Chapter 3 Resource Masters - Math Class - Home · ©Glencoe/McGraw-Hill v Glencoe Algebra 1...

Chapter 3Resource Masters

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks in both English and Spanish.

Study Guide and Intervention Workbook 0-07-827753-1Study Guide and Intervention Workbook (Spanish) 0-07-827754-XSkills Practice Workbook 0-07-827747-7Skills Practice Workbook (Spanish) 0-07-827749-3Practice Workbook 0-07-827748-5Practice Workbook (Spanish) 0-07-827750-7

ANSWERS FOR WORKBOOKS The answers for Chapter 3 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 1. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-827727-2 Algebra 1Chapter 3 Resource Masters

2 3 4 5 6 7 8 9 10 024 11 10 09 08 07 06 05 04 03

Glencoe/McGraw-Hill

© Glencoe/McGraw-Hill iii Glencoe Algebra 1

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Lesson 3-1Study Guide and Intervention . . . . . . . . 137–138Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 139Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Reading to Learn Mathematics . . . . . . . . . . 141Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 142









Chapter 3 AssessmentChapter 3 Test, Form 1 . . . . . . . . . . . . 191–192Chapter 3 Test, Form 2A . . . . . . . . . . . 193–194Chapter 3 Test, Form 2B . . . . . . . . . . . 195–196Chapter 3 Test, Form 2C . . . . . . . . . . . 197–198Chapter 3 Test, Form 2D . . . . . . . . . . . 199–200Chapter 3 Test, Form 3 . . . . . . . . . . . . 201–202Chapter 3 Open-Ended Assessment . . . . . . 203Chapter 3 Vocabulary Test/Review . . . . . . . 204Chapter 3 Quizzes 1 & 2 . . . . . . . . . . . . . . . 205Chapter 3 Quizzes 3 & 4 . . . . . . . . . . . . . . . 206Chapter 3 Mid-Chapter Test . . . . . . . . . . . . 207Chapter 3 Cumulative Review . . . . . . . . . . . 208Chapter 3 Standardized Test Practice . . 209–210Unit 1 Test/Review (Ch. 1–3) . . . . . . . . 211–212

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A39

© Glencoe/McGraw-Hill iv Glencoe Algebra 1

Teacher’s Guide to Using theChapter 3 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 3 Resource Masters includes the core materials neededfor Chapter 3. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theAlgebra 1 TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 3-1.Encourage them to add these pages to theirAlgebra Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Algebra 1 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

© Glencoe/McGraw-Hill v Glencoe Algebra 1

Assessment OptionsThe assessment masters in the Chapter 3Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 1. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 186–187. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

33

© Glencoe/McGraw-Hill vii Glencoe Algebra 1

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 3.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Algebra Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

consecutive integers

kuhn-SEH-kyuh-tihv

defining a variable

dimensional analysis

duh·MEHNCH·nuhl

equivalent equation

ih·KWIHV·luhnt

extremes

formula

identity

means

multi-step equations

(continued on the next page)

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

© Glencoe/McGraw-Hill viii Glencoe Algebra 1

Vocabulary Term Found on Page Definition/Description/Example

number theory

percent of change

percent of decrease

percent of increase

proportion

pruh·POHR·shun

ratio

rate

scale

solve an equation

weighted average

work backward

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

33

Study Guide and InterventionWriting Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

© Glencoe/McGraw-Hill 137 Glencoe Algebra 1

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Write Equations Writing equations is one strategy for solving problems. You can use avariable to represent an unspecified number or measure referred to in a problem. Then youcan write a verbal expression as an algebraic expression.

Translate eachsentence into an equation or aformula.

a. Ten times a number x is equal to2.8 times the difference y minus z.10 ! x " 2.8 ! ( y # z)The equation is 10x " 2.8( y # z).

b. A number m minus 8 is the sameas a number n divided by 2.m # 8 " n $ 2The equation is m # 8 " .

c. The area of a rectangle equals thelength times the width. Translatethis sentence into a formula.Let A " area, ! " length, and w " width.Formula: Area equals length timeswidth.A " ! ! wThe formula for the area of arectangle is A " !w.

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Use the Four-Step Problem-Solving Plan.The population of the United States in 2001was about 284,000,000, and the land area ofthe United States is about 3,500,000 squaremiles. Find the average number of peopleper square mile in the United States.Source: www.census.gov

Step 1 Explore You know that there are284,000,000 people. You want to knowthe number of people per square mile.

Step 2 Plan Write an equation to represent thesituation. Let p represent the number ofpeople per square mile.3,500,000 ! p " 284,000,000

Step 3 Solve 3,500,000 ! p " 284,000,000.3,500,000p " 284,000,000 Divide each side by

p ! 81.14 3,500,000.

There about 81 people per square mile.Step 4 Examine If there are 81 people per

square mile and there are 3,500,000square miles, 81 ! 3,500,000 "283,500,000, or about 284,000,000 people.The answer makes sense.

Example 1Example 1 Example 2Example 2

ExercisesExercises

Translate each sentence into an equation or formula.

1. Three times a number t minus twelve equals forty.

2. One-half of the difference of a and b is 54.

3. Three times the sum of d and 4 is 32.

4. The area A of a circle is the product of & and the radius r squared.

WEIGHT LOSS For Exercises 5–6, use the following information.

Lou wants to lose weight to audition for a part in a play. He weighs 160 pounds now. Hewants to weigh 150 pounds.

5. If p represents the number of pounds he wants to lose, write an equation to representthis situation.

6. How many pounds does he need to lose to reach his goal?


Write Verbal Sentences You can translate equations into verbal sentences.

Translate each equation into a verbal sentence.

a. 4n ! 8 " 12.

4n # 8 " 12Four times n minus eight equals twelve.

b. a2 # b2 " c2

a2 ' b2 " c2

The sum of the squares of a and b is equal to the square of c.


1. 4a # 5 " 23 2. 10 ' k " 4k

3. 6xy " 24 4. x2 ' y2 " 8

5. p ' 3 " 2p 6. b " (h # 1)

7. 100 # 2x " 80 8. 3(g ' h) " 12

9. p2 # 2p " 9 10. C " (F # 32)

11. V " Bh 12. A " hb1%2

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Study Guide and Intervention (continued)

Writing Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

ExampleExample

ExercisesExercises

Skills PracticeWriting Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1


Less

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Translate each sentence into an equation.

1. Two added to three times a number m is the same as 18.

2. Twice a increased by the cube of a equals b.

3. Seven less than the sum of p and q is as much as 6.

4. The sum of x and its square is equal to y times z.

5. Four times the sum of f and g is identical to six times g.

Translate each sentence into a formula.

6. The perimeter P of a square equals four times the length of a side s.

7. The area A of a square is the length of a side s squared.

8. The perimeter P of a triangle is equal to the sum of the lengths of sides a, b, and c.

9. The area A of a circle is pi times the radius r squared.

10. The volume V of a rectangular prism equals the product of the length !, the width w,and the height h.


11. g ' 10 " 3g 12. 2p ' 4q " 20

13. 4(a ' b) " 9a 14. 8 # 6x " 4 ' 2x

15. ( f ' y) " f # 5 16. s2 # n2 " 2b

Write a problem based on the given information.

17. c " cost per pound of plain coffee beans 18. p " cost of dinnerc ' 3 " cost per pound of flavored coffee beans 0.15p " cost of a 15% tip2c ' (c ' 3) " 21 p ' 0.15p " 23

1%2


Translate each sentence into an equation.

1. Fifty-three plus four times c is as much as 21.

2. The sum of five times h and twice g is equal to 23.

3. One fourth the sum of r and ten is identical to r minus 4.

4. Three plus the sum of the squares of w and x is 32.

Translate each sentence into a formula.

5. Degrees Kelvin K equals 273 plus degrees Celsius C.

6. The total cost C of gas is the price p per gallon times the number of gallons g.

7. The sum S of the measures of the angles of a polygon is equal to 180 times the differenceof the number of sides n and 2.


8. q # (4 ' p) " q 9. t ' 2 " t

10. 9(y2 ' x) " 18 11. 2(m # n) " v ' 7

Write a problem based on the given information.

12. a " cost of one adult’s ticket to zoo 13. c " regular cost of one airline ticketa # 4 " cost of one children’s ticket to zoo 0.20c " amount of 20% promotional discount2a ' 4(a # 4) " 38 3(c # 0.20c) " 330

14. GEOGRAPHY About 15% of all federally-owned land in the 48 contiguous states of theUnited States is in Nevada. If F represents the area of federally-owned land in thesestates, and N represents the portion in Nevada, write an equation for this situation.

FITNESS For Exercises 15–17, use the following information.Deanna and Pietra each go for walks around a lake a few times per week. Last week,Deanna walked 7 miles more than Pietra.15. If p represents the number of miles Pietra walked, write an equation that represents the

total number of miles T the two girls walked.

16. If Pietra walked 9 miles during the week, how many miles did Deanna walk?

17. If Pietra walked 11 miles during the week, how many miles did the two girls walktogether?

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Practice Writing Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

Reading to Learn MathematicsWriting Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1


Less

on

3-1

Pre-Activity How are equations used to describe heights?

Read the introduction to Lesson 3-1 at the top of page 120 in your textbook.

Does the equation 305 # s " 154 also represent the situation? Explain.

Reading the Lesson

1. Translate each sentence into an equation.

a. Two times the sum of x and three minus four equals four times x.

b. The difference of k and 3 is two times k divided by five.

2. A 1 oz serving of chips has 140 calories. There are about 14 servings of chips in a bag.How many calories are there in a bag of chips? Write what your solution would be as youuse each step in the Four-Step Problem-Solving Plan.

Explore What do you know?

What do you want to know?

Plan Write an equation.

Solve Solve the problem.

Examine Does your answer make sense?

Helping You Remember

3. If you cannot remember all the steps of the Four-Step Problem-Solving Plan, try toremember the first letters of the first word in each step. Write those letters here withtheir associated words.


Rep-TilesA rep-tile is a figure that can be subdivided into smaller copies of itself. The large figure is similar to the small ones and the small figures are allcongruent.

Show that each figure is a rep-tile by subdividing it into four smaller and similar figures.

1. 2. 3.

4. 5. 6.

Subdivide each rep-tile into nine smaller and similar figures.

7. 8.

9. 10.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

Study Guide and InterventionSolving Equations by Using Addition and Subtraction

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2


Less

on

3-2

Solve Using Addition If the same number is added to each side of an equation, theresulting equation is equivalent to the original one. In general if the original equationinvolves subtraction, this property will help you solve the equation.

Addition Property of Equality For any numbers a, b, and c, if a " b, then a ' c " b ' c.

Solve m ! 32 " 18.

m # 32 " 18 Original equation

m # 32 ' 32 " 18 ' 32 Add 32 to each side.

m " 50 Simplify.

The solution is 50.

Solve !18 " p !12.

#18 " p #12 Original equation

#18 ' 12 " p #12 ' 12 Add 12 to each side.

p " #6 Simplify.

The solution is #6.


ExercisesExercises

Solve each equation. Then check your solution.

1. h # 3 " #2 2. m # 8 " #12 3. p # 5 " 15

4. 20 " y # 8 5. k # 0.5 " 2.3 6. w # "

7. h # 18 " #17 8. #12 " #24 ' k 9. j # 0.2 " 1.8

10. b # 40 " #40 11. m # (#12) " 10 12. w # "

Write an equation for each problem. Then solve the equation and check thesolution.

13. Twelve subtracted from a number equals 25. Find the number.

14. What number decreased by 52 equals #12?

15. Fifty subtracted from a number equals eighty. Find the number.

16. What number minus one-half is equal to negative one-half ?

17. The difference of a number and eight is equal to 14. What is the number?

18. A number decreased by fourteen is equal to eighteen. What is the number?

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Solve Using Subtraction If the same number is subtracted from each side of anequation, the resulting equation is equivalent to the original one. In general if the originalequation involves addition, this property will help you solve the equation.

Subtraction Property of Equality For any numbers a, b, and c, if a " b, then a # c " b # c.

Solve 22 # p " !12.

22 ' p " #12 Original equation

22 ' p # 22 " #12 # 22 Subtract 22 from each side.

p " #34 Simplify.

The solution is #34.


1. x ' 12 " 6 2. z ' 2 " #13 3. #17 " b ' 4

4. s ' (#9) " 7 5. #3.2 " ! ' (#0.2) 6. # ' x "

7. 19 ' h " #4 8. #12 " k ' 24 9. j ' 1.2 " 2.8

10. b ' 80 " #80 11. m ' (#8) " 2 12. w ' "

Write an equation for each problem. Then solve the equation and check thesolution.

13. Twelve added to a number equals 18. Find the number.

14. What number increased by 20 equals #10?

15. The sum of a number and fifty equals eighty. Find the number.

16. What number plus one-half is equal to four?

17. The sum of a number and 3 is equal to #15. What is the number?

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Solving Equations by Using Addition and Subtraction

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2

ExampleExample

ExercisesExercises

Skills PracticeSolving Equations by Using Addition and Subtraction

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2


Less

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1. y # 7 " 8 2. w ' 14 " #8

3. p # 4 " 6 4. #13 " 5 ' x

5. 98 " b ' 34 6. y # 32 " #1

7. s ' (#28) " 0 8. y ' (#10) " 6

9. #1 " s ' (#19) 10. j # (#17) " 36

11. 14 " d ' (#10) 12. u ' (#5) " #15

13. 11 " #16 ' y 14. c # (#3) " 100

15. 47 " w # (#8) 16. x # (#74) " #22

17. 4 # (#h) " 68 18. #56 " 20 # (#e)

Write an equation for each problem. Then solve the equation and check yoursolution.

19. A number decreased by 14 is #46. Find the number.

20. Thirteen subtracted from a number is #5. Find the number.

21. The sum of a number and 67 is equal to #34. Find the number.

22. What number minus 28 equals #2?

23. A number plus #73 is equal to 27. What is the number?

24. A number plus #17 equals #1. Find the number.

25. What number less 5 is equal to #39?



1. d # 8 " 17 2. v ' 12 " #5 3. b # 2 " #11

4. #16 " s ' 71 5. 29 " a # 76 6. #14 ' y " #2

7. 8 # (#c) " 1 8. 78 ' r " #15 9. f ' (#3) " #9

10. 4.2 " n ' 7.3 11. w ' 1.9 " #2.5 12. 4.6 # (#b) " #0.4

13. y # (#1.5) " 0.5 14. a # 0.13 " #0.58 15. k ' (#4.21) " #19

16. r ' " 17. ' q " 18. " h '

19. ' x " # 20. y ' " 21. # # (#n) " #

Write an equation for each problem. Then solve the equation and check yoursolution.

22. What number minus 9 is equal to #18?

23. A number plus 15 equals #12. What is the number?

24. The sum of a number and #3 is equal to #91. Find the number.

25. Negative seventeen equals 63 plus a number. What is the number?

26. The sum of negative 14, a number, and 6 is #5. What is the number?

27. What number plus one half is equal to three eighths?

HISTORY For Exercises 28 and 29, use the following information.

Galileo Galilei was born in 1564. Many years later, in 1642, Sir Isaac Newton was born.

28. Write an addition equation to represent the situation.

29. How many years after Galileo was born was Isaac Newton born?

HURRICANES For Exercises 30 and 31, use the following information.

The day after a hurricane, the barometric pressure in a coastal town has risen to 29.7 inches of mercury, which is 2.9 inches of mercury higher than the pressure when theeye of the hurricane passed over.

30. Write an addition equation to represent the situation.

31. What was the barometric pressure when the eye passed over?

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Practice Solving Equations by Using Addition and Subtraction

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2

Reading to Learn MathematicsSolving Equations by Using Addition and Subtraction

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2


Less

on

3-2

Pre-Activity How can equations be used to compare data?


In the equation m # 66 " 5, the number 5 represents

and the number 66 represents

Reading the Lesson

1. To solve x ' 17 " 46 using the Subtraction Property of Equality, you would subtract from each side.

2. To solve y # 9 " #30 using the Addition Property of Equality, you would add to each side.

3. Write an equation that you could solve by subtracting 32 from each side.

4. A student used the Subtraction Property of Equality to solve an equation. Explain why itwould also be possible to use the Addition Property of Equality to solve the equation.


5. Explain how you decide whether to use the Addition Property or the SubtractionProperty of Equality to solve an equation.


Counting-Off PuzzlesSolve each puzzle.

1. Twenty-five people are standing in a circle.Starting with person 1, they count off from 1 to 7 and then start over with 1. Each person who says “7” drops out of the circle.Who is the last person left?

2. Forty people stand in a circle. They count off so that every third person drops out. Which two people are the last ones left?

3. Only half of the 30 students in Sharon’s class can go on a field trip. Sharon arranges the boys and girls as shown. They count off from 1 to 9 and every ninthperson drops out until only 15 people are left. Who gets to go on the field trip.

A group of people stand in a circle and count off 1, 2, 1, 2, 1 and so on. Every second person drops out. Person number 1 is the last person left.

4. Draw a diagram to show why the number of people in the circle must be even. Then, explain your answer.

5. When the count returns to person number 1 for the first time, how many people have dropped out?

6. Find the number of people in the circle if the number is between 10 and 20. Do the same if the number is between 30 and 40. What can you conclude about the original number of people?

G

GG

G

G G GG

GG

G

GGG

G B

B

B B B

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BB

B

B1

23 4 5 6 7 8 9 10

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171819202122232425

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1 2 3 45

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Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2

Study Guide and InterventionSolving Equations by Using Multiplication and Division

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3


Less

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Solve Using Multiplication If each side of an equation is multiplied by the samenumber, the resulting equation is equivalent to the given one. You can use the property tosolve equations involving multiplication and division.

Multiplication Property of Equality For any numbers a, b, and c, if a " b, then ac " bc.

Solve 3 p " 1 .

3 p " 1 Original equation

p " Rewrite each mixed number as animproper fraction.

" p# " " # Multiply each side by .

p " Simplify.

The solution is .3%7

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1$2

1$2 Solve ! n " 16.

# n " 16 Original equation

#4"# n# " #4(16) Multiply each side by #4.

n " #64 Simplify.


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1$4


ExercisesExercises


1. " #2 2. m " 6 3. p "

4. 5 " 5. # k " #2.5 6. # "

7. #1 h " 4 8. #12 " # k 9. "

10. #3 b " 5 11. m " 10 12. " #

Write an equation for each problem. Then solve the equation.

13. One-fifth of a number equals 25. Find the number.

14. What number divided by 2 equals #18?

15. A number divided by eight equals 3. Find the number.

16. One and a half times a number equals 6. Find the number.

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Solve Using Division To solve equations with multiplication and division, you can alsouse the Division Property of Equality. If each side of an equation is divided by the samenumber, the resulting equation is true.

Division Property of Equality For any numbers a, b, and c, with c ( 0, if a " b, then " .b%c

a%c


Solving Equations by Using Multiplication and Division

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3

Solve 8n " 64.

8n " 64 Original equation

" Divide each side by 8.

n " 8 Simplify.

The solution is 8.

64%8

8n%8

Solve !5n " 60.

#5n " 60 Original equation

" Divide each side by #5.

n " #12 Simplify.


60%#5

#5n%#5


ExercisesExercises


1. 3h " #42 2. 8m " 16 3. #3t " 51

4. #3r " #24 5. 8k " #64 6. #2m " 16

7. 12h " 4 8. #2.4p " 7.2 9. 0.5j " 5

10. #25 " 5m 11. 6m " 15 12. #1.5p " #75


13. Four times a number equals 64. Find the number.

14. What number multiplied by #4 equals #16?

15. A number times eight equals #36. Find the number.

Skills PracticeSolving Equations by Using Multiplication and Division

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3


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1. 12z " 108 2. #7t " 49

3. 18e " #216 4. #22 " 11v

5. #6d " #42 6. 96 " #24a

7. " 16 8. " 9

9. #84 " 10. # " #13

11. " #13 12. 31 " # n

13. #6 " z 14. q " #4

15. p " #10 16. "

17. #0.4b " 5.2 18. 1.6m " #4


19. The opposite of a number is #9. What is the number?

20. Fourteen times a number is #42. Find the number.

21. Eight times a number equals 128. What is the number?

22. Negative twelve times a number equals #132. Find the number.

23. Negative eighteen times a number is #54. What is the number?

24. One sixth of a number is #17. Find the number.

25. Negative three fifths of a number is #15. What is the number?

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5%9

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1. 8j " 96 2. #13z " #39 3. #180 " 15m

4. 243 " 27c 5. " #8 6. # " #8

7. " 8. " 9. "

10. #1 " # t 11. # w " #9 12. # s " 4

13. #3x " 14. a " 15. h "

16. 5n " 17. 2.5k " 20 18. #3.4e " #3.74

19. #1.7b " 2.21 20. 0.26p " 0.104 21. 4.2q " #3.36


22. Negative nine times a number equals #117. Find the number.

23. Negative one eighth of a number is # . What is the number?

24. Five sixths of a number is # . Find the number.

25. 2.7 times a number equals 8.37. What is the number?

26. One and one fourth times a number is one and one third. What is the number?

27. PUBLISHING Two units of measure used in publishing are the pica and the point. A picais one sixth of an inch. There are 12 points in a pica, so Points " 12 · Picas. How manypicas are equivalent to 108 points?

ROLLER COASTERS For Exercises 28 and 29, use the following information.

Superman the Escape in California is the fastest roller coaster in the world. Riders fall 415 feet in 7 seconds. Speeds reach a maximum of 100 miles per hour.

28. If x represents the average rate of fall of the roller coaster, write an expression torepresent the situation (Hint: Use the distance formula d " rt.)

29. What is the average rate that riders fall in feet per second?

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Practice Solving Equations by Using Multiplication and Division

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3

Reading to Learn MathematicsSolving Equations by Using Multiplication and Division

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3


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Pre-Activity How can equations be used to find how long it takes light to reachEarth?


• In the equation d " rt, shown in the introduction, what number is usedfor r? for d?

• What equation could you use to find the time it takes light to reach Earthfrom the farthest star in the Big Dipper?

Reading the Lesson

Complete the sentence after each equation to tell how you would solve theequation.

1. " 16 each side by .

2. 5x " 125 each side by , or multiply each side by .

3. #8k " 96 Divide each side by , or multiply each side by

4. Explain how rewriting 4 x " 2 as x " helps you solve the equation.


5. One way to remember something is to explain it to someone else. Write how you would explain to a classmate how to solve the equation x " 12.2

%3

17%8

13%3

1%8

1%3

x%7


Dissection Puzzles: Make the SquareIn a dissection puzzle, you are to cut apart one figure using onlystraight cuts and then rearrange the pieces to make a new figure.Usually the puzzle-solver must figure out where to make the givennumber of cuts. However, for these puzzles, the cut lines are shown.You must discover how to rearrange the pieces.

Cut apart each figure. Then rearrange the pieces to form a square.

1.

2.

3.

4. Hint: Cut one of the triangles into two pieces to make this square.

A AA A

A

2x 2x 2x

x

A AA A

B

2x 2x x

x

A

C

B

D

A

BB

BB

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3

Study Guide and InterventionSolving Multi-Step Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

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Work Backward Working backward is one of many problem-solving strategies thatyou can use to solve problems. To work backward, start with the result given at the end of aproblem and undo each step to arrive at the beginning number.

A number is dividedby 2, and then 8 is subtracted fromthe quotient. The result is 16. Whatis the number?

Solve the problem by working backward.The final number is 16. Undosubtracting 8 by adding 8 to get 24. Toundo dividing 24 by 2, multiply 24 by 2to get 48.The original number is 48.

A bacteria culture doubleseach half hour. After 3 hours, there are6400 bacteria. How many bacteria werethere to begin with?

Solve the problem by working backward.The bacteria have grown for 3 hours. Since thereare 2 one-half hour periods in one hour, in 3 hoursthere are 6 one-half hour periods. Since thebacteria culture has grown for 6 time periods, ithas doubled 6 times. Undo the doubling byhalving the number of bacteria 6 times.

6,400 ! ! ! ! ! ! " 6,400 !

" 100

There were 100 bacteria to begin with.

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ExercisesExercises

Solve each problem by working backward.

1. A number is divided by 3, and then 4 is added to the quotient. The result is 8. Find thenumber.

2. A number is multiplied by 5, and then 3 is subtracted from the product. The result is 12.Find the number.

3. Eight is subtracted from a number, and then the difference is multiplied by 2. The resultis 24. Find the number.

4. Three times a number plus 3 is 24. Find the number.

5. CAR RENTAL Angela rented a car for $29.99 a day plus a one-time insurance cost of$5.00. Her bill was $124.96. For how many days did she rent the car?

6. MONEY Mike withdrew an amount of money from his bank account. He spent onefourth for gasoline and had $90 left. How much money did he withdraw?

7. TELEVISIONS In 1999, 68% of households with TV’s subscribed to cable TV. If 8,000 moresubscribers are added to the number of households with cable, the total number ofhouseholds with cable TV would be 67,600,000. How many households were there withTV in 1999? Source: World Almanac


Solve Multi-Step Equations To solve equations with more than one operation, oftencalled multi-step equations, undo operations by working backward. Reverse the usualorder of operations as you work.

Solve 5x # 3 " 23.

5x ' 3 " 23 Original equation.

5x ' 3 # 3 " 23 # 3 Subtract 3 from each side.

5x " 20 Simplify.


x " 4 Simplify.


1. 5x ' 2 " 27 2. 6x ' 9 " 27 3. 5x ' 16 " 51

4. 14n # 8 " 34 5. 0.6x # 1.5 " 1.8 6. p # 4 " 10

7. 16 " 8. 8 ' " 13 9. ' 3 " #13

10. " 10 11. 0.2x # 8 " #2 12. 3.2y # 1.8 " 3

13. #4 " 14. 8 " #12 ' 15. 0 " 10y # 40

Write an equation and solve each problem.

16. Find three consecutive integers whose sum is 96.

17. Find two consecutive odd integers whose sum is 176.

18. Find three consecutive integers whose sum is #93.

k%#4

7x # (#1)%%

#8

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Solving Multi-Step Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4

ExampleExample

ExercisesExercises

Skills PracticeSolving Multi-Step Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4


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1. A number is divided by 2, and then the quotient is added to 8. The result is 33. Find thenumber.

2. Two is subtracted from a number, and then the difference is divided by 3. The result is30. Find the number.

3. A number is multiplied by 2, and then the product is added to 9. The result is 49. Whatis the number?

4. ALLOWANCE After Ricardo received his allowance for the week, he went to the mallwith some friends. He spent half of his allowance on a new paperback book. Then hebought himself a snack for $1.25. When he arrived home, he had $5.00 left. How muchwas his allowance?


5. 5x ' 3 " 23 6. 4 " 3a # 14 7. 2y ' 5 " 19

8. 6 ' 5c " #29 9. 8 # 5w " #37 10. 18 # 4v " 42

11. # 8 " #2 12. 5 ' " 1 13. # # 4 " 13

14. # ' 12 " #7 15. # 2 " 9 16. ' 3 " #1

17. q # 7 " 8 18. g ' 6 " #12 19. z # 8 " #3

20. m ' 2 " 6 21. " 3 22. " 2


23. Twice a number plus four equals 6. What is the number?

24. Sixteen is seven plus three times a number. Find the number.

25. Find two consecutive integers whose sum is 35.

26. Find three consecutive integers whose sum is 36.

b ' 1%3

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1. Three is added to a number, and then the sum is multiplied by 4. The result is 16. Findthe number.

2. A number is divided by 4, and the quotient is added to 3. The result is 24. What is thenumber?

3. Two is subtracted from a number, and then the difference is multiplied by 5. The resultis 30. Find the number.

4. BIRD WATCHING While Michelle sat observing birds at a bird feeder, one fourth of thebirds flew away when they were startled by a noise. Two birds left the feeder to go toanother stationed a few feet away. Three more birds flew into the branches of a nearbytree. Four birds remained at the feeder. How many birds were at the feeder initially?


5. #12n # 19 " 77 6. 17 ' 3f " 14 7. 15t ' 4 " 49

8. ' 6 " 2 9. ' 3 " 15 10. # 6 " #2

11. y # " 12. #32 # f " #17 13. 8 # k " #4

14. " 1 15. " #9 16. " 16

17. # 0.5 " 2.5 18. 2.5g ' 0.45 " 0.95 19. 0.4m # 0.7 " 0.22


20. Seven less than four times a number equals 13. What is the number?

21. Find two consecutive odd integers whose sum is 116.

22. Find two consecutive even integers whose sum is 126.

23. Find three consecutive odd integers whose sum is 117.

24. COIN COLLECTING Jung has a total of 92 coins in his coin collection. This is 8 morethan three times the number of quarters in the collection. How many quarters does Junghave in his collection?

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15 # a%3

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Practice Solving Multi-Step Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4

Reading to Learn MathematicsSolving Multi-Step Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4


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Pre-Activity How can equations be used to estimate the age of an animal?


• Write the equation 8 ' 12a " 124 in words.

• How many operations are involved in the equation?

Reading the Lesson

1. What does the phrase undo the operations mean to you? Give an example.

2. a. If we undo operations in reverse of the order of operations, what operations do we dofirst?

b. What operations do we do last?

3. Suppose you want to solve " 6.

a. What is the grouping symbol in the equation " 6?

b. What is the first step in solving the equation?

c. What is the next step in solving the equation?

4. Write an equation for the problem below.

Seven times k minus five equals negative forty-seven


5. Explain why working backward is a useful strategy for solving equations.

x ' 3%5

x ' 3%5


Consecutive Integer ProblemsMany types of problems and puzzles involve the idea of consecutiveintegers. Knowing how to represent these integers algebraically canhelp to solve the problem.

Find four consecutive odd integers whose sum is !80.

An odd integer can be written as 2n ' 1, where n is any integer.If 2n + 1 is the first odd integer, then add 2 to get the next largest oddinteger, and so on.Now write an equation to solve this problem.(2n ' 1) ' (2n ' 3) ' (2n ' 5) ' (2n ' 7) " #80

Write an equation for each problem. Then solve.

1. Complete the solution to the problem in the example.

2. Find three consecutive even integers whose sum is 132.



5. The lesser of two consecutive even integers is 10 more than one-half the greater. Find the integers.

6. The greater of two consecutive even integers is 6 less than three times the lesser. Find the integers.

7. Find four consecutive integers such that twice the sum of the two greater integers exceeds three times the first by 91.

8. Find a set of four consecutive positive integers such that the greatest integer in the set is twice the least integer in the set.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4

ExampleExample

ExercisesExercises

Study Guide and InterventionSolving Equations with the Variable on Each Side

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5


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Variables on Each Side To solve an equation with the same variable on each side,first use the Addition or the Subtraction Property of Equality to write an equivalentequation that has the variable on just one side of the equation. Then solve the equation.

Solve 5y ! 8 " 3y # 12.

5y # 8 " 3y ' 125y # 8 # 3y " 3y ' 12 # 3y

2y # 8 " 122y # 8 ' 8 " 12 ' 8

2y " 20

"

y " 10

The solution is 10.

20%2

2y%2

Solve !11 ! 3y " 8y # 1.

#11 # 3y " 8y ' 1#11 # 3y ' 3y " 8y ' 1 ' 3y

#11 " 11y ' 1#11 # 1 " 11y ' 1 # 1

#12 " 11y

"

#1 " y

The solution is #1 .1%11

1%11

11y%11

#12%11


ExercisesExercises


1. 6 # b " 5b ' 30 2. 5y # 2y " 3y ' 2 3. 5x ' 2 " 2x # 10

4. 4n # 8 " 3n ' 2 5. 1.2x ' 4.3 " 2.1 # x 6. 4.4s ' 6.2 " 8.8s # 1.8

7. b ' 4 " b ' 88 8. k # 5 " k # 1 9. 8 # 5p " 4p # 1

10. 4b # 8 " 10 # 2b 11. 0.2x # 8 " #2 # x 12. 3y # 1.8 " 3y # 1.8

13. #4 # 3x " 7x # 6 14. 8 ' 4k " #10 ' k 15. 20 # a " 10a # 2

16. n ' 8 " n ' 2 17. y # 8 " 9 # y 18. #4r ' 5 " 5 # 4r

19. #4 # 3x " 6x # 6 20. 18 # 4k " #10 #4k 21. 12 ' 2y " 10y # 12

3%5

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Grouping Symbols When solving equations that contain grouping symbols, first usethe Distributive Property to eliminate grouping symbols. Then solve.

Solve 4(2a ! 1) " !10(a ! 5).

4(2a # 1) " #10(a # 5) Original equation

8a # 4 " #10a ' 50 Distributive Property

8a # 4 ' 10a " #10a ' 50 ' 10a Add 10a to each side.

18a # 4 " 50 Simplify.

18a # 4 ' 4 " 50 ' 4 Add 4 to each side.

18a " 54 Simplify.


a " 3 Simplify.

The solution is 3.


1. #3(x ' 5) " 3(x # 1) 2. 2(7 ' 3t) " #t 3. 3(a ' 1) # 5 " 3a # 2

4. 75 # 9g " 5(#4 ' 2g) 5. 5(f ' 2) " 2(3 # f ) 6. 4(p ' 3) " 36

7. 18 " 3(2c ' 2) 8. 3(d # 8) " 3d 9. 5(p ' 3) ' 9 " 3(p # 2) ' 6

10. 4(b # 2) " 2(5 # b) 11. 1.2(x # 2) " 2 # x 12. "

13. " 14. 2(4 ' 2k) ' 10 " k 15. 2(w # 1) ' 4 " 4(w ' 1)

16. 6(n # 1) " 2(2n ' 4) 17. 2[2 ' 3( y # 1)] " 22 18. #4(r ' 2) " 4(2 # 4r)

19. #3(x # 8) " 24 20. 4(4 # 4k) " #10 #16k 21. 6(2 # 2y) " 5(2y # 2)

2a ' 5%3

a # 8%12

#y %8

3 ' y%4

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Solving Equations with the Variable on Each Side

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5

ExampleExample

ExercisesExercises

Skills PracticeSolving Equations with the Variable on Each Side

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5


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Justify each step.

1. 4k # 3 " 2k ' 54k # 3 # 2k " 2k ' 5 # 2k a.

2k # 3 " 5 b.

2k # 3 ' 3 " 5 ' 3 c.

2k " 8 d.

" e.

k " 4 f.

2. 2(8u ' 2) " 3(2u # 7)16u ' 4 " 6u # 21 a.

16u ' 4 # 6u " 6u # 21 # 6u b.

10u ' 4 " #21 c.

10u ' 4 # 4 " #21 # 4 d.

10u " #25 e.

" f.

u " #2.5 g.


3. 2m ' 12 " 3m # 31 4. 2h # 8 " h ' 17

5. 7a # 3 " 3 # 2a 6. 4n # 12 " 12 # 4n

7. 4x # 9 " 7x ' 12 8. #6y # 3 " 3 # 6y

9. 5 ' 3r " 5r # 19 10. #9 ' 8k " 7 ' 4k

11. 8q ' 12 " 4(3 ' 2q) 12. 3(5j ' 2) " 2(3j # 6)

13. 6(#3v ' 1) " 5(#2v # 2) 14. #7(2b # 4) " 5(#2b ' 6)

15. 3(8 # 3t) " 5(2 ' t) 16. 2(3u ' 7) " #4(3 # 2u)

17. 8(2f # 2) " 7(3f ' 2) 18. 5(#6 # 3d) " 3(8 ' 7d)

19. 6(w # 1) " 3(3w ' 5) 20. 7(#3y ' 2) " 8(3y # 2)

21. v # 6 " 6 # v 22. # x " x '7%2

7%8

5%8

1%2

2%3

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#25%10

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1. 5x # 3 " 13 # 3x 2. #4c # 11 " 4c ' 21

3. 1 # s " 6 # 6s 4. 14 ' 5n " #4n ' 17

5. k # 3 " 2 # k 6. (6 # z) " z

7. 3(#2 # 3x) " #9x # 4 8. 4(4 # w) " 3(2w ' 2)

9. 9(4b # 1) " 2(9b ' 3) 10. 3(6 ' 5y) " 2(#5 ' 4y)

11. #5x # 10 " 2 # (x ' 4) 12. 6 ' 2(3j # 2) " 4(1 ' j)

13. t # t " 3 ' t 14. 1.4f ' 1.1 " 8.3 # f

15. x # " x ' 16. 2 # z " z ' 9

17. (3g # 2) " 18. (c ' 1) " (3c # 5)

19. (5 # 2h) " 20. (2m # 16) " (2m ' 4)

21. 3(d # 8) # 5 " 9(d ' 2) ' 1 22. 2(a # 8) ' 7 " 5(a ' 2) # 3a # 19

23. Two thirds of a number reduced by 11 is equal to 4 more than the number. Find thenumber.

24. Five times the sum of a number and 3 is the same as 3 multiplied by 1 less than twicethe number. What is the number?

25. NUMBER THEORY Tripling the greater of two consecutive even integers gives the sameresult as subtracting 10 from the lesser even integer. What are the integers?

26. GEOMETRY The formula for the perimeter of a rectangle is P " 2! ' 2w, where ! isthe length and w is the width. A rectangle has a perimeter of 24 inches. Find itsdimensions if its length is 3 inches greater than its width.

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Practice Solving Equations with the Variable on Each Side

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5

Reading to Learn MathematicsSolving Equations with the Variable on Each Side

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5


Less

on

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Pre-Activity How can an equation be used to determine when two populationsare equal?


In the equation 12 ' 7.6x " 6 ' 8x, what do 7.6x and 8x represent?

Reading the Lesson

1. Suppose you want to help a friend solve 6k ' 7 " 3k # 8. What would you advise her todo first? Why?

2. When solving 2(3x # 4)" 3(x ' 5), why is it helpful first to use the Distributive Propertyto remove the grouping symbols?

3. On a quiz, Jason solved three equations. His teacher said all the work was correct, butshe asked him to write short sentences to tell what the solutions were. In what follows,you see the last equation in his work for each equation. Write sentences to describe thesolutions.

a. x " #4

b. 6m " 6m

c. 12 " 37

4. In Question 3, one of the equations Jason solved was an identity. Which equation was it?Explain how you know.


5. An equation with variables is an identity when the equation is always true. In otherwords, the expressions on the left and right sides always have the same value. Look upthe word identity in the dictionary. Write all the definitions that are similar to themathematical definition.


IdentitiesAn equation that is true for every value of the variable is called anidentity. When you try to solve an identity, you end up with astatement that is always true. Here is an example.

Solve 8 ! (5 ! 6x) " 3(1 # 2x).

8 # (5 # 6x) " 3(1 ' 2x)

8 # 5 # (#6x) " 3(1 ' 2x)

8 # 5 ' 6x " 3 ' 6x

3 ' 6x " 3 ' 6x

State whether each equation is an identity. If it is not, find its solution.

1. 2(2 # 3x) " 3(3 ' x) ' 4 2. 5(m ' 1) ' 6 " 3(4 ' m) ' (2m # 1)

3. (5t ' 9) # (3t # 13) " 2(11 ' t) 4. 14 # (6 # 3c) " 4c # c

5. 3y # 2( y ' 19) " 9y # 3(9 # y) 6. 3(3h # 1) " 4(h ' 3)

7. Use the true equation 3x # 2 " 3x # 2 to create an identity of your own.

8. Use the false equation 1 " 2 to create an equation with no solution.

9. Create an equation whose solution is x " 3.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5

ExampleExample

ExercisesExercises

Study Guide and InterventionRatios and Proportions

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6


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Ratios and Proportions A ratio is a comparison of two numbers by division. The ratio

of x to y can be expressed as x to y, x:y or . Ratios are usually expressed in simplest form.

An equation stating that two ratios are equal is called a proportion. To determine whethertwo ratios form a proportion, express both ratios in simplest form or check cross products.

x%y

Determine whether the

ratios and form a proportion.

" when expressed in simplest form.

" when expressed in simplest form.

The ratios and form a proportion

because they are equal when expressed insimplest form.

12%18

24%36

2%3

12%18

2%3

24%36

12$18

24$36

Use cross products to

determine whether and form aproportion.

! Write the proportion.

10(45) ! 18(25) Cross products

450 " 450 Simplify.

The cross products are equal, so " .

Since the ratios are equal, they form aproportion.

25%45

10%18

25%45

10%18

25$45

10$18


ExercisesExercises

Use cross products to determine whether each pair of ratios forms a proportion.

1. , 2. , 3. ,

4. , 5. , 6. ,

7. , 8. , 9. ,

10. 2:3, 20:30 11. 5 to 9, 25 to 45 12. ,

13. 5:5, 30:20 14. 18 to 24, 50 to 75 15. 100:75, 44:33

16. , 17. , 18. , 0.45%0.9

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9%8

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20%30

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0.1%2

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25%49

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Solve Proportions If a proportion involves a variable, you can use cross products to solve

the proportion. In the proportion " , x and 13 are called extremes and 5 and 10 are

called means. In a proportion, the product of the extremes is equal to the product of the means.

Means-Extremes Property of Proportions For any numbers a, b, c, and d, if " , then ad " bc.

Solve " .

" Original proportion

13(x) " 5(10) Cross products

13x " 50 Simplify.


x " 3 Simplify.

The solution is 3 .

Solve each proportion.

1. " 2. " 3. "

4. " 5. " 6. "

7. " 8. " 9. "

10. " 11. " 12. "

13. " 14. " 15. "

Use a proportion to solve each problem.

16. MODELS To make a model of the Guadeloupe River bed, Hermie used 1 inch of clay for5 miles of the river’s actual length. His model river was 50 inches long. How long is theGuadeloupe River?

17. EDUCATION Josh finished 24 math problems in one hour. At that rate, how manyhours will it take him to complete 72 problems?

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x$5

c%d

a%b

10%13

x%5


Ratios and Proportions

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6

ExampleExample

ExercisesExercises

Skills PracticeRatios and Proportions

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6


Less

on

3-6

Use cross products to determine whether each pair of ratios forms a proportion.Write yes or no.

1. , 2. ,

3. , 4. ,

5. , 6. ,

7. , 8. ,

Solve each proportion. If necessary, round to the nearest hundredth.

9. " 10. "

11. " 12. "

13. " 14. "

15. " 16. "

17. " 18. "

19. " 20. "

21. " 22. "

23. " 24. "

25. BOATING Hue’s boat used 5 gallons of gasoline in 4 hours. At this rate, how manygallons of gasoline will the boat use in 10 hours?

11%30

22%x

s%84

4%21

20%g

5%12

27%39

9%c

n%60

11%15

30%m

10%14

1%9

7%b

6%f

42%56

y%69

6%23

36%s

12%7

35%21

5%e

3%5

6%z

1%6

3%a

15%10

9%g

3%9

5%b

2%14

1%a

50%85

12%17

21%98

3%14

26%38

13%19

42%90

7%16

72%81

8%9

24%28

6%7

7%11

5%9

20%25

4%5


Use cross products to determine whether each pair of ratios forms a proportion.Write yes or no.

1. , 2. , 3. ,

4. , 5. , 6. ,

7. , 8. , 9. ,

Solve each proportion. If necessary, round to the nearest hundredth.

10. " 11. " 12. "

13. " 14. " 15. "

16. " 17. " 18. "

19. " 20. " 21. "

22. " 23. " 24. "

25. " 26. " 27. "

28. " 29. " 30. "

31. " 32. " 33. "

34. PAINTING Ysidra paints a room that has 400 square feet of wall space in 2 hours. At

this rate, how long will it take her to paint a room that has 720 square feet of wallspace?

35. VACATION PLANS Walker is planning a summer vacation. He wants to visit PetrifiedNational Forest and Meteor Crater, Arizona, the 50,000-year-old impact site of a largemeteor. On a map with a scale where 2 inches equals 75 miles, the two areas are about

1 inches apart. What is the distance between Petrified National Forest and Meteor

Crater?

1%2

1%2

x # 2%6

3%7

5%7

r ' 2%7

x ' 1%4

5%12

2%4

m # 1%8

2%y ' 6

3%12

14%6

7%a # 4

3%0.51

6%n

12%b

3%0.72

7%1.61

v%0.23

5%8

m%6

5%6

3%q

8%c

7%9

2%a

14%49

6%4

g%16

12%h

6%61

z%17

3%51

35%x

5%11

10%60

2%y

48%9

y%3

27%162

3%u

4%w

28%49

k%7

40%56

34%23

v%46

30%54

5%a

3.9%0.9

7.6%1.8

2.9%2.4

1.7%1.2

7.14%10.92

3.4%5.2

1%6

1.5%9

72%81

8%9

108%99

12%11

36%48

18%24

15%66

3%11

52%48

7%6

Practice Ratios and Proportions

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6

Reading to Learn MathematicsRatios and Proportions

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6


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Pre-Activity How are ratios used in recipes?


• How many servings of honey frozen yogurt are made by this recipe?

• How many recipes would be needed to make enough honey frozen yogurtfor all the students in your class?

Reading the Lesson

1. Complete the following sentence.

A ratio is a comparison of two numbers by .

2. Describe two ways to decide whether the sentence " is a proportion.

3. For each proportion, tell what the extremes are and what the means are.

a. " Extremes: Means:

b. " Extremes: Means:

4. A jet flying at a steady speed traveled 825 miles in 2 hours. If you solved the proportion

" , what would the answer tell you about the jet?


5. Write how you would explain solving a proportion to a friend who missed Lesson 3-6.

x%1.5

825%2

12%16

6%8

6%15

14%35

8%20

2%5


Angles of a TriangleIn geometry, many statements about physical space are proven to betrue. Such statements are called theorems. Here are two examples ofgeometric theorems.

a. The sum of the measures of the angles of a triangle is 180°.b. If two sides of a triangle have equal measure, then the two angles

opposite those sides also have equal measure.

For each of the triangles, write an equation and then solve for x. (A tick mark ontwo or more sides of a triangle indicates that the sides have equal measure.)

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. Two angles of a triangle have the same 12. The measure of one angle of a triangle is measure. The sum of the measures of twice the measure of a second angle. The these angles is one-half the measure of measure of the third angle is 12 less than the third angle. Find the measures of the sum of the other two. Find the the angles of the triangle. measures of the angles of the triangle.

x2x

30°

x

40°

x

x # 15°

3x

x ' 30°

90°

4x

5x

5x

2x x

x ' 30°

x5x ' 10°

90° x

90° 45°

x70°

50°x

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6

Study Guide and InterventionPercent of Change

NAME ______________________________________________ DATE ____________ PERIOD _____

3-73-7


Less

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Percent of Change When an increase or decrease in an amount is expressed as apercent, the percent is called the percent of change. If the new number is greater than theoriginal number, the percent of change is a percent of increase. If the new number is lessthan the original number, the percent of change is the percent of decrease.

Find the percent of increase.original: 48new: 60

First, subtract to find the amount ofincrease. The amount of increase is 60 # 48 " 12.Then find the percent of increase by usingthe original number, 48, as the base.

" Percent proportion

12(100) " 48(r) Cross products

1200 " 48r Simplify.


25 " r Simplify.

The percent of increase is 25%.

48r%48

1200%48

r%100

12%48

Find the percent of decrease.original: 30new: 22

First, subtract to find the amount ofdecrease. The amount of decrease is 30 # 22 " 8.Then find the percent of decrease by usingthe original number, 30, as the base.

" Percent proportion

8(100) " 30(r) Cross products

800 " 30r Simplify.


26 " r Simplify.

The percent of decrease is 26 %, or about27%.

2%3

2%3

30r%30

800%30

r%100

8%30


ExercisesExercises

State whether each percent of change is a percent of increase or a percent ofdecrease. Then find each percent of change. Round to the nearest whole percent.

1. original: 50 2. original: 90 3. original: 45new: 80 new: 100 new: 20

4. original: 77.5 5. original: 140 6. original: 135new: 62 new: 150 new: 90

7. original: 120 8. original: 90 9. original: 27.5new: 180 new: 270 new: 25

10. original: 84 11. original: 12.5 12. original: 250new: 98 new: 10 new: 500


Solve Problems Discounted prices and prices including tax are applications of percentof change. Discount is the amount by which the regular price of an item is reduced. Thus,the discounted price is an example of percent of decrease. Sales tax is amount that is addedto the cost of an item, so the price including tax is an example of percent of increase.

A coat is on sale for 25% off the original price. If the original priceof the coat is $75, what is the discounted price?

The discount is 25% of the original price.

25% of $75 " 0.25 ! 75 25% " 0.25

" 18.75 Use a calculator.

Subtract $18.75 from the original price.$75 # $18.75 " $56.25

The discounted price of the coat is $56.25.

Find the final price of each item. When a discount and a sales tax are listed,compute the discount price before computing the tax.

1. Compact disc: $16 2. Two concert tickets: $28 3. Airline ticket: $248.00Discount: 15% Student discount: 28% Superair discount: 33%

4. Shirt: $24.00 5. CD player: $142.00 6. Celebrity calendar: $10.95Sales tax: 4% Sales tax: 5.5% Sales tax: 7.5%

7. Class ring: $89.00 8. Software: $44.00 9. Video recorder: $110.95Group discount: 17% Discount: 21% Discount: 20%Sales tax: 5% Sales tax: 6% Sales tax: 5%

10. VIDEOS The original selling price of a new sports video was $65.00. Due to the demandthe price was increased to $87.75. What was the percent of increase over the originalprice?

11. SCHOOL A high school paper increased its sales by 75% when it ran an issue featuringa contest to win a class party. Before the contest issue, 10% of the school’s 800 studentsbought the paper. How many students bought the contest issue?

12. BASEBALL Baseball tickets cost $15 for general admission or $20 for box seats. Thesales tax on each ticket is 8%, and the municipal tax on each ticket is an additional 10%of the base price. What is the final cost of each type of ticket?


Percent of Change

NAME ______________________________________________ DATE ____________ PERIOD _____

3-73-7

ExampleExample

ExercisesExercises

Skills PracticePercent of Change

NAME ______________________________________________ DATE ____________ PERIOD _____

3-73-7


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1. original: 25 2. original: 50new: 10 new: 75




Find the total price of each item.

9. dress: $69.00 10. binder: $14.50tax: 5% tax: 7%

11. hardcover book: $28.95 12. groceries: $47.52tax: 6% tax: 3%

13. filler paper: $6.00 14. shoes: $65.00tax: 6.5% tax: 4%

15. basketball: $17.00 16. concert tickets: $48.00tax: 6% tax: 7.5%

Find the discounted price of each item.

17. backpack: $56.25 18. monitor: $150.00discount: 20% discount: 50%

19. CD: $15.99 20. shirt: $25.50discount: 20% discount: 40%

21. sleeping bag: $125 22. coffee maker: $102.00discount: 25% discount: 45%





7. original: 15 8. original: 98.6 9. original: 58.8new: 35.5 new: 64 new: 65.7

Find the total price of each item.

10. concrete blocks: $95.00 11. crib: $240.00 12. jacket: $125.00tax: 6% tax: 6.5% tax: 5.5%

13. class ring: $325.00 14. blanket: $24.99 15. kite: $18.90tax: 6% tax: 7% tax: 5%

Find the discounted price of each item.

16. dry cleaning: $25.00 17. computer game: $49.99 18. luggage: $185.00discount: 15% discount: 25% discount: 30%

19. stationery: $12.95 20. prescription glasses: $149 21. pair of shorts: $24.99 discount: 10% discount: 20% discount: 45%

Find the final price of each item.

22. television: $375.00 23. DVD player: $269.00 24. printer: $255.00discount: 25% discount: 20% discount: 30%tax: 6% tax: 7% tax: 5.5%

25. INVESTMENTS The price per share of an internet-related stock decreased from $90 pershare to $36 per share early in 2001. By what percent did the price of the stockdecrease?

26. HEATING COSTS Customers of a utility company received notices in their monthly billsthat heating costs for the average customer had increased 125% over last year becauseof an unusually severe winter. In January of last year, the Garcia’s paid $120 for heating.What should they expect to pay this January if their bill increased by 125%?

Practice Percent of Change

NAME ______________________________________________ DATE ____________ PERIOD _____

3-73-7

Reading to Learn MathematicsPercent of Change

NAME ______________________________________________ DATE ____________ PERIOD _____

3-73-7


Less

on

3-7

Pre-Activity How can percents describe growth over time?


• How many area codes were in use in 1947?

• How many more area codes were in use in 1999?

Reading the Lesson

1. If you use (original amount) — (new amount) to find the change for a percent of change problem, then the problem involves a percent of (increase/decrease).

2. If you use (new amount) — (original amount) to find the change for a percent of change problem, then the problem involves a percent of (increase/decrease).

Complete the chart.

Original New Percent Increase or Amount Amount

Percent ProportionPercent Decrease?

3. 10 13

4. 10 7

5. 50 42

6. 50 58

7. When you find a discount price, do you add to or subtract from the original price?


8. If you remember only two things about the ratio used for finding percent of change, whatshould they be?


Using PercentUse what you have learned about percent to solve each problem.

A TV movie had a “rating” of 15 and a 25 “share.” The rating is the percentageof the nation’s total TV households that were tuned in to this show. The shareis the percentage of homes with TVs turned on that were tuned to the movie.How many TV households had their TVs turned off at this time?To find out, let T ! the number of TV households

and x ! the number of TV households with the TV off.Then T " x ! the number of TV households with the TV on.

Since 0.15T and 0.25(T " x) both represent the number of households tunedto the movie,

0.15T ! 0.25(T " x)0.15T ! 0.25T " 0.25x.

Solve for x. 0.25x ! 0.10Tx ! ! 0.40T

Forty percent of the TV households had their TVs off when the movie was aired.

Answer each question.

1. During that same week, a sports broadcast had a rating of 22.1 and a 43 share.Show that the percent of TV households with their TVs off was about 48.6%.0.221T ! 0.43T " 0.43x

x !

! 0.486T2. Find the percent of TV households with their TVs turned off during a show

with a rating of 18.9 and a 29 share. 34.8%

3. Show that if T is the number of TV households, r is the rating, and s is the

share, then the number of TV households with the TV off is .Solve rT ! s(T " x) for x.

4. If the fraction of TV households with no TV on is then show that the

fraction of TV households with TVs on is .

5. Find the percent of TV households with TVs on during the most watched serial program in history: the last episode of M*A*S*H, which had a 60.3 rating and a 77 share. ! 78.3%

6. A local station now has a 2 share. Each share is worth $50,000 in advertising revenue per month. The station is thinking of going commercial free for the three months of summer to gain more listeners. What would its new share have to be for the last 4 months of the year to make more money for the year than it would have made had it not gone commercial free? greater than 3.5

60.3#77

r#s

s – r#s

(s " r)T#s

0.221T " 0.43T##

"0.43

0.10T#0.25

Enrichment

NAME ______________________________________________ DATE______________ PERIOD _____

3-73-7

1 " ! r#s

s " r#s

Study Guide and InterventionSolving Equations and Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

3-83-8


Less

on

3-8

Solve for Variables Sometimes you may want to solve an equation such as V " !wh forone of its variables. For example, if you know the values of V, w, and h, then the equation

! " is more useful for finding the value of !. If an equation that contains more than one

variable is to be solved for a specific variable, use the properties of equality to isolate thespecified variable on one side of the equation.

V%wh

Solve 2x ! 4y " 8 for y.

2x # 4y " 82x # 4y # 2x " 8 # 2x

#4y " 8 # 2x

"

y " or

The value of y is .2x # 8%4

2x # 8%4

8 # 2x%

#4

8 # 2x%

#4#4y%#4

Solve 3m ! n " km ! 8 for m.

3m # n " km # 83m # n # km " km # 8 # km3m # n # km " # 8

3m # n # km ' n " # 8 ' n3m # km " #8 ' nm(3 # k) " #8 ' n

"

m " , or

The value of m is . Since division by 0 is

undefined, 3 # k ( 0, or k ( 3.

n # 8%3 # k

n # 8%3 # k

#8 ' n%3 # k

#8 ' n%3 # k

m(3 # k)%%3 # k


ExercisesExercises

Solve each equation or formula for the variable specified.

1. ax # b " c for x 2. 15x ' 1 " y for x 3. (x ' f) ' 2 " j for x

4. xy ' z " 9 for y 5. x(4 # k) " p for k 6. 7x ' 3y " m for y

7. 4(c ' 3) " t for c 8. 2x ' b " c for x 9. x(1 ' y) " z for x

10. 16z ' 4x " y for x 11. d " rt for r 12. A " for h

13. C" (F # 32) for F 14. P " 2! ' 2w for w 15. A " !w for !5%9

h(a ' b)%%2


Use Formulas Many real-world problems require the use of formulas. Sometimes solvinga formula for a specified variable will help solve the problem.

The formula C " %d represents the circumference of a circle, or thedistance around the circle, where d is the diameter. If an airplane could fly aroundEarth at the equator without stopping, it would have traveled about 24,900 miles.Find the diameter of Earth.

C " &d Given formula

d " Solve for d.

d " Use & " 3.14.

d ! 7930 Simplify.

The diameter of Earth is about 7930 miles.

1. GEOMETRY The volume of a cylinder V is given by the formula V " &r2h, where r isthe radius and h is the height.

a. Solve the formula for h.

b. Find the height of a cylinder with volume 2500& feet and radius 10 feet.

2. WATER PRESSURE The water pressure on a submerged object is given by P " 64d,where P is the pressure in pounds per square foot, and d is the depth of the object in feet.

a. Solve the formula for d.

b. Find the depth of a submerged object if the pressure is 672 pounds per square foot.

3. GRAPHS The equation of a line containing the points (a, 0) and (0, b) is given by the

formula ' " 1.

a. Solve the equation for y.

b. Suppose the line contains the points (4, 0), and (0, #2). If x " 3, find y.

4. GEOMETRY The surface area of a rectangular solid is given by the formula S " 2!w ' 2!h ' 2wh, where ! " length, w " width, and h " height.

a. Solve the formula for h.

b. The surface area of a rectangular solid with length 6 centimeters and width 3 centimeters is 72 square centimeters. Find the height.

y%b

x%a

24,900%3.14

C%&


Solving Equations and Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

3-83-8

ExampleExample

ExercisesExercises

Skills PracticeSolving Equations and Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

3-83-8


Less

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3-8


1. 7t " x, for t 2. e " wp, for p

3. q # r " r, for r 4. 4m # n " m, for m

5. 7a # b " 15a, for a 6. #5c ' d " 2c, for c

7. x # 2y " 1, for y 8. m ' 3n " 1, for n

9. 7f ' g " 5, for f 10. ax # c " b, for x

11. rt # 2n " y, for t 12. bc ' 3g " 2k, for c

13. kn ' 4f " 9v, for n 14. 8c ' 6j " 5p, for c

15. " d, for x 16. " d, for c

17. " q, for p 18. " a, for b

Write an equation and solve for the variable specified.

19. Five more than a number g is six less than twice a number h. Solve for g.

20. One fourth of a number q is three more than three times a number w. Solve for q.

21. Eight less than a number s is three more than four times a number t. Solve for s.

b # 4z%7

p ' 9%5

x # c%2

x # c%2



1. d " rt, for r 2. 6w # y " 2z, for w

3. mx ' 4y " 3c, for x 4. 9s # 5g " #4u, for s

5. ab ' 3c " 2d, for b 6. 2p " kx # q, for x

7. m ' a " a ' c, for m 8. h ' g " d, for h

9. y ' v " s, for y 10. a # q " k, for a

11. " h, for x 12. " c, for b

13. 2w # y " 7w # 2, for w 14. 3! ' y " 5 ' 5!, for !


15. Three times a number s plus 4 times a number y is 1 more than 6 times the number s.Solve for s.

16. Five times a number k minus 9 is two thirds of a number j. Solve for j.

ELECTRICITY For Exercises 17 and 18, use the following information.

The formula for Ohm’s Law is E " IR, where E represents voltage measured in volts,I represents current measured in amperes, and R represents resistance measured in ohms.

17. Solve the formula for R.

18. Suppose a current of 0.25 ampere flows through a resistor connected to a 12-volt battery.What is the resistance in the circuit?

MOTION For Exercises 19 and 20, use the following information.

In uniform circular motion, the speed v of a point on the edge of a spinning disk is v " r,

where r is the radius of the disk and T is the time it takes the point to travel once aroundthe circle.

19. Solve the formula for r.

20. Suppose a merry-go-round is spinning once every 3 seconds. If a point on the outsideedge has a speed of 12.56 feet per second, what is the radius of the merry-go-round? (Use 3.14 for &.)

2&%T

3b # 4%2

rx ' 9%5

3%4

2%3

2%5

2%3

Practice Solving Equations and Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

3-83-8

Reading to Learn MathematicsSolving Equations and Formulas

NAME ______________________________________________ DATE______________ PERIOD _____

3-83-8


Less

on 3

-8

Pre-Activity How are equations used to design roller coasters?


The equation g(195 ! h) " v2 contains several variables. What

number values do you know for these variables in this situation?

32 for g and 49 for v

Reading the Lesson

1. Suppose you have an equation with several variables. You want to solve for a particularvariable. How does the procedure compare with that for solving an equation with justone variable? How does the solution compare with the solution for an equation with onevariable?

Sample answer: The procedure is basically the same. You use propertiesof operations and equality to isolate the variable in which you areinterested. The solution will probably contain variables instead of just anumber.

2. Describe what dimensional analysis involves.

Sample answer: You use the units along with number values as you docalculations. You treat the units pretty much the same way you wouldvariables. For example, you can divide them and use exponents with them.

3. What do you have to be careful about when you use variables in denominators offractions?

You have to be sure that you do not use values that would make thedenominator 0.


4. When you give the dimensions of a rectangle, you have to tell how many units long it isand how many units wide it is. How can this help you remember what dimensionalanalysis involves.

Sample answer: Keep the words dimension and unit in mind. Indimensional analysis, you include units for dimensions in calculations.

1#2


Dr. Bernardo Houssay Even though researchers have been studying the disease diabetesmellitus for hundreds of years, scientists have only recently discoveredthe cause of the disease and developed methods for reducing itsseverity. Dr. Bernardo Houssay, an Argentine physiologist, was one ofthe pioneers of this more modern research. He studied the relationshipbetween diabetes and the pituitary gland, and in 1947 became the firstLatin American to win the Nobel Prize in Medicine and Physiology.

Though there is no cure for diabetes, specific diets and exercise can helppeople control the disease. The American Diabetes Association (ADA)has helped establish flexible dietary guidelines for consumers to follow.These guidelines include some of the following general nutrition rules.

• Fat intake should be equal to or less than 30% of daily calories.

• Saturated fat intake should be equal to or less than 10% of dailycalories.

• Protein should be limited to 10% to 20% of daily calories. Personsshowing the initial signs of diabetes-induced kidney disease shouldlimit protein to 10% of daily calories.

• Cholesterol intake should be 300 milligrams or less daily.

Refer to the information above for Exercises 1–4.

1. Robert consumed 2100 calories on Tuesday. His fat intake totaled70 grams, and of that 70 grams, 14 were saturated.

a. What percentage of his calorie consumption was fat, and whatpercentage of that fat was saturated? (To find the percentage ofcalories from fat, multiply the number of fat grams by 9 beforedividing by the number of calories.) 30%; 20%

b. Did Robert stay within the recommended allowance of fats? yes

2. Anna's cholesterol intake was 330 milligrams on Sunday. By whatpercentage does she need to reduce her cholesterol consumption toremain within the guidelines? 10%

3. What number of fat grams is 30% of 240 calories? 8

4. Sharon follows a diet that provides about 50 grams of protein eachday. Sharon's doctor has just told her to reduce her daily proteinintake by 30%. About how much protein will be in her reducedprotein diet? 35 grams

Enrichment

NAME ______________________________________________ DATE______________ PERIOD _____

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Study Guide and InterventionWeighted Averages

NAME ______________________________________________ DATE ____________ PERIOD _____

3-93-9


Less

on

3-9

Mixture Problems

Weighted AverageThe weighted average M of a set of data is the sum of the product of each number in the set and its weight divided by the sum of all the weights.

Mixture Problems are problems where two or more parts are combined into a whole. Theyinvolve weighted averages. In a mixture problem, the weight is usually a price or a percentof something.

Delectable Cookie Company sells chocolate chip cookies for $6.95per pound and white chocolate cookies for $5.95 per pound. How many pounds ofchocolate chip cookies should be mixed with 4 pounds of white chocolate cookiesto obtain a mixture that sells for $6.75 per pound.

Let w " the number of pounds of chocolate chip cookies

Number of Pounds Price per Pound Total Price

Chocolate Chip w 6.95 6.95w

White Chocolate 4 5.95 4(5.95)

Mixture w ' 4 6.75 6.75(w ' 4)

Equation: 6.95w ' 4(5.95) " 6.75(w ' 4)Solve the equation.

6.95w ' 4(5.95) " 6.75(w ' 4) Original equation6.95w ' 23.80 " 6.75w ' 27 Simplify.

6.95w ' 23.80 # 6.75w " 6.75w ' 27 # 6.75w Subtract 6.75w from each side.0.2w ' 23.80 " 27 Simplify.

0.2w ' 23.80 # 23.80 " 27 # 23.80 Subtract 23.80 from each side.0.2w " 3.2 Simplify.

w " 16 Simplify.

16 pounds of chocolate chip cookies should be mixed with 4 pounds of white chocolate cookies.

1. SOLUTIONS How many grams of sugar must be added to 60 grams of a solution that is32% sugar to obtain a solution that is 50% sugar?

2. NUTS The Quik Mart has two kinds of nuts. Pecans sell for $1.55 per pound andwalnuts sell for $1.95 per pound. How many pounds of walnuts must be added to 15pounds of pecans to make a mixture that sells for $1.75 per pound?

3. INVESTMENTS Alice Gleason invested a portion of $32,000 at 9% interest and thebalance at 11% interest. How much did she invest at each rate if her total income fromboth investments was $3,200.

4. MILK Whole milk is 4% butterfat. How much skim milk with 0% butterfat should beadded to 32 ounces of whole milk to obtain a mixture that is 2.5% butterfat?

ExampleExample

ExercisesExercises


Uniform Motion Problems Motion problems are another application of weightedaverages. Uniform motion problems are problems where an object moves at a certainspeed, or rate. Use the formula d " rt to solve these problems, where d is the distance, r isthe rate, and t is the time.

Bill Gutierrez drove at a speed of 65 miles per hour on anexpressway for 2 hours. He then drove for 1.5 hours at a speed of 45 miles perhour on a state highway. What was his average speed?

M " Definition of weighted average

! 56.4 Simplify.

Bill drove at an average speed of about 56.4 miles per hour.

1. TRAVEL Mr. Anders and Ms. Rich each drove home from a business meeting. Mr. Anderstraveled east at 100 kilometers per hour and Ms. Rich traveled west at 80 kilometers perhours. In how many hours were they 100 kilometers apart.

2. AIRPLANES An airplane flies 750 miles due west in 1 hours and 750 miles due south

in 2 hours. What is the average speed of the airplane?

3. TRACK Sprinter A runs 100 meters in 15 seconds, while sprinter B starts 1.5 secondslater and runs 100 meters in 14 seconds. If each of them runs at a constant rate, who isfurther in 10 seconds after the start of the race? Explain.

4. TRAINS An express train travels 90 kilometers per hour from Smallville to Megatown.A local train takes 2.5 hours longer to travel the same distance at 50 kilometers perhour. How far apart are Smallville and Megatown?

5. CYCLING Two cyclists begin traveling in the same direction on the same bike path. Onetravels at 15 miles per hour, and the other travels at 12 miles per hour. When will thecyclists be 10 miles apart?

6. TRAINS Two trains leave Chicago, one traveling east at 30 miles per hour and onetraveling west at 40 miles per hour. When will the trains be 210 miles apart?

1%2

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Weighted Averages

NAME ______________________________________________ DATE ____________ PERIOD _____

3-93-9

ExampleExample

ExercisesExercises

Skills PracticeWeighted Averages

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Less

on

3-9

SEASONING For Exercises 1–4, use the following information.

A health food store sells seasoning blends in bulk. One blend contains 20% basil. Sheilawants to add pure basil to some 20% blend to make 16 ounces of her own 30% blend. Let brepresent the amount of basil Sheila should add to the 20% blend.

1. Complete the table representing the problem.

Ounces Amount of Basil

20% Basil Blend

100% Basil

30% Basil Blend

2. Write an equation to represent the problem.

3. How many ounces of basil should Sheila use to make the 30% blend?

4. How many ounces of the 20% blend should she use?

HIKING For Exercises 5–7, use the following information.

At 7:00 A.M., two groups of hikers begin 21 miles apart and head toward each other. The firstgroup, hiking at an average rate of 1.5 miles per hour, carries tents, sleeping bags, andcooking equipment. The second group, hiking at an average rate of 2 miles per hour, carriesfood and water. Let t represent the hiking time.

5. Copy and complete the table representing the problem.

r t d " rt

First group of hikers

Second group of hikers

6. Write an equation using t that describes the distances traveled.

7. How long will it be until the two groups of hikers meet?

SALES For Exercises 8 and 9, use the following information.

Sergio sells a mixture of Virginia peanuts and Spanish peanuts for $3.40 per pound. Tomake the mixture, he uses Virginia peanuts that cost $3.50 per pound and Spanish peanutsthat cost $3.00 per pound. He mixes 10 pounds at a time.

8. How many pounds of Virginia peanuts does Sergio use?

9. How many pounds of Spanish peanuts does Sergio use?


GRASS SEED For Exercises 1–4, use the following information.

A nursery sells Kentucky Blue Grass seed for $5.75 per pound and Tall Fescue seed for$4.50 per pound. The nursery sells a mixture of the two kinds of seed for $5.25 per pound.Let k represent the amount of Kentucky Blue Grass seed the nursery uses in 5 pounds ofthe mixture.


Number of Pounds Price per Pound Cost

Kentucky Blue Grass

Tall Fescue

Mixture

2. Write an equation to represent the problem.

3. How much Kentucky Blue Grass does the nursery use in 5 pounds of the mixture?

4. How much Tall Fescue does the nursery use in 5 pounds of the mixture?

TRAVEL For Exercises 5–7, use the following information.

Two commuter trains carry passengers between two cities, one traveling east, and the otherwest, on different tracks. Their respective stations are 150 miles apart. Both trains leave atthe same time, one traveling at an average speed of 55 miles per hour and the other at anaverage speed of 65 miles per hour. Let t represent the time until the trains pass each other.


r t d " rt

First Train

Second Train

6. Write an equation using t that describes the distances traveled.

7. How long after departing will the trains pass each other?

8. TRAVEL Two trains leave Raleigh at the same time, one traveling north, and the othersouth. The first train travels at 50 miles per hour and the second at 60 miles per hour.In how many hours will the trains be 275 miles apart?

9. JUICE A pineapple drink contains 15% pineapple juice. How much pure pineapple juiceshould be added to 8 quarts of the drink to obtain a mixture containing 50% pineapplejuice?

Practice Weighted Averages

NAME ______________________________________________ DATE ____________ PERIOD _____

3-93-9

Reading to Learn MathematicsWeighted Averages

NAME ______________________________________________ DATE ____________ PERIOD _____

3-93-9


Less

on

3-9

Pre-Activity How are scores calculated in a figure skating competition?


Why is the sum of Ilia Kulik’s scores divided by 3?

Reading the Lesson

1. Read the definition of weighted average on page 171 of your textbook. What is meant bythe weight of a number in a set of data?

2. Linda’s quiz scores in science are 90, 85, 85, 75, 85, and 90. What is the weight of thescore 85?

3. Suppose Clint drives at 50 miles per hour for 2 hours. Then he drives at 60 miles perhour for 3 hours.

a. Write his speed for each hour of the trip.

Speed

Hour 1 2 3 4 5

b. What is the weight of each of the two speeds?


4. Making a table can be helpful in solving mixture problems. In your own words, explainhow you use a table to solve mixture problems.


Diophantine EquationsThe first great algebraist, Diophantus of Alexandria (about A.D. 300),devoted much of his work to the solving of indeterminate equations.An indeterminate equation has more than one variable and anunlimited number of solutions. An example is x ' 2y " 4.

When the coefficients of an indeterminate equation are integers andyou are asked to find solutions that must be integers, the equation iscalled diophantine. Such equations can be quite difficult to solve,often involving trial and error—and some luck!

Solve each diophantine equation by finding at least one pair of positive integers that makes the equation true. Some hints are given to help you.

1. 2x ' 5y " 32

a. First solve the equation for x.

b. Why must y be an even number?

c. Find at least one solution.

2. 5x ' 2y " 42

a. First solve the equation for x.

b. Rewrite your answer in the form x " 8 ' some expression.

c. Why must (2 # 2y) be a multiple of 5?

d. Find at least one solution.

3. 2x ' 7y " 29 4. 7x ' 5y " 118

5. 8x # 13y " 100 6. 3x ' 4y " 22

7. 5x # 14y " 11 8. 7x ' 3y " 40

Enrichment

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3-93-9

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equa

ls #

12.W

hat

is t

he n

umbe

r?n

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12;!

27

24.T

he s

um o

f a

num

ber

and

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qual

to

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.Fin

d th

e nu

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63

plus

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umbe

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hat

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r?!

17 "

63 #

n;!

8026

.The

sum

of

nega

tive

14,

a nu

mbe

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is #

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hat

is t

he n

umbe

r?!

14 #

n#

6 "

!5;

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.Wha

t nu

mbe

r pl

us o

ne h

alf

is e

qual

to

thre

e ei

ghth

s?n

#"

;!

HIS

TORY

For

Exe

rcis

es 2

8 an

d 2

9,u

se t

he

foll

owin

g in

form

atio

n.

Gal

ileo

Gal

ilei w

as b

orn

in 1

564.

Man

y ye

ars

late

r,in

164

2,Si

r Is

aac

New

ton

was

bor

n.

28.W

rite

an

addi

tion

equ

atio

n to

rep

rese

nt t

he s

itua

tion

.15

64 #

y"

1642

29.H

ow m

any

year

s af

ter

Gal

ileo

was

bor

n w

as I

saac

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ton

born

?78

HU

RR

ICA

NES

For

Exe

rcis

es 3

0 an

d 3

1,u

se t

he

foll

owin

g in

form

atio

n.

The

day

aft

er a

hur

rica

ne,t

he b

arom

etri

c pr

essu

re in

a c

oast

al t

own

has

rise

n to

29

.7 in

ches

of

mer

cury

,whi

ch is

2.9

inch

es o

f m

ercu

ry h

ighe

r th

an t

he p

ress

ure

whe

n th

eey

e of

the

hur

rica

ne p

asse

d ov

er.

30.W

rite

an

addi

tion

equ

atio

n to

rep

rese

nt t

he s

itua

tion

.b

#2.

9 "

29.7

31.W

hat

was

the

bar

omet

ric

pres

sure

whe

n th

e ey

e pa

ssed

ove

r?26

.8 in

.of

mer

cury

1 % 83 % 8

1 % 2

7 % 247 % 12

7 % 81 % 20

3 % 44 % 5

5 % 67 % 12

1 % 4

1 % 152 % 5

1 % 31 % 9

2 % 35 % 9

7 % 109 % 10

1 % 5

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ctic

e (A

vera

ge)

Sol

ving

Equ

atio

ns b

y U

sing

Add

ition

and

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trac

tion

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ME

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____

____

____

____

____

____

____

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____

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____

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___

3-2

3-2



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csS

olvi

ng E

quat

ions

by

Usi

ng A

dditi

on a

nd S

ubtr

actio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

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____

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___

3-2

3-2

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Lesson 3-2

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ow c

an e

quat

ion

s be

use

d t

o co

mp

are

dat

a?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 3-

2 at

the

top

of

page

128

in y

our

text

book

.

In t

he e

quat

ion

m#

66 "

5,th

e nu

mbe

r 5

repr

esen

ts

the

diff

eren

ce b

etw

een

the

perc

ent

of g

row

th fo

r m

edic

al

assi

stan

ts a

nd t

he p

erce

nt o

f gr

owth

for

trav

el a

gent

s,an

d th

e nu

mbe

r 66

rep

rese

nts

the

rate

of

grow

th fo

r tr

avel

age

nts.

Rea

din

g t

he

Less

on

1.To

sol

ve x

'17

"46

usi

ng t

he S

ubtr

acti

on P

rope

rty

of E

qual

ity,

you

wou

ld s

ubtr

act

from

eac

h si

de.

2.To

sol

ve y

#9

"#

30 u

sing

the

Add

itio

n P

rope

rty

of E

qual

ity,

you

wou

ld a

dd

to e

ach

side

.

3.W

rite

an

equa

tion

tha

t yo

u co

uld

solv

e by

sub

trac

ting

32

from

eac

h si

de.

Sam

ple

answ

er:m

#32

"50

4.A

stu

dent

use

d th

e Su

btra

ctio

n P

rope

rty

of E

qual

ity

to s

olve

an

equa

tion

.Exp

lain

why

itw

ould

als

o be

pos

sibl

e to

use

the

Add

itio

n P

rope

rty

of E

qual

ity

to s

olve

the

equ

atio

n.S

ubtr

actin

g on

e nu

mbe

r fr

om a

noth

er g

ives

the

sam

e re

sult

as a

ddin

gth

e op

posi

te o

f th

e nu

mbe

r th

at w

as s

ubtr

acte

d.

Hel

pin

g Y

ou

Rem

emb

er

5.E

xpla

in h

ow y

ou d

ecid

e w

heth

er t

o us

e th

e A

ddit

ion

Pro

pert

y or

the

Sub

trac

tion

Pro

pert

y of

Equ

alit

y to

sol

ve a

n eq

uati

on.

Sam

ple

answ

er:I

f th

e gi

ven

equa

tion

has

a nu

mbe

r ad

ded

to t

heva

riab

le,t

hen

use

the

Sub

trac

tion

Pro

pert

y of

Equ

ality

.If

the

equa

tion

has

a nu

mbe

r su

btra

cted

fro

m t

he v

aria

ble,

then

use

the

Add

ition

Pro

pert

y of

Equ

ality

.

917

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Cou

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g-O

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uzzl

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olve

eac

h p

uzz

le.

1.T

wen

ty-f

ive

peop

le a

re s

tand

ing

in a

cir

cle.

Star

ting

wit

h pe

rson

1,t

hey

coun

t of

f fr

om

1 to

7 a

nd t

hen

star

t ov

er w

ith

1.E

ach

pers

on w

ho s

ays

“7”

drop

s ou

t of

the

cir

cle.

Who

is t

he la

st p

erso

n le

ft?

num

ber

15

2.Fo

rty

peop

le s

tand

in a

cir

cle.

The

y co

unt

off

so t

hat

ever

y th

ird

pers

on d

rops

out

.Whi

ch t

wo

peop

le a

re t

he la

st o

nes

left

?13

th a

nd 2

8th

peop

le

3.O

nly

half

of

the

30 s

tude

nts

in

Shar

on’s

cla

ss c

an g

o on

a f

ield

tr

ip.S

haro

n ar

rang

es t

he b

oys

and

girl

s as

sho

wn.

The

y co

unt

off

from

1 t

o 9

and

ever

y ni

nth

pers

on d

rops

out

unt

il on

ly

15 p

eopl

e ar

e le

ft.W

ho g

ets

to

go o

n th

e fi

eld

trip

.th

e gi

rls

A g

rou

p o

f p

eop

le s

tan

d i

n a

cir

cle

and

cou

nt

off

1,2,

1,2,

1 an

d

so o

n.E

very

sec

ond

per

son

dro

ps

out.

Per

son

nu

mbe

r 1

is t

he

last

per

son

lef

t.

4.D

raw

a d

iagr

am t

o sh

ow w

hy t

he n

umbe

r of

peo

ple

in t

he

circ

le m

ust

be e

ven.

The

n,ex

plai

n yo

ur a

nsw

er.

If th

e nu

mbe

r is

odd

,per

son

1 w

ould

dro

p ou

t af

ter

the

first

rou

nd.

5.W

hen

the

coun

t re

turn

s to

per

son

num

ber

1 fo

r th

e fi

rst

tim

e,ho

w m

any

peop

le h

ave

drop

ped

out?

half

of t

he o

rigi

nal n

umbe

r

6.F

ind

the

num

ber

of p

eopl

e in

the

cir

cle

if t

he n

umbe

r is

be

twee

n 10

and

20.

Do

the

sam

e if

the

num

ber

is b

etw

een

30 a

nd 4

0.W

hat

can

you

conc

lude

abo

ut t

he o

rigi

nal

num

ber

of p

eopl

e?16

;32;

The

num

ber

mus

t be

a p

ower

of

2.

G

GG

G

GG

GG

GG

G

GG

G

GB

B

BB

B

BB

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23

45

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1213

14

1516

1718

1920

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2324

2526

2728

2930

12

34

56

7

8 9 10

11

1213

1415

1617

18

19

20

2122

23

2425

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-2

3-2



Stu

dy

Gu

ide

an

d I

nte

rven

tion

Sol

ving

Equ

atio

ns b

y U

sing

Mul

tiplic

atio

n an

d D

ivis

ion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

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RIO

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___

3-3

3-3

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Lesson 3-3

Solv

e U

sin

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ult

iplic

atio

nIf

eac

h si

de o

f an

equ

atio

n is

mul

tipl

ied

by t

he s

ame

num

ber,

the

resu

ltin

g eq

uati

on is

equ

ival

ent

to t

he g

iven

one

.You

can

use

the

pro

pert

y to

solv

e eq

uati

ons

invo

lvin

g m

ulti

plic

atio

n an

d di

visi

on.

Mul

tiplic

atio

n P

rope

rty

of E

qual

ityF

or a

ny n

umbe

rs a

, b, a

nd c

, if a

"b,

then

ac

"bc

.

Sol

ve 3

p"

1.

3p

"1

Orig

inal

equ

atio

n

p"

Rew

rite

each

mix

ed n

umbe

r as

an

impr

oper

frac

tion.

"p #

""

#M

ultip

ly e

ach

side

by

.

p"

Sim

plify

.

The

sol

utio

n is

.

3 % 7

3 % 7

2 % 73 % 2

2 % 77 % 2

2 % 7

3 % 27 % 2

1 % 21 % 2

1 % 21 % 2

Sol

ve !

n"

16.

#n

"16

Orig

inal

equ

atio

n

#4 "#

n #"

#4(

16)

Mul

tiply

eac

h si

de b

y #

4.

n"

#64

Sim

plify

.

The

sol

utio

n is

#64

.

1 % 41 % 4

1 % 4Ex

ampl

e1Ex

ampl

e1Ex

ampl

e2Ex

ampl

e2

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cises

Exer

cises

Sol

ve e

ach

equ

atio

n.T

hen

ch

eck

you

r so

luti

on.

1."

#2

!6

2.m

"6

483.

p"

3

4.5

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5.#

k"

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"#

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te a

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of

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r eq

uals

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Fin

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3.F

ind

the

num

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24

16.O

ne a

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hal

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mes

a n

umbe

r eq

uals

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ind

the

num

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Solv

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To s

olve

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atio

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mul

tipl

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and

divi

sion

,you

can

als

ous

e th

e D

ivis

ion

Pro

pert

y of

Equ

alit

y.If

eac

h si

de o

f an

equ

atio

n is

div

ided

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the

sam

enu

mbe

r,th

e re

sult

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tion

is t

rue.

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isio

n P

rope

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qual

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nte

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y U

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tiplic

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n an

d D

ivis

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ME

____

____

____

____

____

____

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____

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3-3

3-3

Sol

ve 8

n"

64.

8n"

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rigin

al e

quat

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ivid

e ea

ch s

ide

by 8

.

n"

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impl

ify.

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sol

utio

n is

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64 % 88n % 8

Sol

ve !

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Orig

inal

equ

atio

n

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ivid

e ea

ch s

ide

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5.

n"

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plify

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The

sol

utio

n is

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.

60 % #5

#5n

% #5

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ple1

Exam

ple1

Exam

ple2

Exam

ple2

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cises

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ve e

ach

equ

atio

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hen

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eck

you

r so

luti

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25 "

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ve t

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atio

n.

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our

tim

es a

num

ber

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ls 6

4.F

ind

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num

ber.

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64;1

6

14.W

hat

num

ber

mul

tipl

ied

by #

4 eq

uals

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?!

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;4

15.A

num

ber

tim

es e

ight

equ

als

#36

.Fin

d th

e nu

mbe

r.8n

"!

36;!

41 % 2

1 % 2

1 % 3



An

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s

Skil

ls P

ract

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Sol

ving

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atio

ns b

y U

sing

Mul

tiplic

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n an

d D

ivis

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ME

____

____

____

____

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____

____

____

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ATE

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Lesson 3-3

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ve e

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hen

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eck

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luti

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atio

n.

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he o

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ite

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hat

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9

20.F

ourt

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ight

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22.N

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r eq

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es a

num

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54.W

hat

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24.O

ne s

ixth

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r is

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.Fin

d th

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17;!

102

25.N

egat

ive

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fths

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mbe

r is

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t is

the

num

ber?

!n

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Sol

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Lesson 3-4

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28

17.

q#

7 "

820

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20.

m'

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65

21.

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1722

."

25

Wri

te a

n e

quat

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an

d s

olve

eac

h p

robl

em.

23.T

wic

e a

num

ber

plus

fou

r eq

uals

6.W

hat

is t

he n

umbe

r?2n

#4

"6;

1

24.S

ixte

en is

sev

en p

lus

thre

e ti

mes

a n

umbe

r.F

ind

the

num

ber.

16 "

7 #

3n;3

25.F

ind

two

cons

ecut

ive

inte

gers

who

se s

um is

35.

n#

(n#

1) "

35;1

7,18

26.F

ind

thre

e co

nsec

utiv

e in

tege

rs w

hose

sum

is 3

6.n

#(n

#1)

#(n

#2)

"36

;11

,12,

13

b'

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3c

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4 % 5

5 % 22 % 3

3 % 4

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h % 3x % 4

n % 3

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lenc

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lenc

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lgeb

ra 1

Sol

ve e

ach

pro

blem

by

wor

kin

g ba

ckw

ard

.

1.T

hree

is a

dded

to

a nu

mbe

r,an

d th

en t

he s

um is

mul

tipl

ied

by 4

.The

res

ult

is 1

6.F

ind

the

num

ber.

1

2.A

num

ber

is d

ivid

ed b

y 4,

and

the

quot

ient

is a

dded

to

3.T

he r

esul

t is

24.

Wha

t is

the

num

ber?

84

3.T

wo

is s

ubtr

acte

d fr

om a

num

ber,

and

then

the

dif

fere

nce

is m

ulti

plie

d by

5.T

he r

esul

tis

30.

Fin

d th

e nu

mbe

r.8

4.B

IRD

WA

TCH

ING

Whi

le M

iche

lle s

at o

bser

ving

bir

ds a

t a

bird

fee

der,

one

four

th o

f th

ebi

rds

flew

aw

ay w

hen

they

wer

e st

artl

ed b

y a

nois

e.T

wo

bird

s le

ft t

he f

eede

r to

go

toan

othe

r st

atio

ned

a fe

w f

eet

away

.Thr

ee m

ore

bird

s fl

ew in

to t

he b

ranc

hes

of a

nea

rby

tree

.Fou

r bi

rds

rem

aine

d at

the

fee

der.

How

man

y bi

rds

wer

e at

the

fee

der

init

ially

?12

Sol

ve e

ach

equ

atio

n.T

hen

ch

eck

you

r so

luti

on.

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te a

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olve

eac

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robl

em.

20.S

even

less

tha

n fo

ur t

imes

a n

umbe

r eq

uals

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Wha

t is

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num

ber?

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7 "

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21.F

ind

two

cons

ecut

ive

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inte

gers

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se s

um is

116

.n

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#2)

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6;57

,59

22.F

ind

two

cons

ecut

ive

even

inte

gers

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se s

um is

126

.n

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6;62

,64

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ind

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e co

nsec

utiv

e od

d in

tege

rs w

hose

sum

is 1

17.

n#

(n#

2) #

(n#

4) "

117;

37,3

9,41

24.C

OIN

CO

LLEC

TIN

GJu

ng h

as a

tot

al o

f 92

coi

ns in

his

coi

n co

llect

ion.

Thi

s is

8 m

ore

than

thr

ee t

imes

the

num

ber

of q

uart

ers

in t

he c

olle

ctio

n.H

ow m

any

quar

ters

doe

s Ju

ngha

ve in

his

col

lect

ion?

28

x % 7

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ving

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ti-S

tep

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atio

ns

NA

ME

____

____

____

____

____

____

____

____

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____

____

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ATE

____

____

____

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RIO

D__

___

3-4

3-4



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csS

olvi

ng M

ulti-

Ste

p E

quat

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

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RIO

D__

___

3-4

3-4

©G

lenc

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cGra

w-H

ill15

9G

lenc

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lgeb

ra 1

Lesson 3-4

Pre-

Act

ivit

yH

ow c

an e

quat

ion

s be

use

d t

o es

tim

ate

the

age

of a

n a

nim

al?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 3-

4 at

the

top

of

page

142

in y

our

text

book

.

•W

rite

the

equ

atio

n 8

'12

a"

124

in w

ords

.

Eig

ht p

lus

twel

ve t

imes

aeq

uals

one

hun

dred

tw

enty

-fou

r.•

How

man

y op

erat

ions

are

invo

lved

in t

he e

quat

ion?

two

Rea

din

g t

he

Less

on

1.W

hat

does

the

phr

ase

undo

the

ope

rati

ons

mea

n to

you

? G

ive

an e

xam

ple.

Usi

ng t

he o

ppos

ite o

pera

tions

in t

he o

ppos

ite o

rder

und

oes

the

oper

atio

ns;s

ubtr

actio

n un

does

add

ition

.

2.a.

If w

e un

do o

pera

tion

s in

rev

erse

of

the

orde

r of

ope

rati

ons,

wha

t op

erat

ions

do

we

dofi

rst?

addi

tion

or s

ubtr

actio

n

b.W

hat

oper

atio

ns d

o w

e do

last

?m

ultip

licat

ion

or d

ivis

ion

3.Su

ppos

e yo

u w

ant

to s

olve

"

6.

a.W

hat

is t

he g

roup

ing

sym

bol i

n th

e eq

uati

on

"6?

the

frac

tion

bar

b.W

hat

is t

he f

irst

ste

p in

sol

ving

the

equ

atio

n?M

ultip

ly e

ach

side

by

5.

c.W

hat

is t

he n

ext

step

in s

olvi

ng t

he e

quat

ion?

Sub

trac

t 3

from

eac

h si

de.

4.W

rite

an

equa

tion

for

the

pro

blem

bel

ow.

Sev

entim

esk

min

usfiv

eeq

uals

nega

tive

fort

y-se

ven

7•

k!

5"

!47

Hel

pin

g Y

ou

Rem

emb

er

5.E

xpla

in w

hy w

orki

ng b

ackw

ard

is a

use

ful s

trat

egy

for

solv

ing

equa

tion

s.

Sam

ple

answ

er:Y

ou c

an u

ndo

the

oper

atio

ns t

o ge

t ba

ck t

o th

e va

lue

ofth

e va

riab

le t

hat

will

mak

e th

e eq

uatio

n tr

ue.T

hat

valu

e is

the

sol

utio

n.

x'

3%

5

x'

3%

5

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0G

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Con

secu

tive

Inte

ger

Pro

blem

sM

any

type

s of

pro

blem

s an

d pu

zzle

s in

volv

e th

e id

ea o

f co

nsec

utiv

ein

tege

rs.K

now

ing

how

to

repr

esen

t th

ese

inte

gers

alg

ebra

ical

ly c

anhe

lp t

o so

lve

the

prob

lem

.

Fin

d f

our

con

secu

tive

od

d i

nte

gers

wh

ose

sum

is

!80

.

An

odd

inte

ger

can

be w

ritt

en a

s 2n

'1,

whe

re n

is a

ny in

tege

r.If

2n

+ 1

is t

he f

irst

odd

inte

ger,

then

add

2 t

o ge

t th

e ne

xt la

rges

t od

din

tege

r,an

d so

on.

Now

wri

te a

n eq

uati

on t

o so

lve

this

pro

blem

.(2

n'

1) '

(2n

'3)

'(2

n'

5) '

(2n

'7)

"#

80

Wri

te a

n e

quat

ion

for

eac

h p

robl

em.T

hen

sol

ve.

1.C

ompl

ete

the

solu

tion

to

the

prob

lem

in t

he e

xam

ple.

!23

,!21

,!19

,!17

2.F

ind

thre

e co

nsec

utiv

e ev

en in

tege

rs w

hose

sum

is 1

32.

2n#

(2n

#2)

#(2

n#

4) "

132;

n"

21;4

2,44

,46

3.F

ind

two

cons

ecut

ive

inte

gers

who

se s

um is

19.

n#

(n#

1) "

19;9

,10

4.F

ind

two

cons

ecut

ive

inte

gers

who

se s

um is

100

.

n#

(n#

1) "

100;

no s

olut

ion

5.T

he le

sser

of

two

cons

ecut

ive

even

inte

gers

is 1

0 m

ore

than

on

e-ha

lf t

he g

reat

er.F

ind

the

inte

gers

.

2n"

10 #

(2n

#2)

;22

and

24

6.T

he g

reat

er o

f tw

o co

nsec

utiv

e ev

en in

tege

rs is

6 le

ss t

han

thre

e ti

mes

the

less

er.F

ind

the

inte

gers

.

2n#

2 "

3(2n

) !

6;4,

6

7.F

ind

four

con

secu

tive

inte

gers

suc

h th

at t

wic

e th

e su

m o

f th

e tw

o gr

eate

r in

tege

rs e

xcee

ds t

hree

tim

es t

he f

irst

by

91.

2[(n

#2)

#(n

#3)

] "

3n#

91;8

1,82

,83,

84

8.F

ind

a se

t of

fou

r co

nsec

utiv

e po

siti

ve in

tege

rs s

uch

that

the

gr

eate

st in

tege

r in

the

set

is t

wic

e th

e le

ast

inte

ger

in t

he s

et.

n#

3 "

2n;{

3,4,

5,6}

1 % 2

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-4

3-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Stu

dy

Gu

ide

an

d I

nte

rven

tion

Sol

ving

Equ

atio

ns w

ith t

he V

aria

ble

on E

ach

Sid

e

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-5

3-5

©G

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w-H

ill16

1G

lenc

oe A

lgeb

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Lesson 3-5

Var

iab

les

on

Eac

h S

ide

To s

olve

an

equa

tion

wit

h th

e sa

me

vari

able

on

each

sid

e,fi

rst

use

the

Add

itio

n or

the

Sub

trac

tion

Pro

pert

y of

Equ

alit

y to

wri

te a

n eq

uiva

lent

equa

tion

tha

t ha

s th

e va

riab

le o

n ju

st o

ne s

ide

of t

he e

quat

ion.

The

n so

lve

the

equa

tion

.

Sol

ve 5

y!

8 "

3y#

12.

5y#

8"

3y'

125y

#8

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"3y

'12

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2y#

8"

122y

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"12

'8

2y"

20

"

y"

10

The

sol

utio

n is

10.20 % 2

2y % 2

Sol

ve !

11 !

3y"

8y#

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y'

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11 #

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y

"

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The

sol

utio

n is

#1

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1 % 11

11y

% 11#

12%11

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sol

ve e

ach

equ

atio

n.T

hen

ch

eck

you

r so

luti

on.

1.6

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35

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Gro

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bo

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solv

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equa

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s th

at c

onta

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roup

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sym

bols

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st u

seth

e D

istr

ibut

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Pro

pert

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elim

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oupi

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ymbo

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hen

solv

e.

Sol

ve 4

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impl

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ve e

ach

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hen

ch

eck

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r so

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on.

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sol

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54 % 1818

a% 18St

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ide

and I

nte

rven

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(c

onti

nued

)

Sol

ving

Equ

atio

ns w

ith t

he V

aria

ble

on E

ach

Sid

e

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-5

3-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Sol

ving

Equ

atio

ns w

ith t

he V

aria

ble

on E

ach

Sid

e

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-5

3-5

©G

lenc

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cGra

w-H

ill16

3G

lenc

oe A

lgeb

ra 1

Lesson 3-5

Just

ify

each

ste

p.

1.4k

#3

"2k

'5

4k#

3 #

2k"

2k'

5 #

2ka.

Sub

trac

t 2k

from

eac

h si

de.

2k#

3 "

5b.

Sim

plify

.2k

#3

'3

"5

'3

c.A

dd 3

to

each

sid

e.2k

"8

d.S

impl

ify.

"e.

Div

ide

each

sid

e by

2.

k"

4f.

Sim

plify

.

2.2(

8u'

2) "

3(2u

#7)

16u

'4

"6u

#21

a.D

istr

ibut

ive

Pro

pert

y16

u'

4 #

6u"

6u#

21 #

6ub.

Sub

trac

t 6u

from

eac

h si

de.

10u

'4

"#

21c.

Sim

plify

.10

u'

4 #

4 "

#21

#4

d.S

ubtr

act

4 fr

om e

ach

side

.10

u"

#25

e.S

impl

ify.

"f.

Div

ide

each

sid

e by

10.

u"

#2.

5g.

Sim

plify

.

Sol

ve e

ach

equ

atio

n.T

hen

ch

eck

you

r so

luti

on.

3.2m

'12

"3m

#31

434.

2h#

8 "

h'

1725

5.7a

#3

"3

#2a

6.4n

#12

"12

#4n

3

7.4x

#9

"7x

'12

!7

8.#

6y#

3 "

3 #

6yno

sol

utio

n

9.5

'3r

"5r

#19

1210

.#9

'8k

"7

'4k

4

11.8

q'

12 "

4(3

'2q

)al

l num

bers

12.3

(5j'

2) "

2(3j

#6)

!2

13.6

(#3v

'1)

"5(

#2v

#2)

214

.#7(

2b#

4) "

5(#

2b'

6)!

0.5

or !

15.3

(8 #

3t) "

5(2

't)

116

.2(3

u'

7) "

#4(

3 #

2u)

13

17.8

(2f

#2)

"7(

3f'

2)!

618

.5(#

6 #

3d) "

3(8

'7d

)!

1.5

or !

1

19.6

(w#

1) "

3(3w

'5)

!7

20.7

(#3y

'2)

"8(

3y#

2)

21.

v#

6 "

6 #

v9

22.

#x

"x

'!

27 % 2

7 % 85 % 8

1 % 22 % 3

2 % 3

2 % 3

1 % 21 % 2

2 % 3#25

%10

10u

% 10

8 % 22k % 2

©G

lenc

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cGra

w-H

ill16

4G

lenc

oe A

lgeb

ra 1

Sol

ve e

ach

equ

atio

n.T

hen

ch

eck

you

r so

luti

on.

1.5x

#3

"13

#3x

22.

#4c

#11

"4c

'21

!4

3.1

#s

"6

#6s

14.

14 '

5n"

#4n

'17

5.k

#3

"2

#k

46.

(6 #

z) "

z2

7.3(

#2

#3x

) "#

9x#

4no

sol

utio

n8.

4(4

#w

) "3(

2w'

2)1

9.9(

4b#

1) "

2(9b

'3)

10.3

(6 '

5y) "

2(#

5 '

4y)

!4

11.#

5x#

10 "

2 #

(x'

4)!

212

.6 '

2(3j

#2)

"4(

1 '

j)1

13.

t#

t"

3 '

tno

sol

utio

n14

.1.4

f'

1.1

"8.

3 #

f3

15.

x#

"x

'6

16.2

#z

"z

'9

!8

17.

(3g

#2)

"18

.(c

'1)

"(3

c#

5)7

19.

(5 #

2h) "

120

.(2

m#

16)

"(2

m'

4)!

7 all n

umbe

rs21

.3(d

#8)

#5

"9(

d'

2) '

1!

822

.2(a

#8)

'7

"5(

a'

2) #

3a#

19

23.T

wo

thir

ds o

f a

num

ber

redu

ced

by 1

1 is

equ

al t

o 4

mor

e th

an t

he n

umbe

r.F

ind

the

num

ber.

!45

24.F

ive

tim

es t

he s

um o

f a

num

ber

and

3 is

the

sam

e as

3 m

ulti

plie

d by

1 le

ss t

han

twic

eth

e nu

mbe

r.W

hat

is t

he n

umbe

r?18

25.N

UM

BER

TH

EORY

Tri

plin

g th

e gr

eate

r of

tw

o co

nsec

utiv

e ev

en in

tege

rs g

ives

the

sam

ere

sult

as

subt

ract

ing

10 f

rom

the

less

er e

ven

inte

ger.

Wha

t ar

e th

e in

tege

rs?

!8,

!6

26.G

EOM

ETRY

The

for

mul

a fo

r th

e pe

rim

eter

of

a re

ctan

gle

is P

"2!

'2w

,whe

re !

isth

e le

ngth

and

wis

the

wid

th.A

rec

tang

le h

as a

per

imet

er o

f 24

inch

es.F

ind

its

dim

ensi

ons

if it

s le

ngth

is 3

inch

es g

reat

er t

han

its

wid

th.

4.5

in.b

y 7.

5 in

.

1 % 31 % 9

1 % 4h % 2

1 % 4

1 % 61 % 3

3 % 4g % 6

1 % 2

1 % 83 % 4

5 % 61 % 2

1 % 62 % 3

3 % 25 % 2

5 % 6

1 % 23 % 4

1 % 2

1 % 3

Pra

ctic

e (A

vera

ge)

Sol

ving

Equ

atio

ns w

ith t

he V

aria

ble

on E

ach

Sid

e

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-5

3-5



Rea

din

g t

o L

earn

Math

emati

csS

olvi

ng E

quat

ions

with

the

Var

iabl

e on

Eac

h S

ide

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-5

3-5

©G

lenc

oe/M

cGra

w-H

ill16

5G

lenc

oe A

lgeb

ra 1

Lesson 3-5

Pre-

Act

ivit

yH

ow c

an a

n e

quat

ion

be

use

d t

o d

eter

min

e w

hen

tw

o p

opu

lati

ons

are

equ

al?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 3-

5 at

the

top

of

page

149

in y

our

text

book

.

In t

he e

quat

ion

12 !

7.6x

"6

!8x

,wha

t do

7.6

xan

d 8x

repr

esen

t?

7.6x

repr

esen

ts t

he in

crea

se (

in m

illio

ns)

in t

he n

umbe

r of

mal

e In

tern

et u

sers

,and

8x

repr

esen

ts t

he in

crea

se (

inm

illio

ns)

in t

he n

umbe

r of

fem

ale

Inte

rnet

use

rs.

Read

ing

the

Less

on

1.Su

ppos

e yo

u w

ant

to h

elp

a fr

iend

sol

ve 6

k!

7 "

3k#

8.W

hat

wou

ld y

ou a

dvis

e he

r to

do f

irst

? W

hy?

Sub

trac

t 3k

from

eac

h si

de;a

fter

she

doe

s th

at a

nd s

impl

ifies

,all

of t

heva

riab

les

will

be

on t

he le

ft s

ide.

2.W

hen

solv

ing

2(3x

#4)

"3(

x!

5),w

hy is

it h

elpf

ul f

irst

to

use

the

Dis

trib

utiv

e P

rope

rty

to r

emov

e th

e gr

oupi

ng s

ymbo

ls?

Onc

e yo

u ha

ve r

emov

ed th

e gr

oupi

ng s

ymbo

ls,y

ou c

an te

ll w

hat y

ou n

eed

to a

dd o

r su

btra

ct to

eac

h si

de to

get

all

of th

e va

riab

les

on o

ne s

ide.

3.O

n a

quiz

,Jas

on s

olve

d th

ree

equa

tion

s.H

is t

each

er s

aid

all t

he w

ork

was

cor

rect

,but

she

aske

d hi

m t

o w

rite

sho

rt s

ente

nces

to

tell

wha

t th

e so

luti

ons

wer

e.In

wha

t fo

llow

s,yo

u se

e th

e la

steq

uati

on in

his

wor

k fo

r ea

ch e

quat

ion.

Wri

te s

ente

nces

to

desc

ribe

the

solu

tion

s.

a.x

"#

4Th

e so

lutio

n is

!4.

b.6m

"6m

All

num

bers

are

sol

utio

ns.

c.12

"37

Ther

e ar

e no

sol

utio

ns.

4.In

Que

stio

n 3,

one

of t

he e

quat

ions

Jas

on s

olve

d w

as a

n id

enti

ty.W

hich

equ

atio

n w

as it

?E

xpla

in h

ow y

ou k

now

.

The

one

for

whi

ch t

he la

st s

tep

was

6m

"6m

;the

exp

ress

ions

on

the

left

and

righ

t si

des

are

the

sam

e.

Hel

ping

You

Rem

embe

r

5.A

n eq

uati

on w

ith

vari

able

s is

an

iden

tity

whe

n th

e eq

uati

on is

alw

ays

true

.In

othe

rw

ords

,the

exp

ress

ions

on

the

left

and

rig

ht s

ides

alw

ays

have

the

sam

e va

lue.

Loo

k up

the

wor

d id

enti

tyin

the

dic

tion

ary.

Wri

te a

ll th

e de

fini

tion

s th

at a

re s

imila

r to

the

mat

hem

atic

al d

efin

itio

n.S

ee s

tude

nts’

wor

k.

©G

lenc

oe/M

cGra

w-H

ill16

6G

lenc

oe A

lgeb

ra 1

Iden

titie

sA

n eq

uati

on t

hat

is t

rue

for

ever

y va

lue

of t

he v

aria

ble

is c

alle

d an

iden

tity

.Whe

n yo

u tr

y to

sol

ve a

n id

enti

ty,y

ou e

nd u

p w

ith

ast

atem

ent

that

is a

lway

s tr

ue.H

ere

is a

n ex

ampl

e.

Sol

ve 8

!(5

!6x

) "

3(1

#2x

).

8 #

(5 #

6x)

"3(

1 !

2x)

8 #

5 #

(#6x

)"

3(1

!2x

)

8 #

5 !

6x"

3 !

6x

3 !

6x"

3 !

6x

Sta

te w

het

her

eac

h e

quat

ion

is

an i

den

tity

.If

it i

s n

ot,f

ind

its

sol

uti

on.

1.2(

2 #

3x) "

3(3

!x)

!4

2.5(

m!

1) !

6 "

3(4

!m

) !(2

m#

1)

x"

#1

iden

tity

3.(5

t!

9) #

(3t

#13

) "

2(11

!t)

4.14

#(6

#3c

) "4c

#c

iden

tity

no s

olut

ion

5.3y

#2(

y!

19)

"9y

#3(

9 #

y)6.

3(3h

#1)

"4(

h!

3)

y"

#1

h"

3

7.U

se t

he t

rue

equa

tion

3x

#2

"3x

#2

to c

reat

e an

iden

tity

of

your

ow

n.

Sam

ple

answ

er:

6x!

4 "

2(3x

!2)

8.U

se t

he f

alse

equ

atio

n 1

"2

to c

reat

e an

equ

atio

n w

ith

no s

olut

ion.

Sam

ple

answ

er:

2x#

1 "

2x#

2

9.C

reat

e an

equ

atio

n w

hose

sol

utio

n is

x"

3.

Sam

ple

answ

er:

3x#

2(x

#1)

"20

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-5

3-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Stu

dy

Gu

ide

an

d I

nte

rven

tion

Rat

ios

and

Pro

port

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-6

3-6

©G

lenc

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cGra

w-H

ill16

7G

lenc

oe A

lgeb

ra 1

Lesson 3-6

Rat

ios

and

Pro

po

rtio

ns

A r

atio

is a

com

pari

son

of t

wo

num

bers

by

divi

sion

.The

rat

io

of x

to y

can

be e

xpre

ssed

as

xto

y,x

:yor

.R

atio

s ar

e us

ually

exp

ress

ed in

sim

ples

t fo

rm.

An

equa

tion

sta

ting

tha

t tw

o ra

tios

are

equ

al is

cal

led

a p

rop

orti

on.T

o de

term

ine

whe

ther

two

rati

os f

orm

a p

ropo

rtio

n,ex

pres

s bo

th r

atio

s in

sim

ples

t fo

rm o

r ch

eck

cros

s pr

oduc

ts.

x % y

Det

erm

ine

wh

eth

er t

he

rati

os

and

fo

rm a

pro

por

tion

.

"w

hen

expr

esse

d in

sim

ples

t fo

rm.

"w

hen

expr

esse

d in

sim

ples

t fo

rm.

The

rat

ios

and

form

a p

ropo

rtio

n

beca

use

they

are

equ

al w

hen

expr

esse

d in

sim

ples

t fo

rm.

12 % 1824 % 36

2 % 312 % 18

2 % 324 % 36

12 % 1824 % 36

Use

cro

ss p

rod

uct

s to

det

erm

ine

wh

eth

er

and

fo

rm a

pro

por

tion

.

!W

rite

the

prop

ortio

n.

10(4

5) !

18(2

5)C

ross

pro

duct

s

450

"45

0S

impl

ify.

The

cro

ss p

rodu

cts

are

equa

l,so

"

.

Sinc

e th

e ra

tios

are

equ

al,t

hey

form

apr

opor

tion

.

25 % 4510 % 18

25 % 4510 % 18

25 % 4510 % 18

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Use

cro

ss p

rod

uct

s to

det

erm

ine

wh

eth

er e

ach

pai

r of

rat

ios

form

s a

pro

por

tion

.

1.,

2.,

3.,

yes

nono

4.,

5.,

6.,

nono

yes

7.,

8.,

9.,

yes

yes

yes

10.2

:3,2

0:3

011

.5 t

o 9,

25 t

o 45

12.

,

yes

yes

yes

13.5

:5,3

0:2

014

.18

to 2

4,50

to

7515

.100

:75,

44:3

3

nono

yes

16.

,17

.,

18.

,

yes

yes

yes

0.45

% 0.9

0.1

% 0.2

6 % 81.

5%2

1 % 200.

05%

1

9 % 872 % 64

20 % 3014 % 21

9 % 1215 % 20

5% 10

00.

1%2

12 % 274 % 9

3 % 1612 % 32

15 % 2025 % 36

25 % 4910 % 20

10 % 155 % 8

16 % 321 % 2

©G

lenc

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cGra

w-H

ill16

8G

lenc

oe A

lgeb

ra 1

Solv

e Pr

opor

tion

sIf

a p

ropo

rtio

n in

volv

es a

var

iabl

e,yo

u ca

n us

e cr

oss

prod

ucts

to

solv

e

the

prop

orti

on.I

n th

e pr

opor

tion

"

,xan

d 13

are

cal

led

extr

emes

and

5 an

d 10

are

calle

d m

ean

s.In

a p

ropo

rtio

n,th

e pr

oduc

t of

the

ext

rem

es is

equ

al t

o th

e pr

oduc

t of

the

mea

ns.

Mea

ns-E

xtre

mes

Pro

pert

y of

Pro

port

ions

For

any

num

bers

a, b

, c,a

nd d

, if

", t

hen

ad"

bc.

Sol

ve

".

"O

rigin

al p

ropo

rtio

n

13(x

) "5(

10)

Cro

ss p

rodu

cts

13x

"50

Sim

plify

.

"D

ivid

e ea

ch s

ide

by 1

3.

x"

3S

impl

ify.

The

sol

utio

n is

3.

Sol

ve e

ach

pro

por

tion

.

1."

!12

2."

3."

10

4."

25.

"12

6."

1

7."

268.

"9.

"15

10.

"14

11.

"no

sol

utio

n12

."

!2

13.

"68

14.

"no

sol

utio

n15

."

6

Use

a p

rop

orti

on t

o so

lve

each

pro

blem

.

16. M

OD

ELS

To m

ake

a m

odel

of

the

Gua

delo

upe

Riv

er b

ed,H

erm

ie u

sed

1 in

ch o

f cl

ay f

or5

mile

s of

the

riv

er’s

act

ual l

engt

h.H

is m

odel

riv

er w

as 5

0 in

ches

long

.How

long

is t

heG

uade

loup

e R

iver

?25

0 m

i

17.E

DU

CA

TIO

NJo

sh f

inis

hed

24 m

ath

prob

lem

s in

one

hou

r.A

t th

at r

ate,

how

man

yho

urs

will

it t

ake

him

to

com

plet

e 72

pro

blem

s?3

h

12 % 92

' w

%6

24 % k12 % k

15 % 3a

#8

%12

#y

% 83

' y

%4

12 % x1.

5%

x4 % 12

4% b

#2

p % 245 % 8

1 % 218 % 3

3 % d18 % 54

9% y

'1

3 % 63x % 21

8 % x4 % 6

3 % 4x

'1

%4

0.5

%x

0.1

%2

3 % 55 % 3

1 % t2 % 8

#3

%x

11 % 13

11 % 1350 % 1313

x% 13

10 % 13x % 5

10 % 13x % 5

c % da % b

10 % 13x % 5

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Rat

ios

and

Pro

port

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-6

3-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Rat

ios

and

Pro

port

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-6

3-6

©G

lenc

oe/M

cGra

w-H

ill16

9G

lenc

oe A

lgeb

ra 1

Lesson 3-6

Use

cro

ss p

rod

uct

s to

det

erm

ine

wh

eth

er e

ach

pai

r of

rat

ios

form

s a

pro

por

tion

.W

rite

yes

or n

o.

1.,

yes

2.,

no

3.,

yes

4.,

yes

5.,

no6.

,ye

s

7.,

yes

8.,

no

Sol

ve e

ach

pro

por

tion

.If

nec

essa

ry,r

oun

d t

o th

e n

eare

st h

un

dre

dth

.

9."

710

."

15

11.

"6

12.

"18

13.

"10

14.

"3

15.

"21

16.

"18

17.

"8

18.

"63

19.

"42

20.

"44

21.

"13

22.

"48

23.

"16

24.

"60

25.B

OA

TIN

GH

ue’s

boa

t us

ed 5

gal

lons

of

gaso

line

in 4

hou

rs.A

t th

is r

ate,

how

man

yga

llons

of

gaso

line

will

the

boa

t us

e in

10

hour

s?12

.5 g

al

11 % 3022 % x

s % 844 % 21

20 % g5 % 12

27 % 399 % c

n % 6011 % 15

30 % m10 % 14

1 % 97 % b

6 % f42 % 56

y % 696 % 23

36 % s12 % 7

35 % 215 % e

3 % 56 % z

1 % 63 % a

15 % 109 % g

3 % 95 % b

2 % 141 % a

50 % 8512 % 17

21 % 983 % 14

26 % 3813 % 19

42 % 907 % 16

72 % 818 % 9

24 % 286 % 7

7 % 115 % 9

20 % 254 % 5

©G

lenc

oe/M

cGra

w-H

ill17

0G

lenc

oe A

lgeb

ra 1

Use

cro

ss p

rod

uct

s to

det

erm

ine

wh

eth

er e

ach

pai

r of

rat

ios

form

s a

pro

por

tion

.W

rite

yes

or n

o.

1.,

no2.

,no

3.,

yes

4.,

yes

5.,

yes

6.,

yes

7.,

yes

8.,

no9.

,no

Sol

ve e

ach

pro

por

tion

.If

nec

essa

ry,r

oun

d t

o th

e n

eare

st h

un

dre

dth

.

10.

"9

11.

"68

12.

"5

13.

"7

14.

"18

15.

"16

16.

"12

17.

"77

18.

"1

19.

"12

220

."

2421

."

7

22.

"10

23.

"3

24.

"3

25.

"1

26.

"2.

8827

."

1.02

28.

"7

29.

"2

30.

"5

31.

"32

."

333

."

4

34.P

AIN

TIN

GY

sidr

a pa

ints

a r

oom

tha

t ha

s 40

0 sq

uare

fee

t of

wal

l spa

ce in

2ho

urs.

At

this

rat

e,ho

w lo

ng w

ill it

tak

e he

r to

pai

nt a

roo

m t

hat

has

720

squa

re f

eet

of w

all

spac

e?4

h

35.V

AC

ATI

ON

PLA

NS

Wal

ker

is p

lann

ing

a su

mm

er v

acat

ion.

He

wan

ts t

o vi

sit

Petr

ifie

dN

atio

nal F

ores

t an

d M

eteo

r C

rate

r,A

rizo

na,t

he 5

0,00

0-ye

ar-o

ld im

pact

sit

e of

a la

rge

met

eor.

On

a m

ap w

ith

a sc

ale

whe

re 2

inch

es e

qual

s 75

mile

s,th

e tw

o ar

eas

are

abou

t

1in

ches

apa

rt.W

hat

is t

he d

ista

nce

betw

een

Petr

ifie

d N

atio

nal F

ores

t an

d M

eteo

r

Cra

ter?

abou

t 56

.25

mi

1 % 2

1 % 2

1 % 24 % 7x

#2

%6

3 % 75 % 7

r'

2%

72 % 3

x'

1%

45 % 12

2 % 4m

#1

%8

2% y

'6

3 % 1214 % 6

7% a

#4

3% 0.

516 % n

12 % b3

% 0.72

7% 1.

61v

% 0.23

3 % 45 % 8

m % 63 % 5

5 % 63 % q

2 % 78 % c

7 % 9

2 % a14 % 49

6 % 4g % 16

12 % h6 % 61

z % 173 % 51

35 % x5 % 11

10 % 602 % y

48 % 9y % 3

27 % 162

3 % u4 % w

28 % 49

k % 740 % 56

34 % 23v % 46

30 % 545 % a

3.9

% 0.9

7.6

% 1.8

2.9

% 2.4

1.7

% 1.2

7.14

% 10.9

23.

4% 5.

2

1 % 61.

5%9

72 % 818 % 9

108

% 9912 % 11

36 % 4818 % 24

15 % 663 % 11

52 % 487 % 6

Pra

ctic

e (A

vera

ge)

Rat

ios

and

Pro

port

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-6

3-6



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csR

atio

s an

d P

ropo

rtio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-6

3-6

©G

lenc

oe/M

cGra

w-H

ill17

1G

lenc

oe A

lgeb

ra 1

Lesson 3-6

Pre-

Act

ivit

yH

ow a

re r

atio

s u

sed

in

rec

ipes

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 3-

6 at

the

top

of

page

155

in y

our

text

book

.

•H

ow m

any

serv

ings

of

hone

y fr

ozen

yog

urt

are

mad

e by

thi

s re

cipe

?4

serv

ings

•H

ow m

any

reci

pes

wou

ld b

e ne

eded

to

mak

e en

ough

hon

ey f

roze

n yo

gurt

for

all t

he s

tude

nts

in y

our

clas

s?S

ee s

tude

nts’

wor

k.

Rea

din

g t

he

Less

on

1.C

ompl

ete

the

follo

win

g se

nten

ce.

A r

atio

is a

com

pari

son

of t

wo

num

bers

by

.

2.D

escr

ibe

two

way

s to

dec

ide

whe

ther

the

sen

tenc

e "

is a

pro

port

ion.

Exp

ress

the

rat

ios

in s

impl

est

form

to

see

if th

ey a

re e

qual

.Che

ck t

o se

ew

heth

er t

he c

ross

pro

duct

s ar

e eq

ual.

3.Fo

r ea

ch p

ropo

rtio

n,te

ll w

hat

the

extr

emes

are

and

wha

t th

e m

eans

are

.

a."

Ext

rem

es:

Mea

ns:

b."

Ext

rem

es:

Mea

ns:

4.A

jet

flyi

ng a

t a

stea

dy s

peed

tra

vele

d 82

5 m

iles

in 2

hou

rs.I

f yo

u so

lved

the

pro

port

ion

",w

hat

wou

ld t

he a

nsw

er t

ell y

ou a

bout

the

jet?

how

far

the

jet

trav

eled

in 1

.5 h

Hel

pin

g Y

ou

Rem

emb

er

5.W

rite

how

you

wou

ld e

xpla

in s

olvi

ng a

pro

port

ion

to a

fri

end

who

mis

sed

Les

son

3-6.

Use

cro

ss p

rodu

cts.

Wri

te a

n eq

uatio

n w

ith t

he p

rodu

ct o

f th

e ex

trem

eson

the

left

sid

e an

d th

e pr

oduc

t of

the

mea

ns o

n th

e ri

ght

side

.The

nso

lve

this

sec

ond

equa

tion.

x% 1.

582

5%

2

8 an

d 12

6 an

d 16

12 % 166 % 8

35 a

nd 6

14 a

nd 1

56 % 15

14 % 35

8 % 202 % 5divi

sion

©G

lenc

oe/M

cGra

w-H

ill17

2G

lenc

oe A

lgeb

ra 1

Ang

les

of a

Tri

angl

eIn

geo

met

ry,m

any

stat

emen

ts a

bout

phy

sica

l spa

ce a

re p

rove

n to

be

true

.Suc

h st

atem

ents

are

cal

led

theo

rem

s.H

ere

are

two

exam

ples

of

geom

etri

c th

eore

ms.

a.T

he s

um o

f th

e m

easu

res

of t

he a

ngle

s of

a t

rian

gle

is 1

80°.

b.If

tw

o si

des

of a

tri

angl

e ha

ve e

qual

mea

sure

,the

n th

e tw

o an

gles

oppo

site

tho

se s

ides

als

o ha

ve e

qual

mea

sure

.

For

eac

h o

f th

e tr

ian

gles

,wri

te a

n e

quat

ion

an

d t

hen

sol

ve f

or x

.(A

tic

k m

ark

on

two

or m

ore

sid

es o

f a

tria

ngl

e in

dic

ates

th

at t

he

sid

es h

ave

equ

al m

easu

re.)

1.x

"60

)2.

x"

45)

3.x

"45

)4.

x"

20)

5.x

"22

.5)

6.x

"10

)

7.x

"33

)8.

x"

100)

9.x

"60

)10

.x

"50

)

11.T

wo

angl

es o

f a

tria

ngle

hav

e th

e sa

me

12.T

he m

easu

re o

f on

e an

gle

of a

tri

angl

e is

m

easu

re.T

he s

um o

f th

e m

easu

res

of

twic

e th

e m

easu

re o

f a

seco

nd a

ngle

.The

th

ese

angl

es is

one

-hal

f th

e m

easu

re o

f m

easu

re o

f th

e th

ird

angl

e is

12

less

tha

n th

e th

ird

angl

e.F

ind

the

mea

sure

s of

th

e su

m o

f th

e ot

her

two.

Fin

d th

e th

e an

gles

of

the

tria

ngle

.m

easu

res

of t

he a

ngle

s of

the

tri

angl

e.30

),30

),12

0)64

),32

),84

)

x2x

30°

x

40°

x

x #

15°

3 x

x '

30°

90°4 x

5x

5x

2xx

x '

30°

x5x

' 1

0°

90°

x

90°

45°

x70

°

50°

x

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-6

3-6



Stu

dy

Gu

ide

an

d I

nte

rven

tion

Per

cent

of

Cha

nge

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-7

3-7

©G

lenc

oe/M

cGra

w-H

ill17

3G

lenc

oe A

lgeb

ra 1

Lesson 3-7

Perc

ent

of

Ch

ang

eW

hen

an in

crea

se o

r de

crea

se in

an

amou

nt is

exp

ress

ed a

s a

perc

ent,

the

perc

ent

is c

alle

d th

e p

erce

nt

of c

han

ge.I

f th

e ne

w n

umbe

r is

gre

ater

tha

n th

eor

igin

al n

umbe

r,th

e pe

rcen

t of

cha

nge

is a

per

cen

t of

in

crea

se.I

f th

e ne

w n

umbe

r is

less

than

the

ori

gina

l num

ber,

the

perc

ent

of c

hang

e is

the

per

cen

t of

dec

reas

e.

Fin

d t

he

per

cen

t of

in

crea

se.

orig

inal

:48

new

:60

Fir

st,s

ubtr

act

to f

ind

the

amou

nt o

fin

crea

se.T

he a

mou

nt o

f in

crea

se is

60

#48

"12

.T

hen

find

the

per

cent

of

incr

ease

by

usin

gth

e or

igin

al n

umbe

r,48

,as

the

base

.

"P

erce

nt p

ropo

rtio

n

12(1

00)

"48

(r)

Cro

ss p

rodu

cts

1200

"48

rS

impl

ify.

"D

ivid

e ea

ch s

ide

by 4

8.

25 "

rS

impl

ify.

The

per

cent

of

incr

ease

is 2

5%.

48r

% 4812

00%

48

r% 10

012 % 48

Fin

d t

he

per

cen

t of

dec

reas

e.or

igin

al:3

0n

ew:2

2

Fir

st,s

ubtr

act

to f

ind

the

amou

nt o

fde

crea

se.T

he a

mou

nt o

f de

crea

se is

30

#22

"8.

The

n fi

nd t

he p

erce

nt o

f de

crea

se b

y us

ing

the

orig

inal

num

ber,

30,a

s th

e ba

se.

"P

erce

nt p

ropo

rtio

n

8(10

0) "

30(r

)C

ross

pro

duct

s

800

"30

rS

impl

ify.

"D

ivid

e ea

ch s

ide

by 3

0.

26"

rS

impl

ify.

The

per

cent

of

decr

ease

is 2

6%

,or

abou

t27

%.

2 % 3

2 % 3

30r

% 3080

0% 30

r% 10

08 % 30

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sta

te w

het

her

eac

h p

erce

nt

of c

han

ge i

s a

per

cen

t of

in

crea

se o

r a

per

cen

t of

dec

reas

e.T

hen

fin

d e

ach

per

cen

t of

ch

ange

.Rou

nd

to

the

nea

rest

wh

ole

per

cen

t.

1.or

igin

al:5

02.

orig

inal

:90

3.or

igin

al:4

5ne

w:8

0ne

w:1

00ne

w:2

0in

crea

se;6

0%in

crea

se:1

1%de

crea

se;5

6%

4.or

igin

al:7

7.5

5.or

igin

al:1

406.

orig

inal

:135

new

:62

new

:150

new

:90

decr

ease

;20%

incr

ease

:7%

decr

ease

;33%

7.or

igin

al:1

208.

orig

inal

:90

9.or

igin

al:2

7.5

new

:180

new

:270

new

:25

incr

ease

;50%

incr

ease

:200

%de

crea

se;9

%

10.o

rigi

nal:

8411

.ori

gina

l:12

.512

.ori

gina

l:25

0ne

w:9

8ne

w:1

0ne

w:5

00in

crea

se;1

7%de

crea

se:2

0%in

crea

se;1

00%

©G

lenc

oe/M

cGra

w-H

ill17

4G

lenc

oe A

lgeb

ra 1

Solv

e Pr

ob

lem

sD

isco

unte

d pr

ices

and

pri

ces

incl

udin

g ta

x ar

e ap

plic

atio

ns o

f pe

rcen

tof

cha

nge.

Dis

coun

t is

the

am

ount

by

whi

ch t

he r

egul

ar p

rice

of

an it

em is

red

uced

.Thu

s,th

e di

scou

nted

pri

ce is

an

exam

ple

of p

erce

nt o

f de

crea

se.S

ales

tax

is a

mou

nt t

hat

is a

dded

to t

he c

ost

of a

n it

em,s

o th

e pr

ice

incl

udin

g ta

x is

an

exam

ple

of p

erce

nt o

f in

crea

se.

A c

oat

is o

n s

ale

for

25%

off

th

e or

igin

al p

rice

.If

the

orig

inal

pri

ceof

th

e co

at i

s $7

5,w

hat

is

the

dis

cou

nte

d p

rice

?

The

dis

coun

t is

25%

of

the

orig

inal

pri

ce.

25%

of

$75

"0.

25 !

7525

% "

0.25

"18

.75

Use

a c

alcu

lato

r.

Subt

ract

$18

.75

from

the

ori

gina

l pri

ce.

$75

#$1

8.75

"$5

6.25

The

dis

coun

ted

pric

e of

the

coa

t is

$56

.25.

Fin

d t

he

fin

al p

rice

of

each

ite

m.W

hen

a d

isco

un

t an

d a

sal

es t

ax a

re l

iste

d,

com

pu

te t

he

dis

cou

nt

pri

ce b

efor

e co

mp

uti

ng

the

tax.

1.C

ompa

ct d

isc:

$16

2.T

wo

conc

ert

tick

ets:

$28

3.A

irlin

e ti

cket

:$24

8.00

Dis

coun

t:15

%St

uden

t di

scou

nt:2

8%Su

pera

ir d

isco

unt:

33%

$13.

60$2

0.16

$166

.16

4.Sh

irt:

$24.

005.

CD

pla

yer:

$142

.00

6.C

eleb

rity

cal

enda

r:$1

0.95

Sale

s ta

x:4%

Sale

s ta

x:5.

5%Sa

les

tax:

7.5%

$24.

96$1

49.8

1$1

1.77

7.C

lass

rin

g:$8

9.00

8.So

ftw

are:

$44.

009.

Vid

eo r

ecor

der:

$110

.95

Gro

up d

isco

unt:

17%

Dis

coun

t:21

%D

isco

unt:

20%

Sale

s ta

x:5%

Sale

s ta

x:6%

Sale

s ta

x:5%

$77.

56$3

6.85

$93.

20

10.V

IDEO

ST

he o

rigi

nal s

ellin

g pr

ice

of a

new

spo

rts

vide

o w

as $

65.0

0.D

ue t

o th

e de

man

dth

e pr

ice

was

incr

ease

d to

$87

.75.

Wha

t w

as t

he p

erce

nt o

f in

crea

se o

ver

the

orig

inal

pric

e?35

%

11.S

CH

OO

LA

hig

h sc

hool

pap

er in

crea

sed

its

sale

s by

75%

whe

n it

ran

an

issu

e fe

atur

ing

a co

ntes

t to

win

a c

lass

par

ty.B

efor

e th

e co

ntes

t is

sue,

10%

of

the

scho

ol’s

800

stu

dent

sbo

ught

the

pap

er.H

ow m

any

stud

ents

bou

ght

the

cont

est

issu

e?14

0 st

uden

ts

12.B

ASE

BA

LLB

aseb

all t

icke

ts c

ost

$15

for

gene

ral a

dmis

sion

or

$20

for

box

seat

s.T

hesa

les

tax

on e

ach

tick

et is

8%

,and

the

mun

icip

al t

ax o

n ea

ch t

icke

t is

an

addi

tion

al 1

0%of

the

bas

e pr

ice.

Wha

t is

the

fin

al c

ost

of e

ach

type

of

tick

et?

$17.

70;$

23.6

0

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Per

cent

of

Cha

nge

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-7

3-7

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Per

cent

of

Cha

nge

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-7

3-7

©G

lenc

oe/M

cGra

w-H

ill17

5G

lenc

oe A

lgeb

ra 1

Lesson 3-7

Sta

te w

het

her

eac

h p

erce

nt

of c

han

ge i

s a

per

cen

t of

in

crea

se o

r a

per

cen

t of

dec

reas

e.T

hen

fin

d e

ach

per

cen

t of

ch

ange

.Rou

nd

to

the

nea

rest

wh

ole

per

cen

t.

1.or

igin

al:2

52.

orig

inal

:50

new

:10

new

:75

decr

ease

;60%

incr

ease

;50%

3.or

igin

al:5

54.

orig

inal

:25

new

:50

new

:28

decr

ease

;9%

incr

ease

;12%

5.or

igin

al:5

06.

orig

inal

:90

new

:30

new

:95

decr

ease

;40%

incr

ease

;6%

7.or

igin

al:4

88.

orig

inal

:60

new

:60

new

:45

incr

ease

;25%

decr

ease

;25%

Fin

d t

he

tota

l p

rice

of

each

ite

m.

9.dr

ess:

$69.

0010

.bin

der:

$14.

50ta

x:5%

tax:

7%$7

2.45

$15.

5211

.har

dcov

er b

ook:

$28.

9512

.gro

ceri

es:$

47.5

2ta

x:6%

tax:

3%$3

0.69

$48.

9513

.fill

er p

aper

:$6.

0014

.sho

es:$

65.0

0ta

x:6.

5%ta

x:4%

$6.3

9$6

7.60

15.b

aske

tbal

l:$1

7.00

16.c

once

rt t

icke

ts:$

48.0

0ta

x:6%

ta

x:7.

5%$1

8.02

$51.

60

Fin

d t

he

dis

cou

nte

d p

rice

of

each

ite

m.

17.b

ackp

ack:

$56.

2518

.mon

itor

:$15

0.00

disc

ount

:20%

disc

ount

:50%

$45.

00$7

5.00

19.C

D:$

15.9

920

.shi

rt:$

25.5

0di

scou

nt:2

0%di

scou

nt:4

0%$1

2.79

$15.

3021

.sle

epin

g ba

g:$1

2522

.cof

fee

mak

er:$

102.

00di

scou

nt:2

5%di

scou

nt:4

5%$9

3.75

$56.

10

©G

lenc

oe/M

cGra

w-H

ill17

6G

lenc

oe A

lgeb

ra 1

Sta

te w

het

her

eac

h p

erce

nt

of c

han

ge i

s a

per

cen

t of

in

crea

se o

r a

per

cen

t of

dec

reas

e.T

hen

fin

d e

ach

per

cen

t of

ch

ange

.Rou

nd

to

the

nea

rest

wh

ole

per

cen

t.

1.or

igin

al:1

82.

orig

inal

:140

3.or

igin

al:2

00ne

w:1

0ne

w:1

60ne

w:3

20de

crea

se;4

4%in

crea

se;1

4%in

crea

se;6

0%4.

orig

inal

:10

5.or

igin

al:7

66.

orig

inal

:128

new

:25

new

:60

new

:120

incr

ease

;150

%de

crea

se;2

1%de

crea

se;6

%7.

orig

inal

:15

8.or

igin

al:9

8.6

9.or

igin

al:5

8.8

new

:35.

5ne

w:6

4ne

w:6

5.7

incr

ease

;137

%de

crea

se;3

5%in

crea

se;1

2%

Fin

d t

he

tota

l p

rice

of

each

ite

m.

10.c

oncr

ete

bloc

ks:$

95.0

011

.cri

b:$2

40.0

012

.jac

ket:

$125

.00

tax:

6%ta

x:6.

5%ta

x:5.

5%$1

00.7

0$2

55.6

0$1

31.8

813

.cla

ss r

ing:

$325

.00

14.b

lank

et:$

24.9

915

.kit

e:$1

8.90

tax:

6%ta

x:7%

tax:

5%$3

44.5

0$2

6.74

$19.

85

Fin

d t

he

dis

cou

nte

d p

rice

of

each

ite

m.

16.d

ry c

lean

ing:

$25.

0017

.com

pute

r ga

me:

$49.

9918

.lug

gage

:$18

5.00

disc

ount

:15%

disc

ount

:25%

disc

ount

:30%

$21.

25$3

7.49

$129

.50

19.s

tati

oner

y:$1

2.95

20.p

resc

ript

ion

glas

ses:

$149

21.p

air

of s

hort

s:$2

4.99

di

scou

nt:1

0%di

scou

nt:2

0%di

scou

nt:4

5%$1

1.66

$119

.20

$13.

74

Fin

d t

he

fin

al p

rice

of

each

ite

m.

22.t

elev

isio

n:$3

75.0

023

.DV

D p

laye

r:$2

69.0

024

.pri

nter

:$25

5.00

disc

ount

:25%

disc

ount

:20%

disc

ount

:30%

tax:

6%ta

x:7%

tax:

5.5%

$298

.13

$230

.26

$188

.32

25.I

NV

ESTM

ENTS

The

pri

ce p

er s

hare

of

an in

tern

et-r

elat

ed s

tock

dec

reas

ed f

rom

$90

per

shar

e to

$36

per

sha

re e

arly

in 2

001.

By

wha

t pe

rcen

t di

d th

e pr

ice

of t

he s

tock

decr

ease

?60

%

26.H

EATI

NG

CO

STS

Cus

tom

ers

of a

uti

lity

com

pany

rec

eive

d no

tice

s in

the

ir m

onth

ly b

ills

that

hea

ting

cos

ts f

or t

he a

vera

ge c

usto

mer

had

incr

ease

d 12

5% o

ver

last

yea

r be

caus

eof

an

unus

ually

sev

ere

win

ter.

In J

anua

ry o

f la

st y

ear,

the

Gar

cia’

s pa

id $

120

for

heat

ing.

Wha

t sh

ould

the

y ex

pect

to

pay

this

Jan

uary

if t

heir

bill

incr

ease

d by

125

%?

$270

Pra

ctic

e (A

vera

ge)

Per

cent

of

Cha

nge

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-7

3-7



Rea

din

g t

o L

earn

Math

emati

csP

erce

nt o

f C

hang

e

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-7

3-7

©G

lenc

oe/M

cGra

w-H

ill17

7G

lenc

oe A

lgeb

ra 1

Lesson 3-7

Pre-

Act

ivit

yH

ow c

an p

erce

nts

des

crib

e gr

owth

ove

r ti

me?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 3-

7 at

the

top

of

page

160

in y

our

text

book

.

•H

ow m

any

area

cod

es w

ere

in u

se in

194

7?84

are

a co

des

•H

ow m

any

mor

e ar

ea c

odes

wer

e in

use

in 1

999?

201

area

cod

es

Read

ing

the

Less

on

1.If

you

use

(or

igin

al a

mou

nt)

— (

new

am

ount

) to

fin

d th

e ch

ange

for

a p

erce

nt o

f ch

ange

pr

oble

m,t

hen

the

prob

lem

invo

lves

a p

erce

nt o

f (i

ncre

ase/

decr

ease

).

2.If

you

use

(ne

w a

mou

nt)

— (

orig

inal

am

ount

) to

fin

d th

e ch

ange

for

a p

erce

nt o

f ch

ange

pr

oble

m,t

hen

the

prob

lem

invo

lves

a p

erce

nt o

f (i

ncre

ase/

decr

ease

).

Com

ple

te t

he

char

t.

Ori

gina

lN

ew

Per

cent

Incr

ease

or

Am

ount

Am

ount

Per

cent

Pro

port

ion

Per

cent

Dec

reas

e?

3.10

13!

incr

ease

4.10

7!

decr

ease

5.50

42!

decr

ease

6.50

58!

incr

ease

7.W

hen

you

find

a d

isco

unt

pric

e,do

you

add

to

or s

ubtr

act

from

the

ori

gina

l pri

ce?

subt

ract

Hel

ping

You

Rem

embe

r

8.If

you

rem

embe

r on

ly t

wo

thin

gs a

bout

the

rat

io u

sed

for

find

ing

perc

ent

of c

hang

e,w

hat

shou

ld t

hey

be?

Sub

trac

t th

e pr

ices

, the

n di

vide

by

the

orig

inal

num

ber.

r"

8 "→ →

chan

ge"

r"

8 "→ →

chan

ge"

r"

3 "→ →

chan

ge"

r"

3 "→ →

chan

ge"

incr

ease

decr

ease

©G

lenc

oe/M

cGra

w-H

ill17

8G

lenc

oe A

lgeb

ra 1

Usi

ng P

erce

ntU

se w

hat

you

hav

e le

arn

ed a

bou

t p

erce

nt

to s

olve

eac

h p

robl

em.

A T

V m

ovie

had

a “

rati

ng”

of 1

5 an

d a

25 “

shar

e.”T

he r

atin

g is

the

per

cent

age

of t

he n

atio

n’s

tota

l TV

hou

seho

lds

that

wer

e tu

ned

in t

o th

is s

how

.The

sha

reis

the

per

cent

age

of h

omes

wit

h T

Vs

turn

ed o

n th

at w

ere

tune

d to

the

mov

ie.

How

man

y T

V h

ouse

hold

s ha

d th

eir

TV

s tu

rned

off

at

this

tim

e?To

fin

d ou

t,le

t T

!th

e nu

mbe

r of

TV

hou

seho

lds

and

x!

the

num

ber

of T

V h

ouse

hold

s w

ith

the

TV

off.

The

n T

"x

!th

e nu

mbe

r of

TV

hou

seho

lds

wit

h th

e T

V o

n.Si

nce

0.15

Tan

d 0.

25(T

"x)

bot

h re

pres

ent

the

num

ber

of h

ouse

hold

s tu

ned

to t

he m

ovie

, 0.15

T!

0.25

(T"

x)0.

15T

!0.

25T

"0.

25x.

Solv

e fo

r x.

0.25

x!

0.10

Tx

!!

0.40

T

Fort

y pe

rcen

t of

the

TV

hou

seho

lds

had

thei

r T

Vs

off w

hen

the

mov

ie w

as a

ired

.

An

swer

eac

h q

ues

tion

.

1.D

urin

g th

at s

ame

wee

k,a

spor

ts b

road

cast

had

a r

atin

g of

22.

1 an

d a

43 s

hare

.Sh

ow t

hat

the

perc

ent

of T

V h

ouse

hold

s w

ith

thei

r T

Vs

off w

as a

bout

48.

6%.

0.22

1T!

0.43

T#

0.43

xx

! !0.

486T

2.F

ind

the

perc

ent

of T

V h

ouse

hold

s w

ith

thei

r T

Vs

turn

ed o

ff d

urin

g a

show

w

ith

a ra

ting

of

18.9

and

a 2

9 sh

are.

34.8

%

3.Sh

ow t

hat

if T

is t

he n

umbe

r of

TV

hou

seho

lds,

r is

the

rat

ing,

and

s is

the

shar

e,th

en t

he n

umbe

r of

TV

hou

seho

lds

wit

h th

e T

V o

ff is

.

Sol

ve r

T!

s(T

#x)

for

x.

4.If

the

fra

ctio

n of

TV

hou

seho

lds

wit

h no

TV

on

is

then

sho

w t

hat

the

frac

tion

of T

V h

ouse

hold

s w

ith

TV

s on

is

.

5.F

ind

the

perc

ent

of T

V h

ouse

hold

s w

ith

TV

s on

dur

ing

the

mos

t w

atch

ed

seri

al p

rogr

am in

his

tory

:the

last

epi

sode

of

M*A

*S*H

,whi

ch h

ad a

60.

3 ra

ting

and

a 7

7 sh

are.

!78

.3%

6.A

loca

l sta

tion

now

has

a 2

sha

re.E

ach

shar

e is

wor

th $

50,0

00 in

adv

erti

sing

re

venu

e pe

r m

onth

.The

sta

tion

is t

hink

ing

of g

oing

com

mer

cial

free

for

the

thre

e m

onth

s of

sum

mer

to

gain

mor

e lis

tene

rs.W

hat

wou

ld it

s ne

w s

hare

ha

ve t

o be

for

the

last

4 m

onth

s of

the

yea

r to

mak

e m

ore

mon

ey fo

r th

e ye

ar

than

it w

ould

hav

e m

ade

had

it n

ot g

one

com

mer

cial

free

?gr

eate

r th

an 3

.5

60.3

"

r # s

s –

r#

s

(s"

r)T

#s

0.22

1T#

0.43

T"

"

0.10

T# 0.

25

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-7

3-7

1 #

!r "

s#

r"



An

swer

s

Stu

dy

Gu

ide

an

d I

nte

rven

tion

Sol

ving

Equ

atio

ns a

nd F

orm

ulas

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-8

3-8

©G

lenc

oe/M

cGra

w-H

ill17

9G

lenc

oe A

lgeb

ra 1

Lesson 3-8

Solv

e fo

r V

aria

ble

sSo

met

imes

you

may

wan

t to

sol

ve a

n eq

uati

on s

uch

as V

"!w

hfo

ron

e of

its

vari

able

s.Fo

r ex

ampl

e,if

you

kno

w t

he v

alue

s of

V,w

,and

h,t

hen

the

equa

tion

!"

is m

ore

usef

ul f

or f

indi

ng t

he v

alue

of !

.If

an e

quat

ion

that

con

tain

s m

ore

than

one

vari

able

is t

o be

sol

ved

for

a sp

ecif

ic v

aria

ble,

use

the

prop

erti

es o

f eq

ualit

y to

isol

ate

the

spec

ifie

d va

riab

le o

n on

e si

de o

f th

e eq

uati

on.

V % wh

Sol

ve 2

x!

4y"

8 fo

r y.

2x#

4y"

82x

#4y

#2x

"8

#2x

#4y

"8

#2x

"

y"

or

The

val

ue o

f yis

.

2x#

8%

4

2x#

8%

48

#2x

%#

4

8 #

2x%

#4

#4y

% #4

Sol

ve 3

m!

n"

km

!8

for

m.

3m#

n"

km#

83m

#n

#km

"km

#8

#km

3m#

n#

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atio

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orm

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ME

____

____

____

____

____

____

____

____

____

____

____

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____

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3-8

3-8

Exam

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Exam

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©G

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raw-H

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24G

lencoe Algebra 1

Skills PracticeSolving Equations and Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

3-83-8


Less

on

3-8


1. 7t " x, for t t " 2. e " wp, for p p "

3. q # r " r, for r r " 4. 4m # n " m, for m m "

5. 7a # b " 15a, for a a " ! 6. #5c ' d " 2c, for c c "

7. x # 2y " 1, for y y " 8. m ' 3n " 1, for n n "

9. 7f ' g " 5, for f f " 10. ax # c " b, for x x "

11. rt # 2n " y, for t t " 12. bc ' 3g " 2k, for c c "

13. kn ' 4f " 9v, for n n " 14. 8c ' 6j " 5p, for c c "

15. " d, for x x " c # 2d 16. " d, for c c " x ! 2d

17. " q, for p p " 5q ! 9 18. " a, for b b " 7a # 4z


19. Five more than a number g is six less than twice a number h. Solve for g.

g # 5 " 2h ! 6; g " 2h ! 11

20. One fourth of a number q is three more than three times a number w. Solve for q.

q " 3w # 3; q " 12w # 12

21. Eight less than a number s is three more than four times a number t. Solve for s.

s ! 8 " 4t # 3; s " 4t # 11

1%4

b # 4z%7

p ' 9%5

x # c%2

x # c%2

5p ! 6j%8

9v ! 4f%k

2k ! 3g%b

2n # y%r

b # c%a

5 ! g%7

1 ! m%3

x ! 1%2

d%7

b%8

n%3

q%2

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x%7



1. d " rt, for r r " 2. 6w # y " 2z, for w w "

3. mx ' 4y " 3c, for x x " 4. 9s # 5g " #4u, for s s "

5. ab ' 3c " 2d, for b b " 6. 2p " kx # q, for x x "

7. m ' a " a ' c, for m m " c 8. h ' g " d, for h h " (d ! g)

9. y ' v " s, for y y " (s ! v) 10. a # q " k, for a a " (k # q)

11. " h, for x x " 12. " c, for b b "

13. 2w # y " 7w # 2, for w w " 14. 3! ' y " 5 ' 5!, for ! ! "


15. Three times a number s plus 4 times a number y is 1 more than 6 times the number s.Solve for s.3s # 4y " 6s # 1; s "

16. Five times a number k minus 9 is two thirds of a number j. Solve for j.

5k ! 9 " j; j " (5k ! 9)

ELECTRICITY For Exercises 17 and 18, use the following information.

The formula for Ohm’s Law is E " IR, where E represents voltage measured in volts,I represents current measured in amperes, and R represents resistance measured in ohms.

17. Solve the formula for R. R "

18. Suppose a current of 0.25 ampere flows through a resistor connected to a 12-volt battery.What is the resistance in the circuit? 48 ohms

MOTION For Exercises 19 and 20, use the following information.

In uniform circular motion, the speed v of a point on the edge of a spinning disk is v " r,

where r is the radius of the disk and T is the time it takes the point to travel once aroundthe circle.

19. Solve the formula for r. r "

20. Suppose a merry-go-round is spinning once every 3 seconds. If a point on the outsideedge has a speed of 12.56 feet per second, what is the radius of the merry-go-round? (Use 3.14 for &.) 6 ft

Tv%2$

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Practice (Average)

Solving Equations and Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

3-83-8

Answ

ers(Lesson 3-8)


Ans

wer

s

Rea

din

g t

o L

earn

Math

emati

csS

olvi

ng E

quat

ions

and

For

mul

as

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-8

3-8

©G

lenc

oe/M

cGra

w-H

ill18

3G

lenc

oe A

lgeb

ra 1

Lesson 3-8

Pre-

Act

ivit

yH

ow a

re e

quat

ion

s u

sed

to

des

ign

rol

ler

coas

ters

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 3-

8 at

the

top

of

page

166

in y

our

text

book

.

The

equ

atio

n g(

195

!h)

"v2

cont

ains

sev

eral

var

iabl

es.W

hat

num

ber

valu

es d

o yo

u kn

ow f

or t

hese

var

iabl

es in

thi

s si

tuat

ion?

32 f

org

and

49 f

orv

Read

ing

the

Less

on

1.Su

ppos

e yo

u ha

ve a

n eq

uati

on w

ith

seve

ral v

aria

bles

.You

wan

t to

sol

ve f

or a

par

ticu

lar

vari

able

.How

doe

s th

e pr

oced

ure

com

pare

wit

h th

at f

or s

olvi

ng a

n eq

uati

on w

ith

just

one

vari

able

? H

ow d

oes

the

solu

tion

com

pare

wit

h th

e so

luti

on f

or a

n eq

uati

on w

ith

one

vari

able

?

Sam

ple

answ

er:

The

proc

edur

e is

bas

ical

ly t

he s

ame.

You

use

pro

pert

ies

of o

pera

tions

and

equ

ality

to

isol

ate

the

vari

able

in w

hich

you

are

inte

rest

ed. T

he s

olut

ion

will

pro

babl

y co

ntai

n va

riab

les

inst

ead

of ju

st a

num

ber.

2.D

escr

ibe

wha

t di

men

sion

al a

naly

sis

invo

lves

.

Sam

ple

answ

er: Y

ou u

se t

he u

nits

alo

ng w

ith n

umbe

r va

lues

as

you

doca

lcul

atio

ns. Y

ou t

reat

the

uni

ts p

rett

y m

uch

the

sam

e w

ay y

ou w

ould

vari

able

s. F

or e

xam

ple,

you

can

div

ide

them

and

use

exp

onen

ts w

ithth

em.

3.W

hat

do y

ou h

ave

to b

e ca

refu

l abo

ut w

hen

you

use

vari

able

s in

den

omin

ator

s of

frac

tion

s?

You

have

to

be s

ure

that

you

do

not

use

valu

es t

hat

wou

ld m

ake

the

deno

min

ator

0.

Hel

ping

You

Rem

embe

r

4.W

hen

you

give

the

dim

ensi

ons

of a

rec

tang

le,y

ou h

ave

to t

ell h

ow m

any

unit

s lo

ng it

isan

d ho

w m

any

unit

s w

ide

it is

.How

can

thi

s he

lp y

ou r

emem

ber

wha

t di

men

sion

alan

alys

is in

volv

es.

Sam

ple

answ

er:

Kee

p th

e w

ords

dim

ensi

onan

d un

itin

min

d. In

dim

ensi

onal

ana

lysi

s, y

ou in

clud

e un

its f

or d

imen

sion

s in

cal

cula

tions

.

1 # 2

©G

lenc

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cGra

w-H

ill18

4G

lenc

oe A

lgeb

ra 1

Dr.

Ber

nard

o H

ouss

ay

Eve

n th

ough

res

earc

hers

hav

e be

en s

tudy

ing

the

dise

ase

diab

etes

mel

litu

sfo

r hu

ndre

ds o

f ye

ars,

scie

ntis

ts h

ave

only

rec

entl

y di

scov

ered

the

caus

e of

the

dis

ease

and

dev

elop

ed m

etho

ds f

or r

educ

ing

its

seve

rity

.Dr.

Ber

nard

o H

ouss

ay,a

n A

rgen

tine

phy

siol

ogis

t,w

as o

ne o

fth

e pi

onee

rs o

f th

is m

ore

mod

ern

rese

arch

.He

stud

ied

the

rela

tion

ship

betw

een

diab

etes

and

the

pit

uita

ry g

land

,and

in 1

947

beca

me

the

firs

tL

atin

Am

eric

an t

o w

in t

he N

obel

Pri

ze in

Med

icin

e an

d P

hysi

olog

y.

Tho

ugh

ther

e is

no

cure

for

diab

etes

,spe

cific

die

ts a

nd e

xerc

ise

can

help

peop

le c

ontr

ol t

he d

isea

se.T

he A

mer

ican

Dia

bete

s A

ssoc

iati

on (A

DA

)ha

s he

lped

est

ablis

h fle

xibl

e di

etar

y gu

idel

ines

for

cons

umer

s to

follo

w.

The

se g

uide

lines

incl

ude

som

e of

the

follo

win

g ge

nera

l nut

riti

on r

ules

.

•Fa

t in

take

sho

uld

be e

qual

to

or le

ss t

han

30%

of

daily

cal

orie

s.

•Sa

tura

ted

fat

inta

ke s

houl

d be

equ

al t

o or

less

tha

n 10

% o

f da

ilyca

lori

es.

•P

rote

in s

houl

d be

lim

ited

to

10%

to

20%

of

daily

cal

orie

s.Pe

rson

ssh

owin

g th

e in

itia

l sig

ns o

f di

abet

es-i

nduc

ed k

idne

y di

seas

e sh

ould

limit

pro

tein

to

10%

of

daily

cal

orie

s.

•C

hole

ster

ol in

take

sho

uld

be 3

00 m

illig

ram

s or

less

dai

ly.

Ref

er t

o th

e in

form

atio

n a

bove

for

Exe

rcis

es 1

–4.

1.R

ober

t co

nsum

ed 2

100

calo

ries

on

Tue

sday

.His

fat

inta

ke t

otal

ed70

gra

ms,

and

of t

hat

70 g

ram

s,14

wer

e sa

tura

ted.

a.W

hat

perc

enta

ge o

f hi

s ca

lori

e co

nsum

ptio

n w

as f

at,a

nd w

hat

perc

enta

ge o

f th

at f

at w

as s

atur

ated

? (T

o fi

nd t

he p

erce

ntag

e of

calo

ries

fro

m f

at,m

ulti

ply

the

num

ber

of f

at g

ram

s by

9 b

efor

edi

vidi

ng b

y th

e nu

mbe

r of

cal

orie

s.) 3

0%;

20%

b.D

id R

ober

t st

ay w

ithi

n th

e re

com

men

ded

allo

wan

ce o

f fa

ts?

yes

2.A

nna'

s ch

oles

tero

l int

ake

was

330

mill

igra

ms

on S

unda

y.B

y w

hat

perc

enta

ge d

oes

she

need

to

redu

ce h

er c

hole

ster

ol c

onsu

mpt

ion

tore

mai

n w

ithi

n th

e gu

idel

ines

? 10

%

3.W

hat

num

ber

of f

at g

ram

s is

30%

of

240

calo

ries

? 8

4.Sh

aron

fol

low

s a

diet

tha

t pr

ovid

es a

bout

50

gram

s of

pro

tein

eac

hda

y.Sh

aron

's d

octo

r ha

s ju

st t

old

her

to r

educ

e he

r da

ily p

rote

inin

take

by

30%

.Abo

ut h

ow m

uch

prot

ein

will

be

in h

er r

educ

edpr

otei

n di

et?

35 g

ram

s

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-8

3-8


©G

lencoe/McG

raw-H

illA

26G

lencoe Algebra 1

Study Guide and InterventionWeighted Averages

NAME ______________________________________________ DATE ____________ PERIOD _____

3-93-9


Less

on

3-9

Mixture Problems

Weighted AverageThe weighted average M of a set of data is the sum of the product of each number in the set and its weight divided by the sum of all the weights.

Mixture Problems are problems where two or more parts are combined into a whole. Theyinvolve weighted averages. In a mixture problem, the weight is usually a price or a percentof something.

Delectable Cookie Company sells chocolate chip cookies for $6.95per pound and white chocolate cookies for $5.95 per pound. How many pounds ofchocolate chip cookies should be mixed with 4 pounds of white chocolate cookiesto obtain a mixture that sells for $6.75 per pound.

Let w " the number of pounds of chocolate chip cookies

Number of Pounds Price per Pound Total Price

Chocolate Chip w 6.95 6.95w

White Chocolate 4 5.95 4(5.95)

Mixture w ' 4 6.75 6.75(w ' 4)

Equation: 6.95w ' 4(5.95) " 6.75(w ' 4)Solve the equation.

6.95w ' 4(5.95) " 6.75(w ' 4) Original equation6.95w ' 23.80 " 6.75w ' 27 Simplify.

6.95w ' 23.80 # 6.75w " 6.75w ' 27 # 6.75w Subtract 6.75w from each side.0.2w ' 23.80 " 27 Simplify.

0.2w ' 23.80 # 23.80 " 27 # 23.80 Subtract 23.80 from each side.0.2w " 3.2 Simplify.

w " 16 Simplify.

16 pounds of chocolate chip cookies should be mixed with 4 pounds of white chocolate cookies.

1. SOLUTIONS How many grams of sugar must be added to 60 grams of a solution that is32% sugar to obtain a solution that is 50% sugar? 21.6 g

2. NUTS The Quik Mart has two kinds of nuts. Pecans sell for $1.55 per pound andwalnuts sell for $1.95 per pound. How many pounds of walnuts must be added to 15pounds of pecans to make a mixture that sells for $1.75 per pound? 15 lb

3. INVESTMENTS Alice Gleason invested a portion of $32,000 at 9% interest and thebalance at 11% interest. How much did she invest at each rate if her total income fromboth investments was $3,200. $16,000 at 9% and $16,000 at 11%

4. MILK Whole milk is 4% butterfat. How much skim milk with 0% butterfat should beadded to 32 ounces of whole milk to obtain a mixture that is 2.5% butterfat? 19.2 oz

ExampleExample

ExercisesExercises


Uniform Motion Problems Motion problems are another application of weightedaverages. Uniform motion problems are problems where an object moves at a certainspeed, or rate. Use the formula d " rt to solve these problems, where d is the distance, r isthe rate, and t is the time.

Bill Gutierrez drove at a speed of 65 miles per hour on anexpressway for 2 hours. He then drove for 1.5 hours at a speed of 45 miles perhour on a state highway. What was his average speed?

M " Definition of weighted average

! 56.4 Simplify.

Bill drove at an average speed of about 56.4 miles per hour.

1. TRAVEL Mr. Anders and Ms. Rich each drove home from a business meeting. Mr. Anderstraveled east at 100 kilometers per hour and Ms. Rich traveled west at 80 kilometers perhours. In how many hours were they 100 kilometers apart. h

2. AIRPLANES An airplane flies 750 miles due west in 1 hours and 750 miles due south

in 2 hours. What is the average speed of the airplane? about 429 mph

3. TRACK Sprinter A runs 100 meters in 15 seconds, while sprinter B starts 1.5 secondslater and runs 100 meters in 14 seconds. If each of them runs at a constant rate, who isfurther in 10 seconds after the start of the race? Explain.

Sprinter A; since sprinter A runs 100 m in 15 s, this sprinter runs at a

rate of m/s. In 10 seconds, sprinter A will have run (10) " 66.7 m.

Sprinter B’s rate is . In 10 seconds, with the delayed start, sprinter B

has run (10 ! 1.5) " 60.7 m.

4. TRAINS An express train travels 90 kilometers per hour from Smallville to Megatown.A local train takes 2.5 hours longer to travel the same distance at 50 kilometers perhour. How far apart are Smallville and Megatown? 281.25 km

5. CYCLING Two cyclists begin traveling in the same direction on the same bike path. Onetravels at 15 miles per hour, and the other travels at 12 miles per hour. When will thecyclists be 10 miles apart? 3 h

6. TRAINS Two trains leave Chicago, one traveling east at 30 miles per hour and onetraveling west at 40 miles per hour. When will the trains be 210 miles apart? 3 h

1%3

100%14

100%14

100%15

100%15

1%2

5%9

65 ) 2 ' 45 ) 1.5%%2 ' 1.5


Weighted Averages

NAME ______________________________________________ DATE ____________ PERIOD _____

3-93-9

ExampleExample

ExercisesExercises

Answ

ers(Lesson 3-9)

©G

lencoe/McG

raw-H

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27G

lencoe Algebra 1

Answers

Skills PracticeWeighted Averages

NAME ______________________________________________ DATE ____________ PERIOD _____

3-93-9


Less

on

3-9

SEASONING For Exercises 1–4, use the following information.

A health food store sells seasoning blends in bulk. One blend contains 20% basil. Sheilawants to add pure basil to some 20% blend to make 16 ounces of her own 30% blend. Let brepresent the amount of basil Sheila should add to the 20% blend.


Ounces Amount of Basil

20% Basil Blend 16 ! b 0.20(16 ! b)

100% Basil b 1.00b

30% Basil Blend 16 0.30(16)

2. Write an equation to represent the problem. 0.20(16 ! b) # 1.00b " 0.30(16)

3. How many ounces of basil should Sheila use to make the 30% blend? 2 oz

4. How many ounces of the 20% blend should she use? 14 oz

HIKING For Exercises 5–7, use the following information.

At 7:00 A.M., two groups of hikers begin 21 miles apart and head toward each other. The firstgroup, hiking at an average rate of 1.5 miles per hour, carries tents, sleeping bags, andcooking equipment. The second group, hiking at an average rate of 2 miles per hour, carriesfood and water. Let t represent the hiking time.


r t d " rt

First group of hikers 1.5 t 1.5t

Second group of hikers 2 t 2t

6. Write an equation using t that describes the distances traveled. 1.5t # 2t " 21

7. How long will it be until the two groups of hikers meet? 6 h

SALES For Exercises 8 and 9, use the following information.

Sergio sells a mixture of Virginia peanuts and Spanish peanuts for $3.40 per pound. Tomake the mixture, he uses Virginia peanuts that cost $3.50 per pound and Spanish peanutsthat cost $3.00 per pound. He mixes 10 pounds at a time.

8. How many pounds of Virginia peanuts does Sergio use? 8 lb

9. How many pounds of Spanish peanuts does Sergio use? 2 lb


GRASS SEED For Exercises 1–4, use the following information.

A nursery sells Kentucky Blue Grass seed for $5.75 per pound and Tall Fescue seed for$4.50 per pound. The nursery sells a mixture of the two kinds of seed for $5.25 per pound.Let k represent the amount of Kentucky Blue Grass seed the nursery uses in 5 pounds ofthe mixture.


Number of Pounds Price per Pound Cost

Kentucky Blue Grass k $5.75 5.75k

Tall Fescue 5 ! k $4.50 4.50(5 ! k)

Mixture 5 $5.25 5.25(5)

2. Write an equation to represent the problem. 5.75k # 4.50(5 ! k) " 5.25(5)

3. How much Kentucky Blue Grass does the nursery use in 5 pounds of the mixture? 3 lb

4. How much Tall Fescue does the nursery use in 5 pounds of the mixture? 2 lb

TRAVEL For Exercises 5–7, use the following information.

Two commuter trains carry passengers between two cities, one traveling east, and the otherwest, on different tracks. Their respective stations are 150 miles apart. Both trains leave atthe same time, one traveling at an average speed of 55 miles per hour and the other at anaverage speed of 65 miles per hour. Let t represent the time until the trains pass each other.


r t d " rt

First Train 55 t 55t

Second Train 65 t 65t

6. Write an equation using t that describes the distances traveled. 55t # 65t " 150

7. How long after departing will the trains pass each other? 1.25 h

8. TRAVEL Two trains leave Raleigh at the same time, one traveling north, and the othersouth. The first train travels at 50 miles per hour and the second at 60 miles per hour.In how many hours will the trains be 275 miles apart? 2.5 h

9. JUICE A pineapple drink contains 15% pineapple juice. How much pure pineapple juiceshould be added to 8 quarts of the drink to obtain a mixture containing 50% pineapplejuice? 5.6 qt

Practice (Average)

Weighted Averages

NAME ______________________________________________ DATE ____________ PERIOD _____

3-93-9

Answ

ers(Lesson 3-9)


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3-9

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5?

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(4,1

1),(

2,16

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3.2x

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4.7x

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8

(11,

1) o

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100

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7x'

3y"

40

(5,1

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9,6)

or

any

pair

(4

,4)

or (

1,11

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y"

5m!

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d m

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2 !

2y%

5

42 !

2y%

5

5y % 2

En

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____

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3-9

3-9


Chapter 3 Resource Masters - Math Class - Home · ©Glencoe/McGraw-Hill v Glencoe Algebra 1...

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Transcript of Chapter 3 Resource Masters - Math Class - Home · ©Glencoe/McGraw-Hill v Glencoe Algebra 1...