown problems?

Why write the retail price as the sum of two terms?

as one term?

Math Talk

Mathematical Practices

Why write the retail price 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.

Also7.RP.3, 7.EE.3 7.EE.2, 7.RP.3, 7.EE.3 Sample answer: Two terms shows the original cost and the markup. One term allows for quicker calculation.

The expression would change to 1s+ 0.34s or 1.34s.

147 Lesson 5.2 © Hough ton Mifflin Har cour t P ublishing Compan y

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### PROFESSIONAL DEVELOPMENT

Math Background

Not explored in this lesson is a not-so-subtle distinction between the amount used as the basis for a markup and the amount used as the basis for a markdown. As presented in the lesson, a markup is generally based on the cost of an item to the retailer. However, not covered in this lesson, a markdown is generally based on the retail price of an item after a markup has been applied. This means that if an item costing \$100 is marked up 20%, it will retail for \$120. If this item is later placed on sale at a 20% markdown, the sale price is not \$120 - \$20, but \$120 - (20% of

Integrate Mathematical Practices MP.5

This lesson provides an opportunity to address this Mathematical Practice standard. It calls for students to use bar models to model the relationship between a mathematical expression and a real-world context regarding either a markup or a markdown. This gives students the opportunity to read a real-world situation and use that information to write an algebraic expression to represent retail and sale prices. Finally, the students use the expression they write to solve problems regarding markups and

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### LE 2

Questioning Strategies Mathematical Practices

In Step 1, how do you know how much of the 1p bar to shade to show 0.2p? The amount shaded does not need to be a specific amount, just a portion of the bar to represent 0.2p.

How does the bar model for a markdown differ from the bar model for a markup? For a markup, the bar model for the expression is longer than the original cost. For a markdown, the bar model for the expression is shorter than the retail price.

Connect Vocabulary EL

Remind students that both percents in Example 2 are rational numbers. 20% is equivalent to 0.2 or __ 102 , and 80% is equivalent to 0.8 or __ 108 .

### RN

Engagewiththe Whiteboard

Have a student volunteer draw the model in part a. Discuss whether the model needs to be drawn to scale for it to be helpful in solving the problem.

Focuson Math Connections Mathematical Practices

Point out that the Distributive Property also works for subtraction. So, 1b- 0.24b= (1 - 0.24)b= 0.76b.

### Elaborate

Summarizethe Lesson

Ask: How does a bar model showing the expression for a sale price compare to one showing the expression for a retail price? Both show the original price and the percent markup or markdown. The bar model for the retail price shows the percent markup added to the bar model, while the sale price shows the percent markdown subtracted from the bar model.

### CTICE

Engagewiththe Whiteboard

To the right of each row, have students volunteer to write the expression that could be used to find the retail price in Exercises 2–7 and write the expression that could be used to find the sale price for Exercises 8–11 on the write-on lines.

Avoid Common Errors

Exercise 1c Remind students that once \$32 has been substituted for s in the expression 1.35s, they do not need to add \$32 to the value of the expression again.

Exercises 2–7 Remind students that the markup is an amount found by multiplying the cost by the percent markup. The retail price is the cost plus the markup.

Integrating LanguageArts EL

Encourage English learners to ask for clarification on any terms or phrases that they don’t understand.

Interactive Whiteboard

Interactive example available online

A pet store marks down all of its grooming products by 15% off the regular selling price of p. Write an expression for the sale price. 0.85p

my.hrw.com

AnimatedMath Explore Markups and Markdowns

Students discover how markups and markdowns relate to the original cost using virtual manipulatives.

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Guided Practice

1. Dana buys dress shirts from a clothing manufacturer for s dollars each, and then sells the dress shirts in her retail clothing store at a 35% markup. (Example 1)

a. Write the markup as a decimal. b. Write two expressions for the retail price of the dress shirt. c. What is the retail price of a dress shirt that Dana purchased for \$32.00? d. How much was added to the original price of the dress shirt? List the markup and retail price of each item. Round to two decimal places

when necessary. (Example 1)

Item Price Markup % Markup Retail Price

2. Hat \$18 15% 3. Book \$22.50 42% 4. Shirt \$33.75 75% 5. Shoes \$74.99 33% 6. Clock \$48.60 100% 7. Painting \$185.00 125%

Find the sale price of each item. Round to two decimal places when necessary. (Example 2)

8. Original price:\$45.00; Markdown:22% 9. Original price:\$89.00; Markdown:33%

10. Original price:\$23.99; Markdown: 44% 11. Original price:\$279.99, Markdown:75%

12. How can you determine the sale price if you are given the regular price and the percent of markdown?

CHECK-IN ESSENTIAL QUESTION

### ?

0.35s \$2.70 \$9.45 \$25.31 \$24.75 \$48.60 \$231.25 \$20.70 \$31.95 \$59.06 \$99.74 \$97.20 \$416.25 1s+0.35s, 1.35s \$43.20 \$11.20 \$35.10 \$70.00 \$59.63 \$13.43

Write the percent of markdown as a decimal, subtract the product of this decimal and the regular price from the regular price.

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Saleprice = Original price - Markdown

= p - 0.2p

= 1p - 0.2p

= 0.8p

Usetheexpressiontofindthesalepriceofeachdecoration.

p = \$16 Saleprice = 0.8(\$16) = \$12.80

p = \$23 Saleprice = 0.8(\$23) = \$18.40

Reflect

4. Conjecture Comparethesingletermexpressionforretail priceaftera markupfrom Example 1 andthesingletermexpressionforsaleprice afteramarkdownfrom Example 2. Whatdoyou noticeaboutthe coefficientsinthetwoexpressions?

STEP 2

STEP 3

5. Abicycleshopmarksdowneachbicycle’ssellingpricebby 24% for a holidaysale.

a. Drawabarmodel torepresenttheproblem.

1b-0.24b

1b

0.24b

b. Whatisasingletermexpressionforthesaleprice? 6. Janesellspillows. Forasale, shemarksthemdown5%.

a. Writetwoexpressionsthatrepresentthesalepriceofthepillows.

b. Iftheoriginal priceofapillowis \$15.00, whatisthesaleprice? YOUR TURN Isa 20% markupequal toa 20% markdown? Explain. Math Talk Mathematical Practices

Amarkupincludesacoefficientgreaterthan 1 and a markdownincludesacoefficient lessthan 1.

Theamountof a 20% markup anda 20% discountare thesame, but oneisadded andtheother issubtracted. 0.76b 1p− 0.05p, 0.95p \$14.25 149 Lesson5.2 © H ough to n Miff l in H ar cour t Pu b l ishing Compan y Kinesthetic Experience

Have students write expressions that could be used to calculate a 10% increase and a 10% decrease in a distance. Then have each student stand on a start line, toss his/her uniquely decorated cotton ball, measure the distance to the nearest centimeter, and record the data in a table like the one shown below. Then have students use their expressions to calculate a distance that would be 10% more and 10% less than their original distance. Next, have each

student throw his/her cotton ball a second time, trying to throw exactly 10% more or less than their original distance. Finally, have students calculate the percent more or less the cotton ball actually went than their original distance.

Cooperative Learning

Have each student “secretly” think of a percent markup or markdown for a hat. Then on an index card, have each student write a one-term expression that could be used to find the retail price or sale price for the hat. Have students exchange cards and decide if the expression they were just given is a markup or a markdown.

decision is correct. Students then trade the completed cards with a third person who will determine if the bar model and conclusion about the expression are correct.

Success for English Learners EL Reteach

Challenge PRE-AP

### ON

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Math Trainer Online Assessment and Intervention Personal Online homework assignment available my.hrw.com

### CTICE

Concepts & Skills Practice

Example 1 Calculating Markups Exercises 1–7, 15, 16 Example 2 Calculating Markdowns Exercises 8–11, 13–15

AdditionalResources Differentiated Instruction includes: Leveled Practice Worksheets

Lesson Quiz available online

5.2 LESSON QUIZ

Fred buys flags from a manufac-turer for f dollars each and then sells the flags in his store for a 26% markup.

1. Write the markup as a decimal.

2. Write an expression for the retail price of a flag.

3. What is the retail price of a flag for which Fred paid \$40?

4. How much was added to the cost of the flag?

List the sale price of each item. Round to two decimal places when necessary.

5. Original price: \$25; Markdown: 12%

6. Original price: \$16.45; Markdown: 33%

my.hrw.com

Focus | Coherence | Rigor

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

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

Exercise Depth of Knowledge (D.O.K.) Mathematical Practices

13 2 Skills/Concepts MP.4 Modeling

14 2 Skills/Concepts MP.2 Reasoning

15 3 Strategic Thinking MP.7 Using Structure

16 2 Skills/Concepts MP.4 Modeling

17 3 Strategic Thinking MP.6 Precision

18 3 Strategic Thinking MP.8 Patterns

19 2 Skills/Concepts MP.7 Using Structure

20 3 Strategic Thinking MP.4 Modeling

Exercise 18 combines concepts from the California Common Core

cluster “Analyze proportional relationships and use them to solve real-world and mathematical problems.”

Answers 1. 0.26 2. 1f+ 0.26f or 1.26f 3. \$50.40 4. \$10.40 5. \$22 6. \$11.02

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Work Area 16. RepresentReal-WorldProblems Haroldworksatamen’sclothingstore,

whichmarks upitsretail clothingby 27%. Thestorepurchasespantsfor \$74.00, suit jacketsfor \$325.00, anddressshirtsfor \$48.00. Howmuchwill Haroldchargeacustomerfortwopairsofpants, threedressshirts, anda suit jacket?

17. AnalyzeRelationships Yourfamilyneedsasetof 4 tires. Whichofthe followingdealswouldyou prefer? Explain.

(I) Buy 3, getonefree (II) 20% off (III) 1 __ 4 off

18. CritiqueReasoningMargopurchasesbulk teasfromawarehouseand marks upthosepricesby 20% forretail sale. Whenteasgo unsoldfor morethantwomonths, Margomarksdowntheretail priceby 20%. She saysthatsheisbreaking even, thatis, sheisgettingthesamepriceforthe teathatshepaidforit. Isshecorrect? Explain.

19. Problem Solving Gradymarksdownsome \$2.49 pensto \$1.99 foraweek andthenmarksthemback upto \$2.49. Findthepercentofincreaseand thepercentofdecreasetothenearesttenth.Arethepercentsofchange thesameforbothpricechanges? Ifnot, whichisagreaterchange?

20. Perseverein Problem SolvingAtDanielle’sclothingboutique, ifan itemdoesnotsell foreightweeks, shemarksitdownby 15%. Ifitremains unsoldafterthat, shemarksitdownanadditional 5% eachweek until she canno longermakeaprofit. Thenshedonatesittocharity.

FOCUS ON HIGHER ORDER THINKING

Eitherbuy 3, getonefreeor _ 14 off. Eithercasewould

No; sheistakinga loss. Hercostfortheteaist, sothe retail priceis 1.2t. Thediscountedpriceis 0.8 × 1.2t,

or 0.96t, whichis lessthant.

No; firstchange: 20.1% decrease; secondchange: 25.1%

increase. Thesecondpercentchangeisgreater.

11 or 12 weeks; after 11 weeks, thepriceis \$109.32,

after 12 weeks, thepriceis \$103.85, andafterthat Danielledonatesthecoat.

\$783.59

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### 5.2

13. Abookstoremanagermarksdownthepriceofolderhardcoverbooks, whichoriginallysell forbdollars, by 46%.

c. Whatisthesalepriceofahardcoverbook forwhichtheoriginal retail

pricewas \$29.00? d. Ifyou buythebook inpartc, howmuchdoyou savebypayingthe

markeddownby 35%.Matcheach RegularPricetothecorrect SalePrice, ifpossible.Notall salestagsmatchanitem.

RegularPrice \$3.29 SalePrice \$2.01 RegularPrice \$4.19 SalePrice \$2.45 RegularPrice \$2.79 SalePrice \$1.15 RegularPrice \$3.09 SalePrice \$2.72 RegularPrice \$3.77 SalePrice \$2.24 15. CommunicateMathematical Ideas Foreachsituation, giveanexample

thatincludestheoriginal priceandfinal priceaftermarkupormarkdown.

a. Amarkdownthatisgreaterthan 99% but lessthan 100%

b. Amarkdownthatis lessthan 1%

c. Amarkupthatismorethan 200% 7.RP.3, 7.EE.2, 7.EE.3

Sampleanswer: original price: \$100; final price: \$0.50 Sampleanswer: original price: \$100; final price: \$99.50 Sampleanswer: original price: \$100; final price: \$350

1b− 0.46b, 0.54b 0.46b \$15.66 \$13.34 151 Lesson5.2 © Hough

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Activity available online my.hrw.com

### TH

PRE-AP

Activity A shirt is on sale now for \$20. Starting today, a morning sales clerk decreases the price by 30%, and then an afternoon sales clerk increases the price by 20%. This pattern continues for several days. Provided the shirt is never purchased, on which day is the shirt marked down to about 75% off the price it is now?

75% off the current price would make the shirt’s sale price \$5. On the morning of Day 6, the shirt will be priced at \$4.90.

Updating...