Parent and Student Study Guide Workbook

Transcription

Parent and Student
Study Guide Workbook
Course 3
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
Mathematics: Applications and Concepts, Course 3. Any other reproduction, for use or sale, is
prohibited without prior written permission of the publisher.
Send all inquiries to:
The McGraw-Hill Companies
8787 Orion Place
Columbus, OH 43240-4027
ISBN: 0-07-860165-7
Mathematics: Applications and Concepts, Course 3
Parent and Student Study Guide
1 2 3 4 5 6 7 8 9 10 024 09 08 07 06 05 04 03
Contents
Chapter
Title
Page
To the parents of Glencoe Mathematics Students . . . . . . . . iv
1
2
3
4
5
6
7
8
9
10
11
12
Algebra: Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Algebra: Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . 11
Algebra: Real Numbers and the Pythagorean Theorem . . . 21
Proportions, Algebra, and Geometry . . . . . . . . . . . . . . . . . 28
Percent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Geometry: Measuring Area and Volume. . . . . . . . . . . . . . . 56
Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Statistics and Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Algebra: More Equations and Inequalities . . . . . . . . . . . . . 83
Algebra: Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . 91
Algebra: Nonlinear Functions and Polynomials . . . . . . . . 100
Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
iii
To
To the
the Parents
Parents of
of Glencoe
Glencoe Mathematics
Mathematics Students:
Students:
ou teach your children all the time. You
taught language to your infants and you
read to your son or daughter. You taught
them how to count and use basic arithmetic.
Here are some ways you can continue to
reinforce mathematics learning.
Y
•
•
Online Resources
Encourage a positive attitude toward
mathematics.
•
Set aside a place and a time for homework.
•
Be sure your child understands the
importance of mathematics
achievement.
For your convenience, these worksheets are also
available in a printable format at
msmath3.net/parent_student.
Online Study Tools can help your student
succeed.
The Glencoe Parent and Student Study
Guide Workbook is designed to help you
support, monitor, and improve your child’s math
performance. These worksheets are written so
that you do not have to be a mathematician to
help your child.
•
msmath3.net/extra_examples shows you
additional worked-out examples that mimic
the ones in the textbook.
•
msmath3.net/self_check_quiz provides a
self-checking practice quiz for each lesson.
•
msmath3.net/vocabulary_review checks
your understanding of the terms and
definitions used in each chapter.
•
msmath3.net/chapter_test allows you to
take a self-checking test before the actual test.
•
msmath3.net/standardized_test is
another way to brush up on your
standardized test-taking skills.
The Parent and Student Study Guide
Workbook includes:
•
A 1-page chapter review (12 in all) for each
chapter. These worksheets review the skills
and concepts needed for success on tests
and quizzes. Answers are located on pages
108–113.
A 1-page worksheet for every lesson in the
Student Edition (95 in all). Completing a
worksheet with your child will reinforce the
concepts and skills your child is learning in
math class. Upside-down answers are
provided right on the page.
iv
NAME ________________________________________ DATE ______________ PERIOD _____
A Plan for Problem Solving (pages 6–10)
You can use a four-step plan to solve a problem.
Explore
Determine what information is given in the problem and what you need to find.
Do you have all of the information you need? Is there too much information?
Plan
Select a strategy for solving the problem. There may be several strategies that
you could use. Estimate the answer.
Solve
Solve the problem by carrying out your plan. If your plan does not work, try
another, and maybe even a third plan.
Examine
Examine the answer carefully. See if it fits the facts given in the problem.
Compare it to your estimate. If your answer is not reasonable, make a new plan
and start again.
Gwen must get to the airport in two hours. If she takes two busses that each take
75 minutes, will she make it in time?
Explore
You need to find out whether Gwen’s bus trips will take two hours or less.
Plan
You need to find the number of hours Gwen’s bus trips will take. Take the sum of
the times of the bus trips and convert the minutes to hours. You estimate that the
bus trips will take longer than two hours.
Solve
75 minutes 75 minutes 150 minutes
150 minutes 60 minutes 2.5 hours
Examine
The bus trips will take 2.5 hours, so Gwen will not make it to the airport in time.
Try This Together
Use the four-step plan to solve each problem.
1. Communication A new telephone company is gaining an average of 75 new
customers a day. How many new customers are they gaining each week?
HINT: Multiply the number of customers per day by the number of days in a week.
2. Recreation Trejon plays basketball 4 days during the week after school and
one day on the weekend. One week he played 2 fewer days than he normally
would in the week. How many days did he play basketball that week?
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
3. Standardized Test Practice Coryn went to buy her textbooks for her
college math course. One book was $35, and a second book was $64.50.
She also bought a third math book. If she spent $130.29, what is a
reasonable estimate for the cost of the third book?
A $30.00
B $35.00
C $40.00
D $25.00
Answers: 1. 525 2. 3 3. A
3.
©
Glencoe/McGraw-Hill
1
NAME ________________________________________ DATE ______________ PERIOD _____
Variables, Expressions, and Properties
(pages 11–15)
Variables, usually letters, are used to represent numbers in some expressions.
Algebraic expressions are combinations of variables, numbers, and at least
one operation. A mathematical sentence that contains an “” is called an
equation. An equation that contains a variable is an open sentence.
Properties are open sentences that are true for any numbers.
Order of
Operations
1. Do all operations within grouping symbols first; start with the innermost
grouping symbols.
2. Evaluate all powers before other operations.
3. Multiply and divide in order from left to right.
4. Add and subtract in order from left to right.
Property
Algebra
abba
abba
a (b c) (a b) c
a (b c) (a b) c
Arithmetic
6116
7337
2 (3 8) (2 3) 8
3 (4 5) (3 4) 5
Distributive
a(b c) ab ac
a(b c) ab ac
4(6 2) 4 6 4 2
3(7 5) 3 7 3 5
Identity
a0a
a1a
909
515
Commutative
Associative
A Evaluate 3(2ab) if a 3 and b 5.
B Name the property shown by the
statement 4 8 8 4.
3(2ab) 3 (2 3 5)
3 (30)
90
The order of the numbers changed. This is the
Commutative Property of Addition.
Evaluate each expression if a 2, b 8, c 4, and d 12.
1. 2a (bc 12)
2. 5a 2b 3c
3. (d c) (2b a)
Name the property shown by each statement.
4. 1 6xy 6xy
5. 12 (3 7) (12 3) 7
B
3.
C
C
A
B
5.
C
B
B
A
7. Standardized Test Practice Prathna needs to figure out how many
people can watch the class play. There are 10 rows that each have
12 seats. Solve the equation 10 12 s to find the number of seats.
A 100
B 120
C 110
D 90
4. Identity () 5. Associative () 6. Commutative () 7. B
8.
©
Glencoe/McGraw-Hill
2
3. 17
A
7.
2. 14
6.
Answers: 1. 24
4.
6. 5 4 4 5
NAME ________________________________________ DATE ______________ PERIOD _____
Integers and Absolute Value (pages 17–21)
The set of integers consists of positive whole numbers, negative whole
numbers, and zero: {…, 3, 2, 1, 0, 1, 2, 3, …}. You can write
positive integers with or without the sign.
Graphing
Integers on a
Number Line
To graph an integer, locate the number and draw a dot at that point on a
number line.
The integer that corresponds to that point is called the coordinate
of the point.
The distance on the number line from a number to 0 is called the
absolute value of the number.
A Find the absolute value of 5.
B Find the absolute value of 7.
The absolute value of 5 is written |5|.
|5| is the distance of 5 from zero.
The point 5 is 5 units from zero.
So |5| 5.
The absolute value of 7 is written |7| or |7|.
|7| is the distance of 7 from zero.
The point 7 is 7 units from zero.
So |7| 7.
Try These Together
1. Name the coordinate of point A
graphed on the number line below.
2. Find |10|.
HINT: How far is 10 from 0?
HINT: What integer corresponds to A?
Name the coordinate of each point graphed on
the number line.
3. G
4. C
5. B
6. E
Graph each set of points on a number line.
9. {3, 5, 8}
10. {4, 1, 2}
12. {5, 2, 2, 5}
13. {3, 7, 9}
Evaluate each expression.
15. |8|
16. |5| |3|
18. |20 10|
19. |12|
G F A
C B
E
D
–5 –4 –3 –2 –1 0 1 2 3 4 5
7. F
8. D
11. {9, 4, 2, 6}
14. {1, 0, 5}
17. |22|
20. |65| |15|
21. Travel Dixonville is 8 miles farther north than Huntland. Express
8 miles farther as an integer.
B
C
C
B
C
A
7.
8.
B
A
22. Standardized Test Practice Cherise and Audra both do the high jump in
track and field. Audra jumps 5 inches lower than Cherise. Express
5 inches lower as an integer.
A 5
B 5
C |5|
D |5|
19. 12
B
6.
©
18. 10
A
5.
3. 4 4. 0 5. 1 6. 3 7. 3 8. 5 9–14. See Answer Key. 15. 8 16. 8 17. 22
4.
Glencoe/McGraw-Hill
3
Answers: 1. 2 2. 10
20. 50 21. 8 22. A
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Adding Integers (pages 23–27)
You can model adding integers with counters or on a number line.
Adding
Integers
• To add integers with the same sign, add their absolute values. Give the result the
same sign as the integers.
• To add integers with different signs, subtract their absolute values. Give the
result the same sign as the integer with the greater absolute value.
A Solve 3 (7) p.
B Solve q 7 3.
The integers have the same sign. They are both
negative so their sum will be negative.
Add the absolute values (3 and 7) and give the
result a negative sign.
10 p
The integers have different signs. |7| is 7;
|3| is 3. The integer with the greater absolute
value is 7, so the result will be negative.
Subtract the absolute values: 7 3 4.
q 4
Try These Together
1. Find 5 (4).
2. Find 18 26.
HINT: Which integer has the greater
absolute value?
Add.
3. 12 5
6. 36 (29) 10
9. (14) (6)
HINT: Are the signs of the integers the same?
4. (25) (3)
7. 7 (30)
10. 17 (11)
12. What is the value of 10 (20)?
5. 15 (6) ( 4)
8. 49 11
11. (3) (8) ( 5)
13. Find the sum 75 (25).
Evaluate each expression if a 5, b 2, and c 8.
14. a b
15. |c b|
16. |a| c
17. Games Mark got to move 13 spaces forward on a game board. Then on
his next turn, he had to move 8 spaces back. Write an addition equation
involving integers to show how far on the game board Mark actually
moved in these two turns.
B
C
C
B
C
18. Standardized Test Practice A store that sells wooden chairs bought
25 chairs from the manufacturer. The next day they sold 8 of the chairs.
Which addition equation shows how to find how many chairs they had left?
A c 25 8
B c (25) (8)
C c 25 8
D c 25 (8)
13. 100 14. 7
B
A
©
Glencoe/McGraw-Hill
12. 10
8.
10. 6 11. 16
A
7.
9. 20
B
6.
8. 60
A
5.
4
7. 23
4.
Answers: 1. 1 2. 44 3. 7 4. 28 5. 5 6. 17
15. 6 16. 13 17. x 13 (8) 18. D
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Subtracting Integers (pages 28–31)
The opposite of an integer is the number that is the same distance from
zero but in the opposite direction. The opposite of any number is called its
additive inverse. The sum of a number and its additive inverse is zero.
a (a) 0.
Subtracting Integers
To subtract an integer, add its additive inverse.
A Find 7 (3).
B Find 5 4.
Subtracting 3 is the same as adding the
inverse of 3.
7 (3) 7 3
10
You can think of this as “taking away a debt
of $3 is the same as adding $3.”
To subtract 4, add 4.
5 4 5 (4)
9
Try These Together
1. What is the additive inverse of 5?
2. What is the additive inverse of 8?
HINT: What number is the same distance from zero
but on the opposite side of zero on a number line?
HINT: What number added to 8 gives zero?
3. Write the additive inverse of 21.
Subtract.
4. 30 (5)
7. 4 16
10. 10 2
13. 0 18
16.
17.
18.
19.
A
7.
8.
C
B
A
8 2
12 (6)
62 (3)
14 (2)
20. Standardized Test Practice Solve the equation x 91 (102).
A 11
B 193
C 11
D 193
14. 41
C
B
B
6.
©
13. 18
C
A
5.
6.
9.
12.
15.
Find the value of y for y 6 (15).
Find the value of x for 15 30 x.
Evaluate 10 b c if b 5 and c 5.
Money Matters In 1999, an Internet company had a balance for the
year of $200,000. In 2000, they lost another $150,000. Write a
subtraction equation to show how to find the total amount of money
they lost in 1999 and 2000.
Glencoe/McGraw-Hill
10. 8 11. 270 12. 65
4.
20 (1)
16 8
120 (150)
26 15
Answers: 1. 5 2. 8 3. 21 4. 35 5. 19 6. 10 7. 12 8. 24 9. 18
15. 12 16. 9 17. 15 18. 10 19. $200,000 $150,000 t 20. C
B
3.
5.
8.
11.
14.
5
NAME ________________________________________ DATE ______________ PERIOD _____
Multiplying and Dividing Integers (pages 34–38)
Multiplying
Integers
The product of two integers with the same sign is positive.
The product of two integers with different signs is negative.
Since division is the inverse operation for multiplication, the rules for dividing integers are
the same as for multiplying integers.
Dividing
Integers
The quotient of two integers with the same sign is positive.
The quotient of two integers with different signs is negative.
A Find the product of 5 and 8.
B Find 36 (12).
The signs of the two factors are the same.
The sign of the product is positive.
(5)(8) 40
The two integers have the same sign.
The quotient is positive.
36 (12) 3
Try These Together
1. Find 3(2).
2. Find 20 (2)
HINT: Are the signs of the integers the
same or different?
HINT: Will the solution be positive or negative?
Multiply or divide.
3. 36 3
5. 3(4)(9)
4. 56 8
6. 5(9)
16
7. 8
9. 42 (6)
8. 11( 15)(5)
10. 6(5)
30
12. 11. 16(2)
5
Evaluate each expression if a 3, b 2, and c 5.
ab
13. 14. 2c b
15. 6abc
16. 3bc
b
17. Taxes In 1995, the Albanos owed $2,000 in taxes. For 2000, they only
owed $1,500 in taxes. What was the average change in the amount of
taxes they owed each of these 5 years?
B
C
C
B
C
A
7.
8.
B
A
18. Standardized Test Practice The value of Mr. Herrera’s stock changed by
$55.00 a day for 5 days. What was the total change in the value of his stock?
A $50.00
B $275.00
C $50.00
D $275.00
12. 6 13. 3 14. 5
B
6.
©
Glencoe/McGraw-Hill
11. 32
A
5.
7. 2 8. 825 9. 7 10. 30
4.
6
Answers: 1. 6 2. 10 3. 12 4. 7 5. 108 6. 45
15. 180 16. 30 17. $100 18. B
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Writing Expressions and Equations (pages 39–42)
There are many words and phrases that suggest arithmetic operations. Any
variable can be used to represent a number.
Translating
Words into
Expressions
Verbal Phrase
five less than a number
a number increased by 12
twice a number decreased by 3
Algebraic Expression
a5
b 12
2d 3
g
4
the quotient of a number and 4
Addition
Common
Phrases that
Indicate the
Four Operations
plus, sum,
more than,
increased by,
total, in all
Subtraction
Multiplication
minus, difference, times, product,
less than,
multiplied,
subtract,
each, of, factors
decreased by
Translating Verbal
Verbal Sentence
Sentences into
24 is 6 more than a number.
Equations
Five times a number is 60.
Division
divided,
quotient,
separate, an, in,
per, rate, ratio
Algebraic Equation
24 h 6
5k 60
A Write 16 plus 7 as an expression.
B Write a minus b as an expression.
16 plus 7
16 7 Plus indicates addition, so
write an addition expression.
a minus b
a b Minus indicates subtraction, so
write a subtraction expression.
Try These Together
Write each verbal phrase as an algebraic expression or equation.
1. 5 more than a number
2. half of the total
HINT: Use the chart of common phrases above to help you write each expression or equation.
Write each verbal phrase as an algebraic expression or equation.
3. g less than 14 is 8
4. the product of 6 and y is 42
5. 13 less a is 5
6. 3 times h is 12
7. 17 decreased by x is 15
8. 5 more than Eric’s score
9. Money Matters Darcey gets 3 times as much allowance every month as her
younger sister Devin. Suppose Darcey gets $18.00 allowance every month.
Write an equation to find out how much allowance Devin gets every month.
B
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
10. Standardized Test Practice Which expression shows how to find the
price per gallon of gasoline if 15 gallons costs $19.65?
A 15 $19.65p
B $19.65 15p
C $19.65 p 15
D $19.65 15 p
1
4.
Answers: 1. n 5 2. t 3. 14 g 8 4. 6y 42 5. 13 a 5 6. 3h 12 7. 17 x 15 8. e 5 9. $18.00 3d 10. B
2
3.
©
Glencoe/McGraw-Hill
7
NAME ________________________________________ DATE ______________ PERIOD _____
Solving Addition and Subtraction Equations
(pages 45–49)
You can use the properties of algebra to find the solution to an equation.
Addition
Property
of Equality
If you add the same number to each side of an equation, the two sides
remain equal.
Arithmetic
Algebra
33
x82
3535
x8828
88
x 10
Subtraction
Property of
Equality
If you subtract the same number from each side of an equation, the two
sides remain equal.
Arithmetic
Algebra
33
x28
3232
x2282
11
x6
To solve an equation in which a number is added to or subtracted from the
variable, you can use the opposite, or inverse, operation. Addition and
subtraction are inverse operations.
A Solve y 5 13.
B Solve b 6 72.
y 5 13
y 5 5 13 5 Use the subtraction
y8
property of equality.
Then check your solution.
The solution to the equation is 8.
b 6 72
b 6 6 72 6 Use the addition
b 78
property of equality.
Then check your solution.
Try These Together
Solve each equation. Check your solution.
1. 15 z 26
2. x 12 7
3. y 34 8
4. 39 25 w
HINT: Remember to use the inverse operation.
5. 36 24 r
6. q 8 17
7. p 5 18
9. 120 t 65
10. j 64 50
11. 1.5 h 3
13. 45 2
14. 41 5 m
15. n 8.1 3.1
B
C
17. Standardized Test Practice Solve the equation 48.2 z 25.1.
A 23.1
B 24.3
C 23.5
9. 185 10. 114 11. 1.5
12. 1 13. 43
14. 46
15. 11.2
Glencoe/McGraw-Hill
8. 16
©
8
7. 13
C
B
A
6. 25
8.
5. 12
A
7.
4. 14
B
B
6.
D 24.8
3. 42
C
A
5.
2. 19
4.
Answers: 1. 11
16. 2.9 17. A
3.
8. 10 s 26
12. k 0.7 0.3
16. a 1.6 1.3
NAME ________________________________________ DATE ______________ PERIOD _____
Solving Multiplication and Division Equations
(pages 50–53)
You can use the properties of algebra to find the solution to an equation.
If you divide each side of an equation by the same nonzero number, the two
sides remain equal.
Arithmetic
Algebra
88
3x 18
3x
18
8282
3
3
44
x 6
Division
Property
of Equality
Multiplication
Property of
Equality
If you multiply each side of an equation by the same number, the two
sides remain equal.
Arithmetic
Algebra
x
4
88
5
x
(4)
4
8282
16 16
5(4)
x 20
To solve a multiplication or division equation, you can use the opposite, or inverse,
operation. Multiplication and division are inverse operations.
B Solve z 5.
9
A Solve 4x 20.
4x 20
4x
4
20
4
z
9
Use the division property of
equality. Then check your solution.
z
9
x 5
5
Use the multiplication property of
9 5 9 equality. Then check your solution.
z = 45
Try These Together
1. 56 8y
2. 30 6p
3. m 9 4
4. 14x 126
HINT: Use the inverse operation to solve each equation.
5. r 9 9
6. 54 6s
B
9. 5
50
10. 9
8
C
C
A
B
5.
C
B
11. Standardized Test Practice Jeremiah’s family pays $35.00 a month for
70 cable television channels. Use the equation $35.00 70y to find out
how much they pay per channel.
A $0.55
B $0.45
C $0.60
D $0.50
11. D
B
A
©
Glencoe/McGraw-Hill
9. 250 10. 72
8.
6. 9 7. 5 8. 75
A
7.
9
4. 9 5. 81
6.
Answers: 1. 7 2. 5 3. 36
4.
k
j
n
8. 3 25
3.
7. 13f 65
NAME ________________________________________ DATE ______________ PERIOD _____
Chapter 1 Review
Equation Football
Solve each equation. Then use your solutions to move the team across the
football field. Positive solutions move the team in the direction of a
touchdown. Negative solutions move the team away from a touchdown. The
goal is to reach the goal line in order to score a touchdown.
Example: Suppose the team starts on the 35-yard line.
x
5
2nd Play:
x 2 (5)
x
10
The team moves back 5 yards to the
40-yard line.
The team moves forward 10 yards to the
30-yard line.
10
x 5 (10)
G
1st Play:
20
30
40
Touchdown!
40
30
20
10
G
Go!
After an interception, Team A starts on the 40-yard line.
1st Play:
a (3)(2)(2) a What yard line is the team on now?
2nd Play:
b 35 (7)
b
What yard line is the team on now?
3rd Play:
c (29) 12
c
4th Play:
3d 48
d
Did Team A score a touchdown? Justify your answer.
Answers are located on page 108.
©
Glencoe/McGraw-Hill
10
NAME ________________________________________ DATE ______________ PERIOD _____
Fractions and Decimals (pages 62–66)
A decimal that ends, such as 0.335, is a terminating decimal.
335
All terminating decimals are rational numbers. 0.335 1,000
A decimal that repeats, such as 0.333… is a repeating decimal.
You can use bar notation to show that the 3 repeats forever. 0.333… 0.3
1
All repeating decimals are rational numbers. 0.333… 3
A Express 0.47 as a fraction in simplest
form.
Let N 0.4
7
Then 100N 47.4
7
1N 0.4
7
B Express 4.5 as a fraction or mixed
number in simplest form.
4.5 is 4 and 5 tenths or
The GCF of 45 and 10 is 5.
Divide numerator and denominator by 5.
Subtract.
The result is 99N 47. Divide each side by 99.
N
45
.
10
45
10
47
99
Try These Together
1. Express 0.757575… using bar notation.
9
2
1
or 4 .
2
2. Express 0.4111… using bar notation.
HINT: Write a bar over the digits that repeat.
HINT: Which digit repeats?.
Express each decimal using bar notation.
3. 6.015015015…
4. 8.222…
5. 0.636363…
Write the first ten decimal places of each decimal.
6. 0.13
7. 1.562
8. 3.498
Express each fraction or mixed number as a decimal.
1
9. 8
2
10. 5
1
7
11. 3 3
12. 5 9
Express each decimal as a fraction or mixed number in simplest form.
13. 0.96
14. 1.25
15. 0.8
16. 4.3
1
17. Sales Jack’s Suit Shop is having a sale on men’s suits. They are 5 off of
1
regular price for one week only. Express 5 as a decimal.
B
C
C
B
C
18. Standardized Test Practice Brandy is 2.75 times as old as her brother
Evan. Express 2.75 as a mixed number.
5
2
3
C 2 5
17. 0.2
B 2 8
D 2
4
1
7
A 2 9
16. 4 3
18. D
Answers: 1. 0.7
5
2. 0.41
3. 6.0
1
5
4. 8.2
5. 0.6
3
6. 0.1313131313 7. 1.5625625625 8. 3.4989898989 9. 0.125 10. 0.4
B
A
8
8.
15. 9
A
7.
1
B
6.
14. 1 4
A
5.
24
4.
11. 3.3
12. 5.7
13. 25
3.
©
Glencoe/McGraw-Hill
11
NAME ________________________________________ DATE ______________ PERIOD _____
Comparing and Ordering Rational Numbers
(pages 67–70)
One way to compare two rational numbers is to write them with fractions
that have the same denominator. You could use any common denominator,
but it is usually easiest to use the least common denominator (LCD). The
LCD is the same as the LCM of the denominators. You can also write the
fractions as decimals and compare the decimals.
1
3
2
A Which is greater, 5 or 3 ?
B Which is greater, 0.3 or 3 ?
The LCD is 15.
Rewrite
2
3
3
5
2
3
10
15
9
15
10
Since 15
and
1
3
5
Rewrite as the decimal 0.3333… .
3
0.333… is greater than 0.3.
with the LCD.
1
3
9 2
, 15 3
is greater than
is greater than 0.3.
3
.
5
Try These Together
3
2
1. Find the LCD for 4 and 3 .
1
3
.
2. Find the LCD for and
15
5
HINT: What is the LCM of 4 and 3?
HINT: What is the LCM of 15 and 5?
Find the LCD for each pair of fractions.
5 7
3. 6 , 8
5 9
4. 7 , 10
5 3
5. 6 , 14
Replace each ● with , , or to make a true sentence.
4
7
1
3
7. 3 ● 8
6. 4 5 ● 4 10
8
8. 8.65 ● 8 9
Order each set of rational numbers from least to greatest.
1 1 1 1
9. 8 , 4 , 5 , 9
5 3
10. , , 0.5, 0.55
12 4
3
5
11. 3.5, 3.65, 3 8 , 3 6
12. Sports The middle school basketball team won 12 out of their 15 games.
The high school volleyball team won 20 out of their 24 games. Which
team had the better record?
B
C
C
B
7
13. Standardized Test Practice Which is greatest, 1.68, 1.6, 1 3 , or 1 9 ?
2
A 1 3
3
©
7
C 1 9
B 1.68
5
B
11. 3 , 3.65, 3.5, 3 6
8
8.
A
Glencoe/McGraw-Hill
3
A
7.
2
C
B
6.
5
A
5.
10. , 0.5, 0.55, 12
4
4.
D 1.6
1 1 1 1
Answers: 1. 12 2. 15 3. 24 4. 70 5. 42 6. 7. 8. 9. , , , 9 8 5 4
12. the volleyball team 13. C
3.
12
NAME ________________________________________ DATE ______________ PERIOD _____
Multiplying Rational Numbers (pages 71–75)
Use the rules of signs for multiplying integers when you multiply rational
numbers.
To multiply fractions, multiply the numerators and multiply the denominators.
Multiplying
Fractions
a
b
1
ac
c
, where b 0, d 0
d
bd
2
A Find 3 2
.
2
5
1
2
3
2
2
5
7
2
12
5
5
1
3
4
Rename the mixed numbers
as improper fractions.
Divide out common factors.
7
12 6
2
3
3
3 3
44
B Find .
4 4
76
3
4
Multiply the numerators.
Multiply the denominators.
9
1
6
Simplify.
Multiply the numerators.
Multiply the denominators.
15
42
2
or 8 Simplify.
5
5
Try These Together
1
4
2
1. Find .
8
7
3
2. Find 4 .
3
HINT: Simplify by dividing numerator
and denominator by 4.
HINT: Will the product be positive or
negative? Simplify before you multiply.
Multiply. Write in simplest form.
3. 4 5 8
2
5
1
4. 2 5 6
5. 8 5 7. 37 6 8. 6 6
1
2
5
4
1
6. 1 5 3 9
5
1
1
5
2
2
Evaluate each expression if k 1 , , m 1
, and n .
2
4
6
3
9. k
11. mn
10. 2m
12. (k)
1
2
Mike and his twin brother ran a 3 6 -mile relay race. The twins ran 3
13. Fitness
of the race. How far did the twins run?
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
2
1
14. Standardized Test Practice Solve 7 4 x.
1
A 14
1
B 14
3
C 28
3
D 28
2
1
1
3
11
2
13
1
3
1
2
5
3
Answers: 1. 2. 3. 2 4. 2 5. 6 6. 3 7. 21 8. 1
9. 10. 3 11. 19 12. 13. 2 miles 14. A
14
2
4
12
5
15
2
8
3
9
8
8
3.
©
Glencoe/McGraw-Hill
13
NAME ________________________________________ DATE ______________ PERIOD _____
Dividing Rational Numbers (pages 76–80)
1
1
Dividing by 2 and multiplying by 2 give the same result. Notice that 2 and 2
are multiplicative inverses.
To divide by a fraction, multiply by its multiplicative inverse.
Dividing
Fractions
d
c
a
, where b, c, d 0
d
b
c
a
b
2
1
A Find 18 3 .
Replace dividing by
18 2
3
18
1
4
B Find 3 2 5 .
2
3
with multiplying by
3
.
2
4
5
Replace dividing by with multiplying by .
5
4
3
2
1
4
3
2
5
27
7
2
5
4
35
3
or 4 8
8
Try These Together
5
2
1. Find 11 1 6 .
4
2. Find 7 9 .
5
HINT: First rename 1 as an improper
6
fraction.
HINT: Change dividing by
4
9
the multiplicative inverse of
to multiplying by
4
.
9
Divide. Write in simplest form.
4. 3 4 8
5
3
6. 6 4 4
7. 2 5
10 1
1
5. 2 5 10 3
3. 4 (12)
2
3
4
1
1
9. 5 6 1 9
1
1
1
7
5
8. 9 3 1 6
2
4
10. 8 9 2 3
7
11. 4 5 10
3
12. 3 2 8
3
4
5
14. 8 25
13. 7 7
7
1
15. Interior Design A hallway that is 4 2 feet across has hardwood floors
1
lined with boards that are 2 inches wide. How many boards fit across
4
the hallway?
1
1
16. Standardized Test Practice What is 16 4 6 ?
2
1
1
9. 3 26
15
10. 3 24
1
11. 6 7
6
12. 4 13. 13
14. 40
1
Glencoe/McGraw-Hill
2
C 2 2
8. 5 11
©
1
B 2 6
14
7. 26
1
A 2 8
10
C
B
A
5. 8 6. 57
8.
D 2 3
4. 26
A
7.
1
C
B
B
6.
3. 16
C
A
5.
9
4.
Answers: 1. 6 2. 14
15. 24 16. C
B
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Adding and Subtracting Like Fractions
(pages 82–85)
Fractions with like denominators are called like fractions.
• To add fractions with like denominators, add the numerators and write the
sum over the denominator.
b
ab
,c
0
c
c
• To subtract fractions with like denominators, subtract the numerators and
write the difference over the denominator.
Adding and
Subtracting
Like Fractions
a
c
a
c
5
ab
b
,c
0
c
c
1
4
A Find .
12
12
5
12
1
12
5 1
12
4
12
1
3
6
B Find 7 7 .
4
7
Subtract the numerators.
6
7
Simplify.
46
7
10
7
3
1
7
Add the numerators.
Rename as a mixed number.
Try These Together
5
3
9
1. Find .
6
6
1
2. Find .
10
10
HINT: After you subtract, simplify
the fraction.
HINT: Find the sign of the sum with the same rules
you use for adding and subtracting integers.
Add or subtract. Write in simplest form.
3
8
3. 7 7 6
4
5
6. 11
11
5
1
2
4. 9 9
5. 2 3 1 3
1
5
7. 8 8 8. 5 5
1
3
5
1
Evaluate each expression if x and y .
12
12
9. y x
10. x y
12. Transportation There is
5
6
11. y (y x)
mile between Ming’s bus stop and the last
1
stop on the way to school. There is 6 mile between the last stop and
school. How many miles does Ming live from school?
13. Standardized Test Practice Solve n 1 4 .
4
1
3
8. 5
4
15
3
9. 2
1
10. 3
1
11. 12
5
12. 1 mile 13. D
Glencoe/McGraw-Hill
7. 4
©
1
C 1 2
B 1
11
3
A 4
1
4. 1 5. 4 6. C
B
A
D 2
4
8.
3. 1 7
A
7.
4
C
B
B
6.
2. 5
C
A
5.
1
4.
Answers: 1. 3
B
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Adding and Subtracting Unlike Fractions
(pages 88–91)
Adding and
Subtracting
Unlike
Fractions
To find the sum or difference of two fractions with unlike denominators,
• rename the fractions with a common denominator,
• add or subtract, and
• simplify if necessary.
7
2
A Find 3 .
9
7
9
2
7
3
Rename each fraction
using the LCD, 9.
6
3 9 9
76
9
1
B Find 2 4 3 2 .
3
1
7
Write the mixed numbers as
fractions.
11
14
Rename using the LCD, 4.
4 4
1
9
11
24 32 4 2
Simplify.
11 14
4
3
4
Simplify.
Try These Together
1
3
2
1. Find 5 .
4
5
2. Find 6 .
9
HINT: Rename both fractions with a
denominator of 20.
HINT: Rename using the LCD, 18.
Add or subtract. Write in simplest form.
3
4. 7 8
5
3
3. 3 4 6
1
3
1
7. 3 8 4
5
2
1
5. 5 7 4 3
1
1
8. 5 7
6
6. 8 5 5
4
1
1
5
9. 8 2 4 9
10. 1 8 1 6
2
11. Subtract 4 from 2.
6
1
12. What is the sum of 5 and ?
7
1
2
4
Evaluate each expression if a , b 1
, and c .
4
3
9
13. b c
14. a b c
15. a (c)
1
1
16. Cooking A recipe uses 1 3 cups wheat flour and 4 cup wheat germ.
What is the sum of these amounts?
B
C
C
1
2
17. Standardized Test Practice Solve t 1 6 5 .
23
23
C 30
B 30
17
D 1
30
7
7
4. 56
45
5. 1 21
1
6. 2 5
4
7. 5 4
1
16
3. 4 12
8. 5 42
13
9. 12 18
17
10. 2 24
19
11. 6 6
1
12. 35
19
13. 1 9
2
Glencoe/McGraw-Hill
2.
©
Answers: 1.
17
A 1 30
11
18
C
B
A
19
20
8.
16. 1 cups 17. B
12
A
7.
7
B
B
6.
15. 36
A
5.
31
4.
14. 1 36
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Solving Equations with Rational Numbers
(pages 92–95)
You can use the skills you have learned for rational numbers as you solve
equations that contain rational numbers.
• To solve an equation, you get the variable alone on one side by using
inverse operations.
• Reverse the order of operations by undoing addition and subtraction first.
• Then undo multiplication and division by doing the same inverse operation
on each side.
• Check your solution by substituting it for the variable to see if it makes the
two sides of the equation equal.
Solving
Equations
with Rational
Numbers
a5
A Solve 7. Check your solution.
3
3
a5
3
a5
3
B Solve 8 b 6. Check your solution.
8 b 6
8 (8) b 6 8 Add 8 to each side.
b 14
Simplify.
(1)(b) 14(1) Multiply each side by 1.
b 14
Simplify.
Check: Does 8 (14) equal 6? Yes.
7
3(7)
Multiply each side by 3.
a 5 21
a 5 5 21 5
a 26
Check: Does
26 5
3
Simplify.
Add 5 to each side.
Simplify.
21
3
equal 7? Yes,
7.
Try These Together
w
1. Solve 15 . Check your solution. 2. Solve 5.8 j 7.3. Check your solution.
8
HINT: Multiply each side by 8 and then by 1.
HINT: Subtract 5.8 from each side.
1
3
4. h (0.09) 4.3
5. 3 3.8
6. 7g 35
7. 2.2 0.8 z
1
1
8. s 4 2
9
9. m (7) 11
10
2
8
12. Standardized Test Practice Solve 5 k 9 .
16
5
A 45
12. D
©
1
B 7
10. 41.6 11. 43
C
B
A
Glencoe/McGraw-Hill
9. 20
8.
2
C 1 8
1
A
7.
8. 4
C
B
B
6.
4. 4.21 5. 11.4 6. 5 7. 1.4
C
A
5.
27 u
11. 8
2
17
D 2 9
1
4.
a 23
10. 9.3
2
Answers: 1. 120 2. 13.1 3. 1 2
B
3.
y
3. 2 5 n 3 10
NAME ________________________________________ DATE ______________ PERIOD _____
Powers and Exponents (pages 98–101)
When you multiply two or more numbers, each number is called a factor
of the product. When the same factor is repeated, you can use an exponent
to simplify the notation. An exponent tells you how many times a number,
called the base, is used as a factor. A power is a number that is expressed
using exponents.
Example of a Power
54 5 5 5 5
Words
Zero and Negative
Exponents
five to the fourth power
Any nonzero number to the zero power is 1. Any nonzero
number to the negative n power is 1 divided by the number
to the nth power.
Symbols Arithmetic
50 1
Algebra
x0 1, x 0
1
73 73
1
xn n , x 0
x
A Write 4 4 7 4 7 using exponents.
B Evaluate 64.
64 6 6 6 6
36 36
1,296
Use the commutative property to rearrange the
factors. Then use the associative property to
group them.
4 4 4 7 7 (4 4 4) (7 7) 43 72
Try These Together
1. Write 5 5 5 using exponents.
2. Evaluate 23.
HINT: How many times is each factor used?
HINT: Write each power as a product.
Write each expression using exponents.
3. 8 8 8 8
4. 1 1
5. 7 7 6 6
6. 2 2 2 4 4
7. 10 10 9 9 9
8. a a a b
Evaluate each expression.
9. 91
10. 35
11. 13 24
12. 62 43
13. 33 22 41
14. 52
15. Sports The Tour de France is one of the most difficult bicycle races in
the world. Cyclists ride about 3.2 103 kilometers through France’s
countryside and mountains. Express this number without exponents.
B
C
B
C
B
6.
A
7.
8.
B
A
16. Standardized Test Practice How can 8 8 8 p p 3 be written using
exponents?
A 3 p 64 p
B 64 p2 3
C 83 p2 3
D 82 p3 3
12. 2,304 13. 432
C
A
5.
25
4.
©
Glencoe/McGraw-Hill
Answers: 1. 53 2. 8 3. 84 4. 12 5. 72 62 6. 23 42 7. 102 93 8. a3 b 9. 9 10. 1 11. 16
243
14. 1 15. 3,200 16. C
3.
18
NAME ________________________________________ DATE ______________ PERIOD _____
Scientific Notation (pages 104–107)
When a number is written in scientific notation, it is expressed as the
product of a number between 1 and 10 and a power of 10.
Converting
Scientific
Notation to
Standard Form
• Multiplying by a positive power of 10 moves the decimal point to the
right the number of places shown by the exponent.
• Multiplying by a negative power of 10 moves the decimal point to the left
the number of places shown by the absolute value of the exponent.
A Write 4.6 103 in standard form.
B Write 89,450 in scientific notation.
The exponent is negative so move the decimal
point 3 places to the left.
4.6 103 0.0046
Try These Together
1. Write 4.5 103 in standard form.
Move the decimal to make a number
between 1 and 10. 8.9450
You moved the decimal point 4 places, so
89,450 8.945 10 4.
2. Write 1.201 105 in standard form.
HINT: Move the decimal point 3 places to
the right.
HINT: Move the decimal point to the right.
Write each number in standard form.
3. 3.65 102
4. 21.549 103
6. 8.95 104
7. 10.567 108
5. 2.3 106
8. 0.505 103
Write each number in scientific notation.
9. 1,200
10. 4,000,000
11. 0.00015
13. 30,300
14. 0.0000068
15. 0.000547
12. 0.0148
16. 702,000
17. Space Science Some satellites orbit Earth at a specific altitude that lets
them stay above one point on Earth’s equator at all times. This is called
a geostationary equatorial orbit and is about 35,800 kilometers above
Earth. Express this number in scientific notation.
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
18. Standardized Test Practice When the space shuttle returns to Earth’s
atmosphere, it needs to withstand tremendous heat. 2.4 104 special
tiles are installed by hand to help protect the shuttle from this heat.
What is 2.4 104 in standard form?
A 24,000
B 2,400
C 240,000
D 240
Answers: 1. 4,500 2. 120,100 3. 0.0365 4. 0.021549 5. 2,300,000 6. 0.000895 7. 1,056,700,000 8. 505 9. 1.2 103
10. 4 106 11. 1.5 104 12. 1.48 102 13. 3.03 104 14. 6.8 106 15. 5.47 104 16. 7.02 105
17. 3.58 104 18. A
3.
©
Glencoe/McGraw-Hill
19
NAME ________________________________________ DATE ______________ PERIOD _____
Chapter 2 Review
Rational Stairway
Climb a stairway made out of the following list of rational numbers. Solve
if necessary, then place the rational numbers in order from least to greatest
on the stairs from bottom to top.
3
5
1. 11
2.
1
11 3
11
2
3
6
3. 5.3
4. 4.7
24
5. 120
1
6. 1 2 3
3
7. 2.03 101
19
8. 4
©
Glencoe/McGraw-Hill
20
NAME ________________________________________ DATE ______________ PERIOD _____
Square Roots (pages 116–119)
Numbers that can be written as p p where p is an integer or a rational
4
36
number, are called perfect squares. For example, 9, 25, 0.09, 9 , and 81
are perfect squares.
Finding
Square
Roots
• When n r 2, then r is a square root of n.
• Notice that 36 6 6 and 36 (6) (6), so both 6 and 6 are square roots of
36. Sometimes we want only the positive square root.
• The positive square root of a number is called the principal square root. The
symbol , called a radical sign, is used to indicate the principal square root.
36 6
• Indicate the negative square root like this. 36 6
A Find 900
.
25
B Find .
121
Ask: what number multiplied by itself gives 900?
Notice that you are finding the negative square
root.
30 30 900, so 900
30.
25
121
5
11
Try These Together
1. Find 49
.
2. Find 16
.
HINT: Find n if n n 49.
Find each square root.
3. 144
36
7. 144
4.
HINT: This root will be a negative integer.
5. 676
9
25
8. 3.61
9.
6. 225
10. 0.81
169
400
Solve each equation.
11. x2 64
B
C
C
C
B
A
14. Standardized Test Practice You are arranging chairs for the school show.
You have 256 chairs to arrange in a square. How many rows of chairs
would you need and how many chairs in each row would you have?
A 16; 16
B 20; 20
C 16; 20
D 4; 4
14. A
8.
©
3
3
10. 0.9 11. 8, 8 12. 2.4, 2.4 13. , 4
4
A
7.
Glencoe/McGraw-Hill
13
8. 1.9 9. 20
B
B
6.
21
1
A
5.
5. 26 6. 15 7. 2
4.
3
Answers: 1. 7 2. 4 3. 12 4. 5
3.
9
13. x2 16
12. x2 5.76
NAME ________________________________________ DATE ______________ PERIOD _____
Estimating Square Roots (pages 120–122)
You can estimate the square roots of numbers that are not perfect squares.
Estimating
Square Roots
To estimate the square root of r, find perfect squares on each side of r. Use
these to estimate.
A Estimate 38
to the nearest whole
number.
B Estimate 21.6
number.
Find a perfect square a little less than 38 and one
a little more than 38. 36
38
49
, so
6 38 7. Since 38 is closer to 36 than 49,
the best whole number estimate for 38 is 6.
Find a perfect square a little less than and a
little more than 21.6. 16
21.6
25
,
so 4 21.6 5. Since 21.6 is closer to 25
than 16, the best whole number estimate for
21.6 is 5.
Try These Together
1. Estimate 69
number.
2. Estimate 7 to the nearest whole
number.
HINT: 69 is between the perfect squares
64 and 81.
HINT: Find the closest perfect squares on each
side of 8.
Estimate to the nearest whole number.
3. 27
4. 147
5. 120
6. 95
7. 254
8. 54
9. 490
10. 313
11. 1.25
12. 101
13. 399
14. 17.4
15. Sewing You are covering the top of a square stool with felt. The area of
the top is 140 square inches. Estimate the length of one side of the top
of the stool.
B
C
C
16. Standardized Test Practice How many whole numbers are there whose
square roots are greater than 9 but less than 10?
A 10
B 15
C 18
D 22
14. 4 15. 12 in.
16. C
Glencoe/McGraw-Hill
13. 20
©
11. 1 12. 10
C
B
A
10. 18
8.
8. 7 9. 22
A
7.
22
7. 16
B
B
6.
6. 10
A
5.
5. 11
4.
Answers: 1. 8 2. 3 3. 5 4. 12
3.
NAME ________________________________________ DATE ______________ PERIOD _____
The Real Number System (pages 125–129)
You have studied whole numbers, integers, and rational numbers. Rational
numbers include terminating and repeating decimals as well as the square
roots of perfect squares. Numbers that do not terminate or repeat are called
irrational numbers.
a
An irrational number is a number that cannot be expressed as , where a and b
b
Irrational
Numbers
are integers and b does not equal 0. The square roots of numbers that are not
perfect squares are irrational. You can use a calculator to find approximate square
roots with numbers such as 11
and 27
.
Real
Numbers
The sets of rational and irrational numbers combine to form the set of real numbers.
The graph of all real numbers is the entire number line.
Rational Numbers
0.7
2
3
25
.
1
Irrational
Numbers
2
Whole
Numbers
5 0
Try These Together
1. Use the letters given below in the
Practice exercises to name the set or
sets of numbers to which 25 belongs.
HINT: You can write 25 as
0.4
Integers
–2
5
–3
5
2. Use the letters given below in the
Practice exercises to name the set or sets
of numbers to which 47 belongs.
HINT: 47 is not a perfect square.
Let R real numbers, Q rational numbers, Z integers,
W whole numbers, and I irrational numbers. Name all sets of
numbers to which each real number belongs.
7
3. 12
5. 16
4. 0.272272227 …
Estimate each square root to the nearest tenth.
6. 10
B
C
8.
C
B
A
10. Standardized Test Practice You are building a fence around your
mother’s square garden. She has told you that she believes that the
garden is about 250 square feet. About how many feet of fence must
you purchase in order to enclose the entire garden?
A 15 ft
B 16 ft
C 50 ft
D 63 ft
9. 11.1 10. D
A
7.
©
Glencoe/McGraw-Hill
8. 8.8
B
B
6.
9. 124
C
A
5.
7. 5.6
4.
8. 77
Answers: 1. W, Z, Q, R 2. I, R 3. Q, R 4. I, R 5. Z, Q, R 6. 3.2
3.
7. 31
23
NAME ________________________________________ DATE ______________ PERIOD _____
The Pythagorean Theorem (pages 132–136)
The longest side of a right triangle is the hypotenuse. The sides that form
the right angle are the legs.
Pythagorean
Theorem
In a right triangle, the square of the length of the hypotenuse
is equal to the sum of the squares of the lengths of the legs.
c2 a2 b2
Converse of
Pythagorean
Theorem
If the sides of a triangle have lengths a, b, and c units such that c2 a2 b2,
then the triangle is a right triangle.
a
c
b
Is a triangle that has sides of 3, 5, and 7 a right triangle?
Is 72 equal to 32 52? No, 49 9 25, so the sides do not fit the
converse of the Pythagorean Theorem. It is not a right triangle.
Try These Together
Round to the nearest tenth.
1. Find the length of the missing side of
the right triangle. a, 7 m; c, 11 m
2. Find the length of the missing side of the
right triangle. b, 24 cm; c, 37 cm
HINT: Use c2 a2 b2. Solve for b.
HINT: Use the Pythagorean Theorem.
Find the missing length in each right triangle. Round to the
nearest tenth if necessary.
3.
4.
5.
c cm
6 cm
15 in.
8 cm
6. a, 19 yd; b, 16 yd
21 in.
17 in.
16 in.
b in.
x in.
7. b, 67 mm; c, 69 mm
8. a, 6.2 m; b, 8.6 m
Determine whether each triangle with sides of given lengths is a
right triangle.
9. 9 in., 12 in., 15 in.
10. 16 ft, 29 ft, 18 ft
11. 9 m, 7 m, 13 m
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
12. Standardized Test Practice The cities of Coldwater, Wayne, and Clinton
form a right triangle on the map. The distance from Wayne to Coldwater is
50 miles. The distance from Coldwater to Clinton is 60 miles. Coldwater
is due north of Wayne, and Clinton is due east of Coldwater. To the nearest
mile, how far is it if you drive directly from Wayne to Clinton?
A 55 mi
B 67 mi
C 78 mi
D 110 mi
Answers: 1. 8.5 m 2. 28.2 cm 3. 10 cm 4. 14.7 in. 5. 15 in. 6. 24.8 yd 7. 16.5 mm 8. 10.6 m 9. yes 10. no 11. no 12. C
3.
©
Glencoe/McGraw-Hill
24
NAME ________________________________________ DATE ______________ PERIOD _____
Using the Pythagorean Theorem (pages 137–140)
You can use the Pythagorean Theorem to find lengths of objects that have
rectangular or right-triangular shapes.
Marcia has a rectangular scarf that measures 36 inches by 48 inches.
She folds it along the diagonal to make a right triangle. How long is
the hypotenuse?
362 482 d2 Pythagorean Theorem
1,296 2,304 d2
3,600 d2
60 d
The hypotenuse is 60 inches long.
Try These Together
1. Determine the length of the second leg
of a right triangle that has a hypotenuse
of 50 inches and a leg of 40 inches.
2. A table top is 3 feet by 4 feet. How long
is its diagonal?
HINT: Draw a sketch. What kind of triangle
does the diagonal make?
HINT: Use the Pythagorean Theorem.
Write an equation that can be used to find the length of the
missing side of each right triangle. Then solve. Round to the
nearest tenth.
3.
4.
5. x m
16 m
x
6 ft
4m
10 ft
bm
ym
10 m
14 m
9m
6. Recreation A sail on a ship is a right triangle. If one leg measures 30 feet
and the other measures 16 ft, find the length of the hypotenuse of the sail.
B
C
C
7. Standardized Test Practice A right triangle has one leg that is
18 centimeters and a hypotenuse that is 30 centimeters. Find the length
of the third side.
A 24 cm
B 48 cm
C 35 cm
D 540 cm
4. 42 b2 162; b 240
15.5 m
C
B
A
©
Glencoe/McGraw-Hill
3. 62 102 x2; x 136
11.7 ft
8.
25
2. 5 ft
A
7.
Answers: 1. 30 in.
B
B
6.
7. A
A
5.
6. 34 ft
4.
5. 142 102 x2; x 296
17.2 m; 92 102 y2; y 181
13.5 m
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Distance on the Coordinate Plane (pages 142–145)
You can use what you know about right triangles to find the distance
between two points on a coordinate grid.
Finding
Distance on
the Coordinate
Plane
To find the distance between two points on the coordinate plane, draw the
segment that joins the points. Then make that segment the hypotenuse of a
right triangle. Use the Pythagorean Theorem to find the length of the
hypotenuse, which is the distance between the two points.
Find the distance between the points (5, 5) and (1, 3).
First draw the segment that joins these two points. Then draw segments
so that this segment is the hypotenuse of a right triangle. Count squares
to find the lengths of the legs, 6 and 8. Since 6 and 8 are the first two parts
of a Pythagorean triple, you know that the length of the hypotenuse is 10.
Check: Does 62 82 102? Yes, because 36 64 100.
The distance between the two points is 10 units.
y
(5, 5)
O
x
(–1, –3)
Try This Together
1. Find the distance between (7, 3) and (2, 1). Round to the
nearest tenth.
HINT: Graph the points and then draw segments down from (7, 3) and
to the right from (2, 1).
Find the distance between each pair of points whose coordinates
are given. Round to the nearest tenth.
y
y
y
2.
3.
4.
(4, 0)
(3, 2)
O
(–1, 2)
x
O
(0, 1)
x
O
(2, –3)
x
(–3, –1)
Find the distance between the points. Round to the nearest tenth.
5. (3, 3), (2, 0)
6. (4, 4), (1, 1)
7. (0, 0), (6, 2)
8. (0, 3), (4, 3)
9. Geometry A right triangle on the coordinate plane has vertices A(3, 2),
B(1, 2), and C(3, 2). Find the length of the hypotenuse.
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
10. Standardized Test Practice Find the distance between A(8, 4) and
B(0,2).
A 48 units
B 100 units
C 10 units
D 64 units
Answers: 1. 6.4 units 2. 3.2 units 3. 3.6 units 4. 3.6 units 5. 5.8 units 6. 7.1 units 7. 6.3 units 8. 7.2 units 9. 5.7 units 10. C
3.
©
Glencoe/McGraw-Hill
26
NAME ________________________________________ DATE ______________ PERIOD _____
Chapter 3 Review
Coordinate Treasure Hunt
Starting at point X on the coordinate plane below, follow the directions to
find the location of a hidden treasure. Record your location at each point.
y
X
x
O
1. Draw a triangle with vertices at X, Y(3, 1) and Z(0, 1). What is the
measure of angle XZY ?
2. To the nearest tenth, what is the length of the hypotenuse of this triangle?
3. From point Z draw a segment to W(0, 3) and from W, draw a segment to
R(2,3). To the nearest tenth, what is the measure of R
Z
?
4. From point R, move 6 units south (or down). Where are you now?
5. From there, move 3 units east (or right) to find the treasure. What are the
coordinates of the hidden treasure?
6. To the nearest tenth, how far is the treasure from point X?
©
Glencoe/McGraw-Hill
27
NAME ________________________________________ DATE ______________ PERIOD _____
Ratios and Rates (pages 156–159)
A ratio compares two numbers by division.
Ratio
27
,
100
27 out of 100, 27 to 100, 27:100
Rate
A rate is a special kind of ratio. A rate compares two quantities with different
units, such as miles to the gallon or cents per pound.
Unit Rate
When a rate is simplified so that it has a denominator of 1, it is called a unit rate.
A Express 12 winners for every 90 people
who enter as a rate in simplest form.
Write a fraction for the rate:
B Express the rate $6 for 3 pounds as a
unit rate.
12
.
90
Write a rate:
$6
.
3 pounds
Divide numerator and denominator by the GCF
to simplify. The GCF of 12 and 90 is 6.
Divide numerator and denominator by 3 to
get a denominator that is 1 unit.
2
15
The unit rate is $2 per pound.
is the rate in simplest form.
Try These Together
1. Express 16 out of 32 in simplest
form.
2. Express 6 wins in 10 games in simplest
form.
HINT: Write a fraction and simplify.
HINT: Write a fraction and simplify.
Express each ratio or rate in simplest form.
3. 3 to 15
4. 3 boys: 24 girls
6. 56 dogs to 48 cats
7. 4 feet: 16 feet
5. 13 meters per second
8. 12 books for 4 students
Express each rate as a unit rate.
9. $18.00 for 3 pounds
10. $19.50 for 15 gallons
12. $2.00 for 10 minutes
13. 8 feet in 2 seconds
11. $1.68 for 8 ounces
14. 25 magazines in 5 days
15. Sports Gloribel ran the 400-meter dash in 80 seconds. How many
meters did she run per second?
B
C
C
16. Standardized Test Practice Suppose that a bottle of peppercorn ranch
salad dressing costs $2.65 at the grocery store. If there are 20 ounces in
the bottle, what is the price of the salad dressing per ounce? Round to
the nearest cent.
A $0.14
B $0.12
C $0.15
D $0.13
5. 1
13
6. 6
7
7. 4
1
8. 1
3
9. $6.00 per pound 10. $1.30 per gallon 11. $0.21 per ounce
28
1
Glencoe/McGraw-Hill
4. 8
©
1
C
B
A
3. 5
8.
3
A
7.
2. 5
B
B
6.
1
A
5.
Answers: 1. 2
4.
12. $0.20 per minute 13. 4 feet per second 14. 5 magazines per day 15. 5 16. D
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Rate of Change (pages 160–164)
A rate of change is a rate that describes how one quantity changes in relation to another.
To find the rate of change, divide the difference in the y-coordinates by the difference in
y y
2
1
. Rates of change
the x-coordinates. The rate of change between (x1, y1) and (x2, y2) is x2 x1
can be positive, negative, or zero.
Rate of Change
positive
zero
negative
Real-Life Meaning
increase
no change
decrease
y
Graph
O
Find the rate of change between 1990 and 2000.
x
x
O
Population of Idaho
Year
588,637
1950
667,191
1960
713,015
1970
944,127
1980
1,006,749
1990
1,293,953
2000
change in population
(1,293,953 1,006,749) people
change in year
(2000 1990) years
287,204 people
10 years
slants
downward
horizontal
line
x
O
y
y
slants
upward
28,720.4 people
1 year
The population of Idaho has grown an average of
28,720.4 people per year.
The World Almanac, 2002, p. 377
For Exercises 1– 4, use the table at the right.
The table shows the number of patrons at the local
swimming pool throughout the day.
1. Find the rate of change from 12 P.M. to 1 P.M.
2. Find the rate of change from 11 A.M. to 2 P.M.
3. Was the rate of change between 1 P.M. and 2 P.M. positive,
negative, or zero?
4. During which time period was the rate of change
in patrons negative?
B
4.
C
B
8.
12
23
25
25
13
C
B
A
7.
11 A.M.
12 P.M.
11 P.M.
12 P.M.
13 P.M.
C
A
5.
6.
Number of
Patrons at the
Swimming Pool
B
A
5. Standardized Test Practice At West High School the T-shirt
sales for the pep club totaled 135 in 1999. In 2002, they totaled 162. If
this rate of change were to continue, what would be the total T-shirt
sales in 2003?
A 171 T-shirts B 153 T-shirts
C 162 T-shirts
D 135 T-shirts
Answers: 1. 2 people/hour 2. 4.3 people/hour 3. zero 4. between 2 P.M. and 3 P.M. 5. A
3.
Time
©
Glencoe/McGraw-Hill
29
NAME ________________________________________ DATE ______________ PERIOD _____
Slope (pages 166–169)
The rate of change between any two points on a line is always the same.
This constant rate of change is called the slope of the line. Slope is the
ratio of the rise, or vertical change, to the run, or horizontal change.
y
Find the slope of the line.
(–3, 3)
Choose two points on the line. The vertical change is down 3
units, or 3, while the horizontal change is right 5 units, or 5.
slope rise
run
3 units
(2, 0)
O
3
5
x
5 units
Find the slope of each line.
1.
2.
y
3.
y
y
(1, 5)
(4, 2)
(–3, 2)
x
O
x
x
O
O
(0, –2)
(3, –2)
(–1, –3)
The points given in each table lie on a line. Find the slope of the
line. Then graph the line.
4.
x
1
0
1
2
y
5
3
1
1
5.
x
8
4
0
4
y
3
0
3
6
Find the slope of each line and interpret its meaning as a rate of change.
Filling a Pool
Shirt Sale
Distance From Home
6.
7.
8.
6
4
2
10
20
30
40
200
100
50
B
30
20
2
3
4
5
Time Traveled (h)
1
2
3
4
5
Number of Shirts
C
C
A
B
5.
C
B
9.
Standardized Test Practice There are two ramps that enter the school.
The first rises 2 feet for every 16-foot run. The second ramp rises 1 foot
for every 7-foot run. Which statement is true?
A The first ramp is steeper than the second. B Both ramps have the same steepness.
C The second ramp is steeper than the first. D This cannot be determined from the
information given.
1
B
A
©
Glencoe/McGraw-Hill
1
8.
3
A
7.
2
6.
Answers: 1. 3 2. 1 3. 4 4–5. See Answer Key for graphs. 4. 2 5. 4 6. 5; The pool fills at a rate of 5 foot per minute.
7. 60; Each hour you get 60 miles closer to home. 8. 20; Each shirt costs $20. 9. C
4.
40
10
1
Time (min)
3.
50
300
Cost ($)
Distance (mi)
Depth (ft)
8
30
NAME ________________________________________ DATE ______________ PERIOD _____
Solving Proportions (pages 170–173)
You can use two equal ratios to write a proportion.
A proportion is an equation stating that two ratios are equivalent.
Solving a
Proportion
a
b
c
, b 0, and d 0
d
c
a
The cross products of a proportion are equal. If , then ad bc.
b
d
2
3
4
Are the cross products for
2
3
and
3
4
Find the cross products.
4 c 5 12
4c 60
equal?
The cross products are 2 4 and 3 3. 8 9.
Since the cross products are not equal,
the ratios do not form a proportion.
2
3
12
B Solve .
5
c
A Determine whether the ratios 3 and 4
form a proportion.
4c
4
3
,
4
60
4
Divide each side by 4.
c 15
Try These Together
3
2
6
1. Determine whether 5 and 4
form a proportion.
3
2. Determine whether 8 and 4
form a proportion.
HINT: Find the cross products.
HINT: See if the cross products are equal.
Determine whether each pair of ratios form a proportion.
10 6
3. ,
20 12
3 1
4. 8 , 5
2 8
5. 6 , 24
5 1
6. ,
25 5
6 2
7. ,
15 5
9
5
8. ,
27 12
Solve each proportion.
2
x
9. 5 20
3
4
10. n 8
3
6
11. p 16
3
6
12. 10
r
15
9
6
a
y
t
3
9
13. 5 14.
15.
16.
25
7
21
4
8
9
k
17. Manufacturing A company manufactures two different types of school
desks. One is a desk with the chair attached and the other is a small desk
with a separate chair. One out of every 3 desks they manufacture has the
chair separate. If they manufactured 90 desks, how many would have the
chairs separate?
B
C
C
18. Standardized Test Practice If a car can travel 60 miles in 1 hour, how
far can it travel in 5 hours?
A 300 mi
B 1,100 mi
C 600 mi
D 550 mi
16. 27
C
B
A
©
8. no 9. 8 10. 6 11. 8 12. 5 13. 3 14. 3 15. 12
8.
Glencoe/McGraw-Hill
31
7. yes
A
7.
6. yes
B
B
6.
4. no 5. yes
A
5.
3. yes
4.
Answers: 1. no 2. yes
17. 30 18. A
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Similar Polygons (pages 178–182)
A polygon is a simple closed figure in a plane formed by three or more
line segments. A quadrilateral is a polygon with four sides. A pentagon is a
polygon with five sides.
Similar
Polygons
Two polygons are similar if their corresponding angles are congruent, and their
corresponding sides are proportional.
In the figure at the right, ABC ~ DEF. Find
E
.
the length of side D
B
AB
corresponds to DE
and BC
corresponds to EF
.
So you can write a proportion.
AB
DE
3
x
BC
EF
4
6
4 cm
3 cm
A
C
6 cm
E
6 cm
x cm
AB 3, DE x, BC 4, EF 6
D
18 4x
4.5 x
Solve for x.
The length of DE
is 4.5 centimeters.
F
9 cm
Each pair of polygons is similar. Write a proportion to find each
missing measure. Then solve.
18 ft
1.
2.
3.
15 in.
8m
x ft
x in.
3 in.
12 ft
5 in.
8 ft
10 m
20 m
xm
4. Hobbies Sean wants to enlarge a 4-inch by 6-inch photo so the shortest
side is 6 inches. How long will the longest side be?
B
C
5. Standardized Test Practice ABC is similar to DEF. If AB 2,
BC 5, and DE 26, then EF is equal to what?
4
2
5. D
Glencoe/McGraw-Hill
10
©
4
B 10 5
x
A 2 5
5
C
B
A
3
8.
2. ; 9 3. ; 4 4. 9 in.
x
15
8
20
A
7.
32
C 20 5
D 65
x
B
B
6.
8
C
A
5.
1. ; 12
12
18
4.
Answers: 1–3. Sample proportions are given.
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Scale Drawings and Models (pages 184–187)
A scale drawing or scale model is used to represent an object that is too
large or too small to be drawn or built at actual size.
Using Scale
Drawings
The scale of a drawing or model is determined by the ratio of a given length on
the drawing or model to its corresponding actual length.
The figure at the right is a scale drawing of a cabin plan.
In the drawing, the side of each square represents 20 inches.
Find the length and width of bedroom 2.
bedroom bath bedroom
1
2
Count the squares in the scale drawing. Bedroom 2 is 6 squares long and
5 squares wide. Use the scale and your counts to write proportions.
1 square
20 in.
6 squares
x in.
1 square
20 in.
kitchen/living room
5 squares
y in.
1 x 20 6
1 y 20 5
x 120
y 100
The length of bedroom 2 is 120 inches, and the width is 100 inches.
Try These Together
1. Use the figure and scale in the Example
to find the length and width of the
kitchen/living room.
porch
2. Use the figure and scale in the
Example to find the length and width
of the porch.
HINT: Write proportions.
HINT: The length is the same as the
kitchen/living room.
3. Find the length and width of the bath in the Example.
4. On a map, the scale is 1 inch 250 miles. Find the actual distance for
each map distance.
B
4.
C
B
8.
Portland, Oregon
about 4 inches
4
c. Portland, Oregon
Minneapolis, Minnesota about 7 inches
1
C
B
A
7.
b. San Diego, California
C
A
5.
6.
Map Distance
about 8 inches
B
A
5. Standardized Test Practice Find the dimensions of the cabin (including
the porch) in the Example.
A 150 in. by 150 in.
B 112 in. by 112 in.
C 300 in. by 280 in.
D 300 in. by 300 in.
Answers: 1. 300 in. by 140 in. 2. 300 in. by 60 in. 3. 60 in. by 100 in. 4a. about 2,000 miles 4b. about 1,062.5 miles
4c. about 1,750 miles 5. D
3.
From
To
a. Minneapolis, Minnesota San Diego, California
©
Glencoe/McGraw-Hill
33
NAME ________________________________________ DATE ______________ PERIOD _____
Indirect Measurements (pages 188–191)
Using proportions to find a measurement is called indirect measurement.
Using Indirect
Measurement
Use the corresponding parts of similar triangles to write a proportion. Solve
the proportion to find the missing measurement.
1
George is 5 2 feet tall. His shadow is 22 inches long at the same time that a
tree has a shadow that is 120 inches long. How many feet tall is the tree?
5.5 feet
22 inches
t feet
120 inches
Write a proportion.
5.5(120) 22t
30 t
Solve for t.
The tree is 30 feet tall.
In Exercises 1–3, the triangles are similar. Write a proportion and
solve the problem.
1. Find the distance across Blue Lake.
1.5 mi
Blue Lake
x mi
0.8 mi
2. The city of Hutchinson plans to build a bridge
over the narrowest part of Stillwater River. Find
the distance across this part of the river.
1 mi
450 m
Stillwater
River
xm
363 m
150 m
3. When Peter stands in front of a 27-foot tree in
front of his apartment building he can barely see
the very top of the building over the tree. How tall
is his apartment building?
x ft
24 ft
56 ft
8.
C
B
A
4. Standardized Test Practice ABC XYZ. AB 45 m, BC 15 m,
and XY 24 m. How long is Y
Z
?
2
2
A 2 3 m
B 7 3 m
4. C
A
7.
©
x
56
3. ; 63 ft
27
24
C
B
B
6.
Glencoe/McGraw-Hill
x
150
2. ; 121 m
363
450
C
A
5.
C 8m
x
1.5
1. ; 1.2 mi
0.8
1
4.
34
D 72 m
Answers: 1–3. Sample proportions are given.
B
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Dilations (pages 194–197)
The image produced by enlarging or reducing a figure is called a dilation.
Working with
Dilations
Since the dilated image has the same shape as the original, the two images are
similar. The ratio of the dilated image to the original is called the scale factor.
A triangle has vertices M(2,2), N(6, 2), and P(2, 4). Find the
5
coordinates of MNP after a dilation with a scale factor of 2 .
Multiply each coordinate in each ordered pair by
M(2, 2) → 2 5
5
, 2 → M(5, 5)
2
2
5
5
6
, 2 → N(15, 5)
2
2
5
5
,4
→ P(5, 10)
2
2
N(6, 2) → P(2, 4) → 2
5
.
2
1. Find the coordinates of the image of point C(12, 4) after a dilation with a
2
scale factor of .
3
Triangle KLM has vertices K(5, 15), L(5, 10), and M(15, 20). Find
the coordinates of its vertices after a dilation with each given scale
factor.
1
3
2. 3
3. 5
4. 5
In each figure, the dashed-lined figure is a dilation of the
solid-lined figure. Find each scale factor.
y
y
5.
6.
7.
x
O
O
y
x
O
B
C
C
8.
C
B
A
8. Standardized Test Practice What are the coordinates of the image of
1
point Q(3,8) after a dilation with a scale factor of 4?
A Q 4 , 2
3
B Q(12, 32)
C Q(3, 2)
4 1
D Q 3, 2
2
A
7.
Answers: 1. C8, 2 2. K(15, 45), L(15, 30), M(45, 60) 3. K(1, 3), L(1, 2), M(3, 4)
3
B
B
6.
8. A
A
5.
2
6. 2 7. 3
4.
1
4. K(3, 9), L(3, 6), M(9, 12) 5. 2
3.
x
©
Glencoe/McGraw-Hill
35
NAME ________________________________________ DATE ______________ PERIOD _____
Chapter 4 Review
Vocabulary Time
Solve each problem. Find the letter from the list at the bottom of the page
that corresponds to your numerical answer. Place the letter in the blank at
the right. When you are finished you will have spelled a vocabulary word
from the chapter.
1. Express the ratio in simplest form: 9 Aspens to 12 trees.
1. ____
2. Express the rate as a unit rate: $12 for 24 donuts.
2. ____
3. Find the slope of the line.
3. ____
4. Find the slope of the line.
y
y
4. ____
x
O
x
O
5. Write a proportion that could be used to solve for m. Then solve.
4 miles run in 30 minutes, 6 miles run in m minutes.
5. ____
6. Segment A'B' is a dilation of segment AB. The endpoints of each segment
1
1
are A2, 2, B12, 3, A'(4, 1), and B'(3, 6). Find the scale factor
of the dilation.
6. ____
7. Corey is 5 feet 6 inches tall. He stands next to a tree that casts a shadow
of 37 feet 6 inches. If Corey’s shadow is 8 feet 3 inches, how tall is the
tree in feet?
7. ____
A B C D E
2 11 4
9
3
5
F
0
G H
I
J
K L M N
6 15
1
2
4
3
7
3
41 45 3
O P
Q
R
S
T U
V W X Y Z
12 30 18 25
3
4
2
3
5 10 1 27 8
7
Answer is located on page 108.
©
Glencoe/McGraw-Hill
36
NAME ________________________________________ DATE ______________ PERIOD _____
Ratios and Percents (pages 206–209)
A percent is a ratio that compares a number to 100.
Writing
a Fraction
or Ratio as
a Percent
As a ratio: 4 out of 5
80
As a fraction with a denominator of 100: 100
As a percent: 80%
A Write 7 students out of 10 as a
percent.
Write the rate as a fraction.
B Write 45% as a fraction in simplest
form.
7
10
% means
Multiply numerator and denominator by 10 to
rename as a fraction with a denominator of
100.
45% is
45% is
70
100
45
.
100
The GCF of 45 and 100 is 5.
70
100
“Percent” means “per 100” so
.
100
9
.
20
is 70%.
Try These Together
1. Write 3 out of 5 as a percent.
1
2. Write 4 as a percent.
HINT: Write as a fraction. Then multiply
numerator and denominator by the same
number to rewrite as a number divided by 100.
HINT: Multiply numerator and denominator by
the same number to rewrite as a number
divided by 100.
Write each ratio or fraction as a percent.
3. 3:10
4. 18:100
5. 3 out of 4
8
6. 10
Write each ratio as a percent.
7. Twelve out of 20 students are involved in after-school activities.
8. One out of 10 instruments in the band is a flute.
Write each percent as a fraction in simplest form.
9. 20%
10. 35%
11. 50%
13. 85%
14. 25%
15. 8%
B
C
C
B
C
17. Standardized Test Practice Jamal surveyed the students in his class. He
found that 2 out of 5 of them read books other than school books for
pleasure. What is this ratio expressed as a percent?
A 15%
B 50%
C 40%
D 80%
7
11. 2
1
12. 5
2
13. 20
17
14. 4
1
Glencoe/McGraw-Hill
10. 20
©
1
B
A
Answers: 1. 60% 2. 25% 3. 30% 4. 18% 5. 75% 6. 80% 7. 60% 8. 10% 9. 5
8.
17. C
A
7.
19
B
6.
16. 50
A
5.
2
4.
15. 25
3.
12. 40%
16. 38%
37
NAME ________________________________________ DATE ______________ PERIOD _____
Fractions, Decimals, and Percents (pages 210–214)
The word percent also means hundredths or per hundred or divided by 100.
Decimals
• To write a percent as a decimal, divide by 100 and remove the % symbol.
and Percents • To write a decimal as a percent, multiply by 100 and add the % symbol.
A Write 47% as a decimal.
47% means
47
100
3
B Write as a percent.
16
or 0.47.
3
16
n
100
Write a proportion.
300 16n
300
16
16n
16
18.75 n
3
16
Try These Together
1. Write 27% as a decimal.
18.75
100
or 18.75%
2. Write 6% as a decimal.
HINT: Divide by 100.
HINT: Divide by 100.
Write each percent as a decimal.
3. 63%
4. 40%
5. 79%
6. 16%
Write each decimal as a percent.
7. 0.12
8. 0.84
9. 0.65
10. 0.04
Write each fraction as a percent.
1
11. 5
7
12. 10
29
13. 50
21
14. 100
15. 58% ● 0.58
16. 8.9 ● 89%
17. 0.04 ● 40%
18. 14% ● 1.4
19. Population In 1997, about 3 out of 25 people in the world lived in
Africa. Express this ratio as a percent.
B
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
20. Standardized Test Practice Jaryn and Blake decorated for a school
party. They used the colors yellow and blue. Three out of 5 balloons
were blue. What percent is this?
A 30%
B 15%
C 60%
D 45%
11. 20% 12. 70% 13. 58%
4.
©
Glencoe/McGraw-Hill
Answers: 1. 0.27 2. 0.06 3. 0.63 4. 0.4 5. 0.79 6. 0.16 7. 12% 8. 84% 9. 65% 10. 4%
14. 21% 15. 16. 17. 18. 19. 12% 20. C
3.
38
NAME ________________________________________ DATE ______________ PERIOD _____
The Percent Proportion (pages 216–219)
In a percent proportion, one of the numbers, called the part, is being
compared to the whole quantity, called the base. The other ratio is the
percent, written as a fraction, whose base is 100.
Percent
Proportion
percent
100
Words
part
base
Symbols
Arithmetic
2
5
40
100
Algebra
a
b
p
,
100
where a is the part, b is
the base, and p is the percent.
A Find 15% of 78.
a
78
15
100
B 30 is 60% of what number?
30
b
b 78, p 15
100a 78(15) Find the cross products.
100a 1170
a 11.7
15% of 78 is 11.7.
60
100
a 30, p 60
30(100) b(60) Find the cross products.
3,000 60b
50 b
30 is 60% of 50.
Try These Together
2
1. Express as a percent.
5
HINT: Solve
2
5
p
100
for p.
2. Write a percent proportion and find 28% of 13.
HINT: Use
a
13
28
.
100
Write each fraction as a percent.
3
3. 10
11
4. 25
17
5. 20
7
6. 50
1
7. 8
13
8. 40
5
9. 16
2
10. 25
Write a percent proportion to solve each problem. Then solve.
Round to the nearest tenth if necessary.
11. What is 8% of 270?
12. 12 is 20% of what number?
13. 48 is what percent of 99?
16. Find 16% of 40.
17. Pet Food A birdseed blend is 65% blackoil sunflower seeds. How
many pounds of blackoil sunflower seeds are in a 40-pound bag?
B
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
18. Standardized Test Practice What is 12% of 60?
A 0.2
B 6.2
C 6.8
11. 21.6 12. 60
4.
©
D 7.2
Answers: 1. 40% 2. 3.64 3. 30% 4. 44% 5. 85% 6. 14% 7. 12.5% 8. 32.5% 9. 31.25% 10. 8%
13. 48.5% 14. 55.6% 15. 20 16. 6.4 17. 26 pounds 18. D
3.
Glencoe/McGraw-Hill
39
NAME ________________________________________ DATE ______________ PERIOD _____
Finding Percents Mentally (pages 220–223)
You can find some percents using mental math. Some common percents
can also be found using the equivalent fractions.
• To find 1% of a number mentally, move the decimal point two places to the left
(which is the same as dividing the number by 100).
• To find 10% of a number mentally, move the decimal point one place to the left
(which is the same as dividing the number by 10).
Finding
Percents
Mentally
Equivalent Fractions, Decimals, and Percents
7
,
8
1
,
2
0.5, 50%
2
,
3
2
2
4
0.66 , 66 % , 0.8, 80%
3
3
5
1
,
4
0.25, 25%
1
,
5
0.2, 20%
1
,
8
3
0.125, 12.5% , 0.3, 30%
10
3
,
4
0.75, 75%
2
,
5
0.4, 40%
3
,
8
7
0.375, 37.5% , 0.7, 70%
10
1
,
3
1
1
3
0.33 , 33 % , 0.6, 60%
3
3
5
5
,
8
9
0.625, 62.5% , 0.9, 90%
10
A Compute 1% of 325 mentally.
Think: 1% is
1
100
B Compute 75% of 12 mentally.
so move the decimal point
Think: 75% is
in 325 two places to the left to make a
smaller number.
1% of 325 is 3.25.
3
4
Try These Together
1. Compute 10% of 200 mentally.
HINT: 10% is
1
.
10
What is
Compute mentally.
3. 12.5% of 56
6. 30% of 120
0.875, 87.5%
3
.
4
of 12 is 9.
2. Compute 50% of 80 mentally.
200
?
10
HINT: What fraction equals 50%?
4. 1% of 21
7. 50% of 46
5. 90% of 300
8. 40% of 40
9. 5 ● 10% of 100
10. 62.5% of 80 ● 45
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
11. Standardized Test Practice An advertising firm has small businesses
and large corporations for clients. If they have 225 clients and 40% of
them are small businesses, how many clients are small businesses?
A 80
B 90
C 70
D 100
Answers: 1. 20 2. 40 3. 7 4. 0.21 5. 270 6. 36 7. 23 8. 16 9. 10. 11. B
3.
©
Glencoe/McGraw-Hill
40
NAME ________________________________________ DATE ______________ PERIOD _____
Percent and Estimation (pages 228–231)
Compatible numbers are two numbers that are easy to divide mentally.
Estimating with
Compatible
Numbers
To estimate a percent using compatible numbers:
• Round to numbers that are easy to divide.
• Use those numbers to make an estimate.
A Estimate 18% of 50.
B Estimate what percent 13 out of 63 represents.
Think: 18% is about 20% and 20% is
1
5
1
.
5
13 out of 63 is about 13 out of 65, or
of 50 is 10.
Since
13
65
1
5
13
.
65
or 20%, 13 out of 63 is about 20%.
18% of 50 is about 10.
Try These Together
1. Estimate 26% of 80.
2. Estimate the percent represented by
11 out of 24.
HINT: 26% is about 25%, which
equals
1
.
4
Estimate.
3. 18% of 50
6. 89% of 10
HINT:
4. 73% of 48
7. 9% of 81
Estimate each percent.
9. 3 out of 23
10. 15 out of 35
12. 11 out of 56
13. 9 out of 16
11
24
is about
12
24
which equals
1
.
2
5. 38% of 31
8. 48% of 52
11. 10 out of 31
14. 32 out of 41
15. Estimate what percent 13 out of 27 represents.
16. Money Matters Gareth’s restaurant bill was $29.65. Estimate how
much a 20% tip would be.
B
C
C
A
7.
8.
C
B
A
17. Standardized Test Practice 78% of the students at Willow Middle
School ride the bus home. If there are 201 students, estimate how many
of them ride the bus home.
A 180
B 160
C 120
D 80
9. 12.5% 10. 40%
B
B
6.
©
Glencoe/McGraw-Hill
6. 9 7. 8 8. 25
A
5.
5. 12
4.
Answers: 1–16. Sample answers are given. 1. 20 2. 50% 3. 10 4. 36
11. 33.3% 12. 20% 13. 50% 14. 80% 15. 50% 16. $6.00 17. B
3.
41
NAME ________________________________________ DATE ______________ PERIOD _____
The Percent Equation (pages 232–235)
Another way to find a percent is to use the percent equation,
Part Percent Base. Express the percent as a decimal and multiply.
Type
Example
Find the Part
The
Percent
Equation
Equation
n 0.25(60)
What number is 25% of 60?
part
Find the Percent
15 n(60)
15 is what percent of 60?
percent
Find the Base
15 0.25n
15 is 25% of what number?
base
A 15% of what number is 3?
B 45 is what percent of 120?
Part Percent Base Use the percent equation.
3 0.15n
The part is 3, and the
percent is 15%. Let n
represent the base.
3
0.15
0.15n
0.15
n 20
15% of 20 is 3.
45
120
Divide each side by 0.15.
Simplify.
Try These Together
1. Find 15.5% of 90 using the percent
equation.
Part Percent Base Use the percent equation.
45 n(120)
The part is 45, and the
base is 120. Let n
represent the percent.
120n
120
n 0.375
n 37.5%
45 is 37.5% of 120.
Simplify.
Write the decimal as a %.
2. Find 33% of 77 using the percent
equation.
HINT: 90 is the base and the percent is 0.155.
HINT: The number following “of” is usually the base.
Solve each problem using the percent equation.
3. 12% of what number is 120?
4. 42% of what number is 21?
5. Find 82% of 30.
7. Find 40% of 37.
10. Find 75% of 98.
11. Find 12% of $1.75.
12. $8.22 is 15% of what amount?
13. Sports Brian had 18 hits in 78 times at bat during the last baseball
season. What percent of his times at bat were hits?
B
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
14. Standardized Test Practice What is 35% of 120?
A 42
B 38
C 34
5. 24.6 6. 25% 7. 14.8 8. 12.5%
4.
©
Glencoe/McGraw-Hill
D 30
Answers: 1. 13.95 2. 25.41 3–12. See Answer Key for equations. 3. 1,000 4. 50
9. 122 10. 73.5 11. $0.21 12. $54.80 13. about 23% 14. A
3.
42
NAME ________________________________________ DATE ______________ PERIOD _____
Percent of Change (pages 236–240)
A ratio that compares the change in quantity to the original amount is called
the percent of change. When the new amount is greater than the original,
the percent of change is a percent of increase. When the new amount is
less than the original, the percent of change is a percent of decrease.
Finding
Percent of
Markup and
Discount
• The increase in price that a store adds to its cost is called the markup. The
percent of markup is a percent of increase. The amount the customer pays is
called the selling price.
• The amount by which a regular price is reduced is called the discount. The
percent of the discount is a percent of decrease. Find the sale price by
subtracting the discount.
A Find the sale price for a $424 item
that is 20% off.
d 0.20(424)
d $84.80
$424 $84.80 $339.20
B A store paid $18 for an item and used a
30% markup. What was the selling price?
First use the percent
equation to find the
discount.
Find the sale price.
First use the percent
m 0.30(18) equation to find
m $5.40
the markup.
$18 $5.40 $23.40 Find the selling price.
Try These Together
1. Find the percent of change (rounded to
the nearest percent) if the original price
is $30 and the new price is $24.
2. Find the percent of change (rounded
to the nearest percent) if the original
is 35 and the new is 45.
HINT: First find the amount of change ($30 $24).
HINT: First find the amount of change.
Find the sale price of each item to the nearest cent.
3. jeans: $28.00, 50% off
1
4. jacket: $48.95, 5 off
5. paperback: $7.50, 10% off
6. watch: $15.30, 15% off
Find the selling price for each item given the cost to the store
and the markup. Round to the nearest cent.
7. CD: $9, 60% markup
8. DVD: $25, 40% markup
9. TV: $400, 45% markup
10. bedroom set: $2,400, 20% markup
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
11. Standardized Test Practice What is the sale price of an $80 CD player
on sale at 25% off?
A $20
B $50
C $60
D $320
Answers: 1. 20% 2. 29% 3. $14.00 4. $39.16 5. $6.75 6. $13.01 7. $14.40 8. $35.00 9. $580.00 10. $2,880.00
11. C
3.
©
Glencoe/McGraw-Hill
43
NAME ________________________________________ DATE ______________ PERIOD _____
Simple Interest (pages 241–244)
Interest is the amount paid or earned for the use of money.
Using the
Simple
Interest
Formula
In the formula for simple interest, I prt,
• I is the interest,
• p is the amount of money invested, or principal,
• r is the annual interest rate, and
• t is the time in years.
A Find the simple interest for $650 at 11%
for 4 months.
B Find the simple interest for $545 at
9.5% for 18 months.
Use the formula I prt. Notice that the time,
1
4 months, is of
3
1
I 650(0.11) 3
I $23.83
Use the formula I prt. Notice that the time,
18 months, is 1.5 years.
I 545(0.095)(1.5)
I $77.66
a year.
Try These Together
1. Find the simple interest for $175 at
12% for 1.5 years.
2. Find the simple interest for $820 at 6.5%
for 16 months.
HINT: r 0.12
HINT: Notice that the time is
16
12
1
or 1 years.
3
Find the simple interest to the nearest cent.
3. $98 at 9.25% for 3 years
4. $340 at 12% for 1.25 years
5. $318 at 8.75% for 6 months
6. $420 at 9% for 6 months
7. $514 at 10% for 2 years
Find the total amount in each account to the nearest cent.
9. $839 at 21% for 1 year
10. $325 at 8.5% for 1 year
12. $100 at 2.5% for 3 months
13. $672 at 5.5% for 2 years
14. $300 at 6.45% for 15 months
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
17. Standardized Test Practice What is the simple interest on $1,000 at 8%
for 2 years?
A $1,600
B $160
C $40
D $20
Answers: 1. $31.50 2. $71.07 3. $27.20 4. $51.00 5. $13.91 6. $18.90 7. $102.80 8. $42.84 9. $1,015.19
10. $352.63 11. $128.10 12. $100.63 13. $745.92 14. $324.19 15. $408.00 16. $255.36 17. B
3.
1
16. $230 at 7.35% for 1 years
2
©
Glencoe/McGraw-Hill
44
NAME ________________________________________ DATE ______________ PERIOD _____
Chapter 5 Review
Carnival Math
You and your parent or a partner can play this game. Your partner asks you
for the information requested in the parentheses under each blank in the
following paragraph. Then your partner writes your answer in each blank.
Read the paragraph and then answer the questions that follow.
out of ______ friends went to a carnival
and ______2. (write
a ratio)
1. (your name)
one afternoon.
3. (name a friend)
only get the bell ringer to raise
spent
5. (name a friend)
bear. At the dunking booth
______ out of ______ times.
8. (write a ratio)
against each other and
11. (decimal less than 1)
12. (percent less than 100)
tried the Test of Strength and could
4. (decimal greater than 1)
6. (dollars and cents)
trying to win a teddy
dunked the heckler
7. (name a friend)
and a friend raced
9. (your name)
10. (your name)
feet high.
won by a margin of
second. By the end of the afternoon, they had all spent
of their money and they decided it was time to go home.
13. Express the ratio in Exercise 8 as a decimal. Estimate,
if necessary.
14. Express the ratio in Exercise 2 as a percent. Estimate,
if necessary.
15. Express the percent in Exercise 12 as a fraction.
16. If 2 drinks at the carnival cost $1.50, how much will
5 drinks cost?
17. If 300 people attended the carnival that day, and 2 out
of 5 of them were adults, how many of the attendees that
day were adults?
18. Suppose you took $20 with you to the carnival and came
home with $5. $5 is what percent of $20?
©
Glencoe/McGraw-Hill
45
NAME ________________________________________ DATE ______________ PERIOD _____
Line and Angle Relationships (pages 256–260)
Parallel lines are lines in a plane that will never intersect. If line p is
parallel to line q, then write p || q. A line that intersects two or more other
lines is called a transversal. Congruent angles formed by parallel lines and
a transversal have special names. Angles formed by parallel lines and a
transversal also have certain special relationships.
Congruent
Angles With
Parallel Lines
If a pair of parallel lines is intersected by a transversal,
these pairs of angles are congruent.
alternate interior angles: 4 6, 3 5
alternate exterior angles: 1 7, 2 8
corresponding angles: 1 5, 2 6,
3 7, 4 8
Vertical
Angles and
Supplementary
Angles
Vertical angles are opposite angles formed by the intersection of two lines.
Vertical angles are congruent. (For example, 1 3 above.)
Supplementary angles are two angles whose measures have a sum of 180°.
(For example, 1 is supplementary to 2 above.)
12
43
56
87
Use the figure above for these examples.
A Find m1 if m5 60°.
B Find m6 if m7 75°.
1 and 5 are corresponding angles.
Corresponding angles are congruent.
Since m5 60°, m1 60°.
6 and 7 are supplementary angles.
So, m6 m7 180°.
m6 75° 180° Substitute 75° for m7.
m6 105° Subtract 75° from each side.
Try These Together
Use the figure at the right for Exercises 1–4. The two lines are
parallel.
1. Find m2 if m8 110°.
2. Find m4 if m6 122°.
12
43
56
87
HINT: Identify the type of angles first.
3. Find m3 if m2 98°.
4. Find m7 if m3 45°.
5. p and q are congruent. Solve for x if mp (2x 5)° and mq 75°.
6. Hobbies Alexis is making a quilt with a pattern that uses parallel
lines and transversals. The pattern is shown at the right. If m1 is
68°, what should m2 be?
B
C
B
5.
C
B
6.
A
7.
8.
1
C
A
B
A
7. Standardized Test Practice a and b are alternate exterior angles of
parallel lines. If ma is 138°, what is mb?
A 180°
B 138°
C 42°
D 48°
6. 68° 7. B
4.
©
Glencoe/McGraw-Hill
46
Answers: 1. 110° 2. 122° 3. 82° 4. 45° 5. 40
3.
2
NAME ________________________________________ DATE ______________ PERIOD _____
Triangles and Angles (pages 262–265)
A polygon is a simple closed figure in a plane formed by three or more line
segments. A polygon formed by three line segments that intersect only at their
endpoints is a triangle. Triangles can be classified by their angles and their sides.
Triangles
Classified
by Angles
• Acute triangles have three acute angles.
• Right triangles have one right angle.
• Obtuse triangles have one obtuse angle.
Triangles
Classified
by Sides
• Scalene triangles have no two sides that are congruent.
• Isosceles triangles have at least two sides congruent.
• Equilateral triangles have three sides congruent.
Classify each triangle by its angles and by its sides.
A ABC has one angle that measures 136°, B EFG has one angle that measures 90°.
and no sides that are the same length.
Since it has one right angle, you know that
EFG is a right triangle. You cannot
determine whether it is scalene or isosceles
without knowing the lengths of the sides of the
triangle.
Because the angle is greater than 90°, this is an
obtuse triangle. Because none of the sides are
the same length, it is also a scalene triangle.
ABC is an obtuse, scalene triangle.
Classify each triangle by its angles and by its sides.
1.
2.
3.
6.2 in.
110
45
5 cm
8 in.
7.1 cm
30
45
5 cm
40
11.7 in.
60
5m
5m
60
60
5m
4. Gift Wrapping Classify the triangles used in the pattern on the
wrapping paper shown at the right.
B
C
C
B
C
B
6.
A
7.
8.
B
A
5. Standardized Test Practice How would you classify a triangle that has
one right angle and two congruent sides?
A right isosceles
B acute scalene
C obtuse isosceles
D right equilateral
5. A
A
5.
©
Glencoe/McGraw-Hill
4. acute, equilateral
4.
Answers: 1. right, isosceles 2. obtuse, scalene 3. acute, equilateral
3.
47
NAME ________________________________________ DATE ______________ PERIOD _____
Special Right Triangles (pages 267–270)
Certain right triangles are called special because they have important
relationships for their sides and angles.
Finding Measures
in Special Right
Triangles
• In a 30°60° right triangle, the length of the hypotenuse is twice the
length of the side opposite the 30° angle (the shortest side).
• In a 45°45° right triangle, the lengths of the legs are equal.
The length of the hypotenuse of a 30°60° right triangle is 15 inches. Find
the lengths of the legs.
The length of the shorter leg (the one opposite the 30° angle) is always half the hypotenuse,
so the shorter leg is 7.5 inches long. Use the Pythagorean Theorem to find the length of the
other leg.
a 2 b2 c2
Pythagorean Theorem
2
2
2
(7.5) b 15
56.25 b2 225
b2 168.75
b 168.7
5
b 13.0
Try These Together
1. Find the missing lengths. Round
to the nearest tenth if necessary.
12 ft
45
2. Find the missing lengths. Round to the
nearest tenth if necessary.
60
c ft
6m
a
30
45
b ft
b
HINT: The legs have equal lengths.
HINT: Find half of the length of the hypotenuse.
Find each missing length. Round to the nearest tenth if necessary.
b
3.
4.
30
a
c
60
6.5 cm
9 yd
45
45
c
B
C
C
B
C
A
7.
8.
B
A
5. Standardized Test Practice Your car has two 30°–60° right triangular
windows. You need a new piece of glass to replace an old window. What are
the lengths of the other sides of the window if the hypotenuse is 14 inches?
A 5 in. by 10 in.
B 7 in. by 10 in.
C 7 in. by 12.1 in.
D 6.5 in. by 12.1 in.
4. a 9 yd; c 12.7 yd 5. C
B
6.
©
Glencoe/McGraw-Hill
3. b 11.3 cm; c 13 cm
A
5.
48
2. a 3 m; b 5.2 m
4.
Answers: 1. b 12 ft; c 17.0 ft
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Classifying Quadrilaterals (pages 272–275)
A quadrilateral is a polygon with four sides and four angles. The sum of
the measures of the angles of a quadrilateral is 360°.
Types of
Quadrilaterals
• A parallelogram is a quadrilateral with both pairs of opposite sides
parallel and congruent.
• A rectangle is a parallelogram with four right angles.
• A rhombus is a parallelogram with all sides congruent.
• A square is a parallelogram with all sides congruent and four right angles.
• A trapezoid is a quadrilateral with exactly one pair of opposite sides that
are parallel.
parallelgram
rectangle
rhombus
square
trapezoid
Classify each quadrilateral using the name that best describes it.
A Quadrilateral ABCD has only one pair
B Quadrilateral HIJK has all sides
of parallel sides.
congruent, with four right angles.
The only quadrilateral with only one pair of
parallel sides is a trapezoid.
Quadrilateral ABCD is a trapezoid.
A quadrilateral with four sides congruent and
four right angles is a square.
Classify each quadrilateral using the name that best describes it.
1.
2.
3.
4.
5. Architecture An architect is designing a rhombus-shaped
window for a new house. A sketch of the window is shown at
the right. Find the value of x so the architect will know the
measures of all four angles.
B
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
6. Standardized Test Practice What is the best way to classify a
quadrilateral that is also a parallelogram with 4 right angles?
A trapezoid
B rhombus
C square
5. 135 6. D
4.
45
x
©
Glencoe/McGraw-Hill
D rectangle
Answers: 1. quadrilateral 2. rhombus 3. trapezoid 4. rectangle
3.
x
45
49
NAME ________________________________________ DATE ______________ PERIOD _____
Congruent Polygons (pages 279–282)
Triangles that have the same size and shape are called congruent
polygons. When two polygons are congruent, the parts that “match” are
called corresponding parts. Two polygons are congruent when all of their
corresponding parts are congruent.
Words
If two polygons are congruent, their corresponding sides are
congruent and their corresponding angles are congruent.
Model
B
Congruent
Polygons
G
A
Symbols
C
F
H
Congruent angles: A F, B G, C H
Congruent sides: BC GH, AC FH, AB FG
Determine whether the polygons shown are congruent. If so,
name the corresponding parts and write a congruence statement.
1.
Z
2.
Y
J
2 ft
K
A
S
C
M
4 ft
Q
3.
S
T
Q
U
2 ft
3 ft
X
B
P
R
R
L
V
Find the value of x in each pair of congruent polygons.
E G
4.
5.
(5x – 5) m
10 m
D
3x
F
J
45
H
6. Flags International code flags are used at sea to signal distress or give
warnings. The flag that corresponds to the letter O, shown at the right,
warns there is a person overboard. How many congruent triangles are
on the flag?
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
7. Standardized Test Practice Sara’s classroom is a square with walls that
are 24 feet long. What are the dimensions of a room congruent to Sara’s
classroom?
A 12 ft by 24 ft
B 24 ft by 18 ft
C 20 ft by 24 ft
D 24 ft by 24 ft
Answers: 1. yes; A X, B Y, C Z, AB XY, BC YZ, AC XZ; ABC XYZ 2. no 3. yes; Q V,
R U, S T, QR VU, RS UT, QS VT; QRS VUT 4. 15 5. 3 6. 2 7. D
3.
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50
NAME ________________________________________ DATE ______________ PERIOD _____
Symmetry (pages 286–289)
Many geometric and other figures have one or more of the types of
symmetry described below.
Types of
Symmetry
• A figure has line symmetry if it can be folded so that one half of the figure
matches the other half. The line that divides the two halves is the line of
symmetry. Some figures have more than one line of symmetry.
• If you can rotate an object less than 360° and it still looks like the original,
the figure has rotational symmetry. The degree measure of the angle through
which the figure is rotated is called the angle of rotation. Some figures have
just one angle of rotation, while others have several.
Identify the type of symmetry.
A A drawing that looks the same if you
turn the paper so that the bottom is now
at the top.
B The brand for Lee’s family cattle ranch
looks like it could be folded in half
and the two sides would match.
Since the drawing looks the same if you turn it
180°, the drawing has rotational symmetry.
Figures that can be folded in half to make
matching sides have line symmetry.
Determine whether each figure has line symmetry. If so, draw
the lines of symmetry.
1.
2.
3.
4.
5. Which of the figures in Exercises 1–4 have rotational symmetry?
6. Sports Sailing is a popular sport in areas near lakes and oceans. Draw
a line of symmetry on the sail of the boat at the right.
B
3.
C
C
A
B
5.
C
B
8.
B
A
7. Standardized Test Practice Which of the following figures shows
correct lines of symmetry?
A
B
C
4. See Answer Key. 5. the star in Exercise 1
A
7.
©
Glencoe/McGraw-Hill
3. no lines of symmetry
6.
51
D
Answers: 1. See Answer Key. 2. no lines of symmetry
6. See Answer Key. 7. B
4.
NAME ________________________________________ DATE ______________ PERIOD _____
Reflections (pages 290–294)
The mirror image produced by flipping a figure over a line is called a
reflection. This line is called the line of reflection. A reflection is one type
of transformation or mapping of a geometric figure.
Reflection
over the x-axis
To reflect a point over the x-axis, use the same x-coordinate and the
opposite of the y-coordinate of the original point. (x, y) becomes (x, y).
Reflection
over the y-axis
To reflect a point over the y-axis, use the opposite of the x-coordinate of
the original point and the same y-coordinate. (x, y) becomes (x, y).
A When you reflect the point A(2, 1) over
the x-axis, what are the new coordinates?
Use 2 for the x-coordinate and the opposite of
the y-coordinate,1. The reflection is A(2, 1).
B When you reflect the point A(2, 1) over
the y-axis, what are the new coordinates?
Use the opposite of the x-coordinate, so 2
becomes 2. Keep the same y-coordinate.
The reflection is A(2, 1).
Try These Together
Name the line of reflection for each pair of figures.
y
y
1.
2.
3.
O
x
x
O
y
O
x
Graph the figure with the given vertices. Then graph the image
of the figure after a reflection over the given axis, and write the
coordinates of its vertices.
4. triangle JKL with vertices J(2, 4), K(4, 1), and L(0, 1); x-axis
5. square QRST with vertices Q(1, 1), R(1, 4), T(4, 1), and S(4, 4);
y-axis
6. trapezoid ABCD with vertices A(2, 4), B(4, 4), C(6, 2), and
D(1, 2); x-axis
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
7. Standardized Test Practice Akela is making a quilt. Her design uses
diamonds. If her first diamond has vertices D(2, 0), E(4, 2), F(2, 4),
and G(0, 2), and her second diamond is the reflection of the first
across the y-axis, what will be the coordinates of E?
A (4, 2)
B (4, 2)
C (0, 2)
D (0, 0)
Answers: 1. x-axis 2. y-axis 3. y-axis 4–6. See Answer Key. 4. J(2, 4), K(4, 1), L(0, 1) 5. Q(1,1), R(1, 4),
S(4, 4), T(4, 1) 6. A(2, 4), B(4, 4), C(6, 2), D(1, 2) 7. B
3.
©
Glencoe/McGraw-Hill
52
NAME ________________________________________ DATE ______________ PERIOD _____
Translations (pages 296–299)
In a coordinate plane, a sliding motion for a figure is called a translation.
A translation down or to the left is negative. A translation up or to the right
is positive.
Graphing
Translations
To translate a point in the way described by an ordered pair, add the
coordinates of the ordered pair to the coordinates of the point.
(x, y) translated by (a, b) becomes (x a, y b).
A What are the coordinates of (2, 3)
translated by (1, 2)?
B What are the coordinates of (3, 5)
translated by (0, 2)?
Add the coordinates of (1, 2) to the
coordinates of (2, 3). The new point is (1, 1).
Try These Together
1. Find the coordinates of D(0, 0),
E(2, 2), and F(1, 3) after they are
translated by (2, 1). Then graph
triangle DEF and its translation,
triangle DEF.
Add the coordinates of (0, 2) to the
coordinates of (3, 5). The new point is (3, 3).
2. Find the coordinates of the square with
vertices A(1, 2), B(1, 4), C(1, 4),
and D(1, 2) after it is translated by
(3, 2). Then graph the square and its
translation.
HINT: Add 3 to the first coordinate and
2 to the second.
HINT: Add 2 to each x-coordinate and add
1 to each y-coordinate.
Graph the figure with the given vertices. Then graph the image
of the figure after the indicated translation, and write the
3. parallelogram BCDE with vertices B(3, 3), C(3, 3), D(1, 1), and
E(5, 1) translated by (4, 3)
4. quadrilateral HIJK with vertices H(1, 0), I(3, 2), J(1, 5), and
K(1, 2) translated by (3, 0)
5. The vertices of triangle KLM are K(1, 2), L(1, 5), and M(5, 0). L has
the coordinates (3, 8)
a. Describe the translation using an ordered pair.
b. Find the coordinates of K and M.
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
6. Standardized Test Practice Manuela is planting a garden with one rectangle
of flowers beside another. If the first has vertices A(2, 3), B(3, 3), C(3, 1),
and D(2, 1), and the second has vertices E(3, 3), F(8, 3), G(8, 5), and
H(3, 5), what is the translation from ABCD to EFGH?
A (10, 6)
B (1, 1)
C (1, 0)
D (5, 6)
Answers: 1–4. See Answer Key for graphs. 1. D(2, 1), E(0, 1), F(3, 2) 2. A(4, 0), B(4, 2), C(2, 2), D(2, 0)
3. B(1, 6), C(7, 6), D(5, 4), E(1, 4) 4. H(2, 0), I(0, 2), J(2, 5), K(4, 2) 5a. (4, 3) 5b. K(3, 1), M(1, 3) 6. D
3.
©
Glencoe/McGraw-Hill
53
NAME ________________________________________ DATE ______________ PERIOD _____
Rotations (pages 300–303)
A rotation moves a figure around a fixed point called the center of
rotation.
• Corresponding points on the original figure and its rotated image are
the same distance from the center of rotation, and the angles formed
by connecting the center of rotation to corresponding points are
congruent.
Properties
of
Rotations
• The image is congruent to the original figure, and the orientation of the
image is the same as that of the original figure.
Graph point A(3, 2). Then graph the point after a
rotation 180° about the origin, and write the
y
A
Step 1 Lightly draw a line connecting point A to the origin.
Step 2. Lightly draw OA
so that mAOA 180° and
O
OA
A
has the same measure as .
Point A has coordinates (3, 2).
x
O
A'
Remember
that an angle
measuring
180° is a
straight line.
Determine whether each pair of figures represents a rotation.
Write yes or no.
y
y
y
1.
2.
3.
O
x
x
O
O
x
4. Graph triangle ABC with vertices A(3,2), B(5,6), and C(1,5).
a. Rotate the triangle 90° counterclockwise about the origin and graph
triangle ABC.
b. Rotate the original triangle 180° about the origin and graph triangle
ABC.
B
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
5. Standardized Test Practice After a figure is rotated 90° counterclockwise
about the origin, one of its vertices is at (2, 3). What were the
coordinates of this vertex before the rotation?
A (3, 2)
B (3, 2)
C (2, 3)
D (3, 2)
2. no 3. no 4. See Answer Key. 5. A
4.
©
Glencoe/McGraw-Hill
54
Answers: 1. yes
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Chapter 6 Review
Find the value of x in each figure. Write each answer in the appropriate
square.
A
L
D
x
R
148°
x
x
32°
111°
99°
103°
x
T
C
18°
U
S
58°
3x
82°
x
x
J
x 4
71°
62°
I
30°
B
E
102°
x
88°
54°
x
8
15
12
91°
x
x
8
P
Y
8
N
K
B
A
C10
92
51°
8
x
18
H
13
G 67°
12
B
43°
D
8
x
16
C
92°
10 A
R
53°
x
R
H
D
x
E
F
ABCD EFGH
A
S
53°
x
ABC RST
T
Now, write the letter from the box that corresponds to each value in the blanks below.
18
78
111
18
78
51
32
9
60
31
99
77 111
99
9
51
78
Answer is located on page 109.
©
Glencoe/McGraw-Hill
55
NAME ________________________________________ DATE ______________ PERIOD _____
Area of Parallelograms, Triangles, and
Trapezoids (pages 314–318)
Any side of a parallelogram or triangle can be used as a base. The altitude of
a parallelogram is a line segment perpendicular to the base with endpoints on
the base and the side opposite the base. The altitude of a triangle is a line
segment perpendicular to the base from the opposite vertex. The length of the
altitude is called the height. A trapezoid is a quadrilateral with exactly one
pair of parallel sides, which are its bases.
b
Area of a
Parallelogram
The area A of a parallelogram is the product of any
base b and its height h. A bh
Area of
a Triangle
The area A of a triangle is equal to half the product of its
h
h
1
bh
base b and height h. A 2
b
The area A of a trapezoid is equal to half the product of the
Area of a
Trapezoid
1
h(b1 b2)
height h and the sum of the bases, b1 and b2. A 2
b1
h
b2
A Find the area of a parallelogram that has B Find the area of a trapezoid with bases of
b 14 in. and h 5 in.
13 cm and 17 cm and a height of 9 cm.
A bh
A
A (14)(5)
Replace b with 14 and h with 5.
A
A 70 in2
Multiply.
A 135 cm2
Try These Together
1. Find the area of a triangle that has
b 16 yd and h 12 yd.
Find the area of each triangle.
3.
Replace the variables.
Multiply.
2. Find the area of a parallelogram that has a base
of 10.5 m and a height of 4.1 m.
Find the area of each trapezoid.
base
height
16 cm
7 cm
6.
14 in.
18 in.
1
6 ft
7.
20 2 m
1
7
m
2
12 m
22 cm
8.
8.6 yd
5.2 yd
7 yd
4. 15 3 ft
5. 20 cm
9.
1
h(b1 b2)
2
1
(9)(13 17)
2
base (b1) base (b2) height
1
6 in.
Standardized Test Practice What is the area of a parallelogram whose
base is 4.5 m and whose height is 3.6 m?
A 5.3 m2
B 8.1 m2
C 10.6 m2
D 16.2 m2
Answers: 1. 96 yd2 2. 43.05 m2 3. 56 cm2 4. 46 ft2 5. 220 cm2 6. 96 in2 7. 168 m2 8. 48.3 yd2 9. D
©
Glencoe/McGraw-Hill
56
NAME ________________________________________ DATE ______________ PERIOD _____
Circumference and Area of Circles (pages 319–323)
The distance from the center to any point on a circle is the radius (r). The
distance across the circle through the center is the diameter (d). The
distance around the circle is the circumference (C). The diameter is twice
the radius, or d 2r.
The circumference C of a circle is equal to its diameter d times , or 2 times
its radius r times . C d or C 2r
Circumference
of a Circle
d
r
22
Use or 3.14 as an approximate value for .
7
C
Area of a Circle The area A of a circle is equal to times the square of the radius r, or
A r2.
A Find C if the diameter is 4.2 meters.
C d
C 3.14(4.2)
C 13.188 m
B Find the area of the circle. Round to the
nearest tenth.
Replace d with 4.2 and with
3.14.
A r2
A 32
A9
A 28.3 yd2
Multiply.
Try These Together
1. Find the area of the circle.
Use a calculator. Round to
the nearest tenth.
r3
3 yd
Use a calculator.
2. Find C if the radius is 23 centimeters.
26 ft
HINT: Use the formula that contains r.
HINT: r 13
Find the circumference of each circle. Round to the nearest tenth.
22
Use 7 or 3.14 for .
1
3. radius, 19.65 cm
4. diameter, 60.2 m 5. diameter, 11.3 yd
6. radius, 8 2 in.
Find the area of each circle. Use a calculator. Round to the nearest tenth.
7. radius, 16 m
8. diameter, 16 in.
9. radius, 10 ft
10. Standardized Test Practice A pizza has a diameter of 18 inches. If two
of the twelve equal pieces are missing, what is the approximate area of
the remaining pizza?
A 254 in2
B 848 in2
C 212 in2
D 424 in2
4. 189.0 m
5. 35.5 yd 6. 53.4 in.
57
3. 123.4 cm
7. 804.2 m2 8. 201.1 in2
9. 314.2 ft2
Glencoe/McGraw-Hill
Answers: 1. 530.9 ft2 2. 144.4 cm
10. C
©
NAME ________________________________________ DATE ______________ PERIOD _____
Area of Complex Figures (pages 326–329)
A complex figure is made up of two or more shapes. To find the area of a complex figure,
separate the figure into shapes whose areas you know how to find. Then find the sum of
those areas.
Find the area of the complex figure.
The figure can be separated into a trapezoid and a semicircle.
Area of trapezoid
A
1
h(a
2
Area of semicircle
1
2
3 cm
A r2
+ b)
1
2
2 cm
1
2
A 2(3 + 5)
A 12
A8
A 1.6
5 cm
The area of the figure is about 8 1.6 or 9.6 square centimeters.
Find the area of each figure. Round to the nearest tenth if necessary.
1.
2.
3.
10 yd
18 cm
10 cm
4 in.
7 yd
2 yd
6 cm
3 yd
6 in.
5 cm
4.
5.
7 in.
15 yd
6.
3 ft
4 cm
2 in.
4 cm
3 ft
2 in.
7 cm
8 in.
4 cm
6 ft
2 in.
8 cm
12 ft
15 ft
11 in.
7. What is the area of a figure that is formed with a rectangle with sides 4 inches and
7 inches and a trapezoid with bases 8 inches and 12 inches, and a height of 3 inches?
B
3.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
8.
Standardized Test Practice What is the area of the
figure at the right?
A 80 in2
C 74 in2
8 in.
1 in.
1 in.
1 in. 1 in.
B 79 in2
D 32 in2
8 in.
4 in.
4 in.
Answers: 1. 30.3 in2 2. 89.1 cm2 3. 87 yd2 4. 48 in2 5. 90 ft2 6. 52 cm2 7. 58 in2 8. B
4.
©
Glencoe/McGraw-Hill
58
NAME ________________________________________ DATE ______________ PERIOD _____
Three-Dimensional Figures (pages 331–334)
Three-dimensional figures are called solids. Prisms are solids that have
flat surfaces. The surfaces of a prism are called faces. All prisms have at
least one pair of faces that are parallel and congruent. These are called
bases, and are used to name the prism.
triangular prism
rectangular prism
triangular pyramid
rectangular pyramid
Use isometric dot paper to draw a three-dimensional figure
that is 3 units high, 1 unit long, and 2 units wide.
1. Lightly draw the bottom of the prism 1 unit by 2 units.
2. Lightly draw the vertical segments at the vertices of the base.
Each segment is three units high.
3. Complete the top of the prism.
4. Go over your light lines. Use dashed lines for the edges of the
prism that you cannot see from your perspective, and solid lines
for edges you can see.
Identify each solid. Name the number and shapes of the faces.
Then name the number of edges and vertices.
1.
2.
3.
4. a. Name the solid at the right.
b. What is the height of the solid?
c. How many faces does the solid have?
d. How many edges does the solid have?
e. How many vertices does the solid have?
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
5. Standardized Test Practice How many vertices does a triangular prism have?
A 3
B 5
C 6
D 8
Answers: 1. rectangular prism; 6 faces, all rectangles; 12 edges; 8 vertices 2. rectangular pyramid; 5 faces, 4 triangles and
1 rectangle; 8 edges; 5 vertices 3. pentagonal prism; 7 faces, 2 pentagons and 5 rectangles; 15 edges; 10 vertices
4a. rectangular prism 4b. 4 4c. 6 4d. 12 4e. 8 5. C
B
3.
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59
NAME ________________________________________ DATE ______________ PERIOD _____
Volume of Prisms and Cylinders (pages 335–339)
Volume is the measure of the space occupied by a solid. It is measured in
cubic units. You can use the following formulas to find the volume of
prisms and circular cylinders. A circular cylinder has circles for its bases.
Volume of
a Prism
The volume V of a prism is equal to the area of the base B times the height h,
or V Bh.
For a rectangular prism, the area of the base B equals the length times the
width w. The formula V Bh becomes V ( w)h.
Volume of
a Cylinder
The volume V of a cylinder is the area of the base B times the height h, or
V Bh. Since the area of the base of a cylinder is the area of a circle, or r 2, the
formula for the volume of a cylinder V becomes V r 2h.
A Find the volume of a rectangular prism
with a length of 4 centimeters, a width of
6 centimeters, and a height of 8 centimeters.
V wh
V468
V 192 cm3
B Find the volume of a cylinder with a
radius of 3 inches and a height of
12 inches
V r 2 h
V 3 2 12
V 339 in 3
4, w 6, h 8
r 3, h 12
Use a calculator.
Find the volume of each solid. Round to the nearest tenth if
necessary.
1.
2. 8 in.
3.
9 mm
16 in.
3 cm
6 cm
9 mm
4 cm
4.
5.
3 in.
9 mm
6.
5 ft
5 ft
B
4.
C
B
C
B
A
7.
8.
15 cm
1.2 cm
C
A
5.
6.
5 in.
12 in.
B
A
7. Standardized Test Practice You just bought a new pot for a plant. The
pot is shaped like a cylinder with a diameter of 12 inches and a height of
12 inches. About how much dirt will you need to fill the pot?
A 144 in3
B 24 in3
C 5,428.7 in3
D 1,357.2 in3
Answers: 1. 72 cm3 2. 804.2 in3 3. 729 mm3 4. 81.3 ft3 5. 180 in3 6. 17.0 cm3 7. D
3.
61–2 ft
©
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60
NAME ________________________________________ DATE ______________ PERIOD _____
Volume of Pyramids and Cones (pages 342–345)
A cone for ice cream is an example of a geometric solid called a circular
cone. A segment that goes from the vertex of the cone to its base and is
perpendicular to the base is called the altitude. The height of a cone is
measured along its altitude.
The volume V of a cone equals one-third the area of the base B times the
Volume of
a Cone
Volume of
a Pyramid
1
height h, or V Bh. Since the base of a cone is a circle, the formula can be
3
1
rewritten as V r 2h.
3
The volume V of a pyramid equals one-third the area of the base B times the
1
height h, or V Bh.
3
A Find the volume of a cone that has a
radius of 1 centimeter and a height
of 6 centimeters.
V
1
r 2h
3
V
1
3
12 6
V 6.3 cm3
B Find the volume of a pyramid that has
an altitude of 10 inches and a square base
with sides of 9 inches.
V
1
Bh
3
r 1, h 6
V
1
3
Use a calculator.
V 270 in3
92 10
The area of a square base is s2.
Find the volume of each solid. Round to the nearest tenth if
necessary.
1.
2.
3.
10 in.
16 ft
8 cm
20 ft
6 cm
14 in.
6 cm
4. The height of a rectangular pyramid is 10 meters. The base is 6 meters by
8.5 meters.
a. Find the volume of the pyramid.
b. Suppose the height is cut in half and the base remains the same. What
is the volume of the new pyramid?
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
5. Standardized Test Practice You are creating a model of the Egyptian
Pyramids for Social Studies class. You make a pyramid with a height of
5 feet and a 2.5-foot by 2.5-foot square base. What is the volume of your
pyramid?
A 4.1 ft3
B 5.2 ft3
C 6.3 ft3
D 10.4 ft3
Answers: 1. 513.1 in3 2. 96 cm3 3. 1,340.4 ft3 4a. 170 m3 4b. 85 m3 5. D
3.
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61
NAME ________________________________________ DATE ______________ PERIOD _____
Surface Area of Prisms and Cylinders
(pages 347–351)
Surface area is the sum of the areas of all faces or surfaces of a solid.
The surface area S of a rectangular prism with length ,
width w, and height h is the sum of the areas of the faces.
Surface Area
of a Prism
h
S 2w 2h 2wh
w
The surface area of a cylinder equals two times the area of the circular bases (2r2)
plus the area of the curved surface (2rh).
S 2r2 2rh
Surface Area
of a Cylinder
A Find the surface area of a cube that has
a side length of 8 centimeters.
B Find the surface area of a cylinder with
a radius of 2 centimeters and a height of
20 centimeters. Round to the nearest
tenth.
A cube has six sides, or faces, that are squares.
The area of one side is 82, or 64 cm2.
Since there are 6 sides, multiply the area of one
side by 6. So, 64 6 384.
The surface area of a cube with a side length of
8 cm is 384 square centimeters.
S 2r 2 2rh
S 2(22) 2(2)(20)
S 276.5 cm2
Try These Together
1. Find the surface area, to the nearest tenth
2.
of a cylinder with a radius of 3 inches
and a height of 5 inches.
6 in.
10 in.
12 in.
r 2, h 20
Use a calculator.
HINT: Find the
surface area of
each face, then
add.
Find the surface area of each cylinder. Round to the nearest
tenth if necessary.
3.
6 cm
4.
5.
1 in.
9 cm
24 in.
24 cm
6 cm
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
6. Standardized Test Practice Selma is wrapping a gift for her friend’s
birthday. She uses a rectangular box that is 20 inches long, 3 inches high,
and 9 inches deep. Find the surface area of the box so she can buy
enough wrapping paper.
A 267 in2
B 534 in2
C 540 in2
D 32 in2
Answers: 1. 150.8 in2 2. 504 in2 3. 452.4 cm2 4. 77.0 in2 5. 1,866.1 cm2 6. B
3.
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62
NAME ________________________________________ DATE ______________ PERIOD _____
Surface Area of Pyramids and Cones
(pages 352–355)
Surface Area
of a Pyramid
Surface Area
of a Cone
The triangular sides of a pyramid are called
lateral faces. The altitude or height of each
lateral face is called the slant height. The
sum of the areas of the lateral faces is the
lateral area. The surface area of a pyramid
is the lateral area plus the area of the base.
Model of Square Pyramid
The surface area of a cone with radius r and
slant height is given by S r r 2.
Model of Cone
lateral face
slant height
base
slant height ()
radius (r)
Find the surface area of the pyramid.
Area of each lateral face
1
1
A 2bh 2(3)(14) 21
3 cm
14 cm
There are 3 faces, so the lateral area is 3(21) or 63 square
3 cm
3 cm
centimeters. The area of the base is given as 3.9 square
centimeters. The surface area of the pyramid is the sum of
the lateral area and the area of the base, 63 + 3.9 or 66.9 square centimeters.
3.9 cm2
Find the surface area of each solid. Round to the nearest tenth if necessary.
1.
2.
3.
2.3 mm
6.7 in.
5.5 cm
7 mm
6.9 cm
8 cm
4.
7 mm
8 cm
1
10 2 mm
2 in.
5.
6.
9 ft
6.1 ft
1
6 4 mm
16 in.
7 ft
7 ft
11 in.
11 in.
7. cone: radius 6.4 in.; slant height, 12 in.
8. triangular pyramid: base area, 10.8 m2; base length, 5 m; slant height, 2.5 m
1
9. square pyramid: base side length, 2 ft; slant height 4 ft
3
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
10. Standardized Test Practice Find the surface area of the complex
solid at the right.
A 285.6 in2
B 187.2 in2
C 250.0 in2
D 249.6 in2
10.4 in.
6 in.
10.4 in.
6 in.
Answers: 1. 93.6 cm2 2. 81.2 mm2 3. 54.7 in2 4. 552.5 mm2 5. 115.9 ft2 6. 473 in2 7. 370.0 in2 8. 29.6 m2 9. 24.1 ft2 10. D
3.
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NAME ________________________________________ DATE ______________ PERIOD _____
Precision and Significant Digits (pages 358–362)
The precision of a measurement is the exactness to which a measurement is made.
Precision depends upon the smallest unit of measure being used, or the precision unit.
The digits you record are significant digits. These digits indicate the precision of the
measurement. When adding, subtracting, multiplying, or dividing measurements, the result
should have the same precision as the least precise measurement.
There are special rules for determining significant digits in a given measurement.
Numbers are analyzed for significant digits by counting digits from left to right, starting
with the first nonzero digit.
Number
A
Number of
Significant Digits
Rule
2.45
3
All nonzero digits are significant.
140.06
5
Zeros between two significant digits are significant.
0.013
2
Zeros used to show place value of the decimal are not
significant.
120.0
4
In a number with a decimal point, all zeros to the right
of a nonzero digit are significant.
350
2
In a number without a decimal point, any zeros to the right
of the last nonzero digit are not significant.
Determine the number of significant
digits in 12.08 cm.
B
The zero is between two significant digits, and
nonzero digits are significant, so there are
4 significant digits in 12.08 cm.
Find 36.5 g 12.24 g using the correct
number of significant digits.
36.5 has the least number of significant digits,
3. Round the quotient so that it has
3 significant digits. The result is 2.98 g.
Determine the number of significant digits in each measure.
1. 20.50
2. 16.8
3. 0.073
Find each sum or difference using the correct precision.
4. 48.25 ft 14.5 ft
5. 3.8 cm 24.05 cm
6. 6.7 yd 0.95 yd
Find each product or quotient using the correct number of significant digits.
7. 3.24 lb 0.75 lb
8. 1.6 mi 2.08 mi
9. 12.40 m 5.36 m
B
3.
C
C
A
B
5.
C
B
10. Standardized Test Practice Television ratings are based on the number
of viewers. A game show had 31.6 million viewers in one evening. How
many significant digits are used in this number?
A 8
B 5
C 3
D 2
10. C
B
A
©
Glencoe/McGraw-Hill
9. 66.5 m
8.
6. 5.8 yd 7. 4.3 lb 8. 3.3 mi
A
7.
64
5. 28 cm
6.
Answers: 1. 4 2. 3 3. 2 4. 33.8 ft
4.
NAME ________________________________________ DATE ______________ PERIOD _____
Chapter 7 Review
Cut-It-Out!
To do this activity, you will need a ruler, scissors, pencil, clear tape, and a
piece of poster board or other thick paper or thin cardboard. Complete the
activity with a parent.
To find the surface area of a prism, you must find the area of each face of
the prism, and then add the areas. Use the materials above to draw and cut
out each face of the prisms below. Once you have cut out the faces, label
them with their individual surface areas. Tape the pieces together to form
the prism. Then add the surface areas from the labels to find the total
surface area of the prism.
1.
2.
5 in.
2 in.
6 in.
4 in.
4 in.
8 in.
6 in.
3.
1 in.
5 in.
12 in.
4. Which prism requires the most paper or cardboard to make?
©
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65
NAME ________________________________________ DATE ______________ PERIOD _____
Probability of Simple Events (pages 374–377)
A list of all possible results, or outcomes, is called a sample space.
Probability is the chance that a specific outcome, or event, will happen.
Probability
number of favorable outcomes
P(event) number of possible outcomes
• When it is impossible for an event to happen, its probability is 0.
• When it is certain that an event will happen, its probability is 1.
A bag contains 4 red and 3 blue marbles. One marble is drawn at random.
What is P(blue)?
P(blue) is the probability of drawing a blue marble.
There are 3 ways that a blue marble can be drawn.
There are 4 3, or 7, possible outcomes.
P(blue) 3
7
or, as a decimal, 0.4
2
8
5
7
1
Try These Together
1. What is the probability that a number cube
is rolled and the outcome is a 3 or a 4?
2. What is the probability that a stone is
randomly tossed onto the first square
of an 8-square hopscotch board?
HINT: Find the number of outcomes that are
3 or 4 and divide this by the total number of
possible outcomes.
HINT: There are 8 possible outcomes.
State the probability of each outcome as a fraction and as a decimal.
3. A person wearing red is randomly picked from a group of 5 people
wearing red and 4 people wearing blue.
4. A green tennis ball is picked from a bag of 4 green, 7 yellow, and
5 white tennis balls.
5. A month picked at random starts with A.
6. A positive one-digit number picked at random is even.
These numbers have been written separately on cards and mixed in
a hat: 1, 2, 2, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 10. A person draws one number
at random without looking. Find the probability of each outcome.
7. P(1)
8. P(3 or 10)
9. P(not 5)
10. P(6)
B
C
C
11. Standardized Test Practice In a deck of 52 playing cards, there are
13 cards in each of the suits: hearts, diamonds, spades, and clubs. What
is the probability that the first card dealt is a spade?
A 0.13
B 0.25
C 0.50
D 0.35
1
8. 7
1
9. 14
11
10. 7
1
11. B
Glencoe/McGraw-Hill
4
©
1
C
B
A
5. ; 0.16
6. ; 0.4
7. 6
9
14
8.
66
1
A
7.
5
B
B
6.
1
A
5.
1
4.
Answers: 1. ; 0.3
2. ; 0.125 3. ; 0.5
4. ; 0.25
3
8
9
4
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Counting Outcomes (pages 380–383)
One way to find the number of possible outcomes is with a tree diagram.
You can also find the total number of outcomes by multiplying with the
Counting Principle.
Counting
Principle
If event M can occur in m ways, and is followed by event N that can occur in
n ways, then the event M followed by the event N can occur in m n ways.
La Donna is going to adopt a puppy from the local animal shelter. The
animal shelter groups their dogs by gender (male or female) and by size
(small, medium, or large). Use a tree diagram and the Counting Principle
to find the number of choices, or possible outcomes, that La Donna has.
Use a tree diagram.
Gender
female
male
Use the Counting Principle.
Size
Outcome
small
medium
large
small
medium
large
small female
medium female
large female
small male
medium male
large male
gender choices size choices outcomes
2
3
6
There are 6 possible outcomes.
Try This Together
1. A restaurant offers three different dinner salads and six types of salad
dressing. How many choices of salad with dressing are there?
HINT: Multiply.
Use a tree diagram or the Counting Principle to find the number
of possible outcomes.
2. Colin has a choice of a black, brown, or blue T-shirt with a choice of
black, blue, or gray pants.
3. Reiko picks millet, oat, thistle, or sunflower seeds for her sparrow, finch,
or dove bird feeders.
4. A restaurant offers eggs cooked three different ways with a choice of
hash browns or fried potatoes.
B
C
C
B
C
B
6.
A
7.
8.
B
A
5. Standardized Test Practice Olga has a choice of five different colored
calligraphy pens, and plain, bond, or parchment paper. How many
possible pen and paper choices does she have?
A 15
B 8
C 10
D 12
4. 6 5. A
A
5.
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67
2. 9 3. 12
4.
Answers: 1. 18
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Permutations (pages 384–387)
An arrangement or listing in which order is important is called a
permutation.
Representing
Permutations
Use P(n, r) to represent a permutation. P(n, r) means the number of
permutations of n things taken r at a time.
P(n, r) n (n 1) (n 2) … (n r 1)
For example, P(8, 3) 8 7 6 or 336.
The notation n! (n factorial) means the product of all counting numbers
beginning with n and counting backward to 1. For example, 4! 4 3 2 1,
or 24. We define 0! as 1.
There are 5 runners in a 400-meter race. The first, second, and third place
runners get ribbons. How many possible ways could the ribbons be
awarded?
You must select 3 runners from the 5.
P(5, 3) 5 4 3
n 5 and r 3, so n r 1 3
60
There are 60 ways the ribbons could be awarded.
Try These Together
Find each value.
1. P(6, 3)
Find each value.
3. P(5, 5)
4. P(8, 4)
8. 5!
9. 2!
2. 6!
5. P(13, 5)
10. 9!
6. 8!
11. P(15, 1)
7. 0!
12. P(10, 5)
13. Pets How many ways can you select 5 dogs from a group of 7 to enter
5 different events at a local dog show?
B
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
14. Standardized Test Practice There are 12 preschoolers waiting to use
4 different pieces of playground equipment. How many ways can the
teacher distribute the equipment to 4 students?
A 11,880
B 479,001
C 24
D 48
12. 30,240
4.
©
Answers: 1. 120 2. 720 3. 120 4. 1,680 5. 154,440 6. 40,320 7. 1 8. 120 9. 2 10. 362,880 11. 15
13. 2,520 14. A
3.
Glencoe/McGraw-Hill
68
NAME ________________________________________ DATE ______________ PERIOD _____
Combinations (pages 388–391)
An arrangement or listing in which order is not important is a combination.
Calculating
Combinations
To find the number of combinations of n items taken r at a time, or C(n, r),
divide the number of permutations P(n, r) by the number of ways r items
can be arranged, which is r!.
P(n, r)
C(n, r) r!
A Find C(3, 2).
C(3, 2) B Find C(5, 3).
P(3, 2)
2!
C(5, 3) P(5, 3)
3!
32
21
543
321
6
2
60
6
or 3
Try These Together
Find each value.
1. C(5, 2)
2. C(12, 4)
or 10
3. C(16, 3)
4. C(8, 5)
HINT: Find the number of permutations first, then divide by r!.
Find each value.
5. C(10, 6)
9. C(6, 3)
6. C(4, 2)
10. C(4, 4)
7. C(7, 4)
11. C(1, 1)
8. C(11, 5)
12. C(100, 1)
Determine whether each situation is a permutation or a
combination.
13. choosing 3 paper clips from a box of 100
14. picking 5 tennis balls from a basket of 10
15. six birds sitting on a telephone wire
16. choosing 4 colored markers from a box of 8 different colors
17. five bicycles parked at a bicycle stand for 10 bikes
18. Purchasing A market carries 15 flavors of gum. Nate buys three
flavors of gum each time he visits the market. How many different
combinations of three flavors of gum could Nate buy?
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
19. Standardized Test Practice Mr. Begay has 8 insects for students to
study. How many different groups of 3 insects can a student study?
A 8
B 70
C 28
D 56
Answers: 1. 10 2. 495 3. 560 4. 56 5. 210 6. 6 7. 35 8. 462 9. 20 10. 1 11. 1 12. 100 13. combination
14. combination 15. permutation 16. combination 17. permutation 18. 455 19. D
3.
©
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69
NAME ________________________________________ DATE ______________ PERIOD _____
Probability of Compound Events (pages 396–399)
When you find probability, you often have to look at two or more events,
known as compound events. In a compound event, if the second event
does not depend on the outcome of the first event, then the events are
independent. If the outcome of one event of a compound event affects the
other event, then the events are dependent.
Probability of Two
Independent Events
The probability of two independent events can be found by multiplying
the probability of the first event by the probability of the second event.
P(A and B) P(A) P(B)
Probability of Two
Dependent Events
If two events, A and B, are dependent, then the probability of both
events occurring is the product of the probability of A and the
probability of B after A occurs.
P(A and B) P(A) P(B following A)
A What is the probability of
tossing heads on a coin twice
in a row?
The first coin toss does not affect the
second coin toss, so these are
independent events.
P(heads and heads) P(heads) P(heads)
P(heads and heads) 1
2
P(heads and heads) 1
4
1
2
The probability of tossing heads twice
in a row is
1
.
4
B A bag contains three pink and two purple
marbles. What is the probability of drawing two
purple marbles in a row from the bag if the first
marble is not replaced?
Drawing the first marble changes the number of marbles
in the bag, which changes the probability of the
second event. These are dependent events.
P(purple and purple) P(purple) P(purple after purple)
P(purple and purple) 2
5
P(purple and purple) 2
20
1
4
or
1
10
The probability of drawing two purple marbles in a row
from the bag is
1
.
10
Twenty game cards are used. Five are red, five are blue, four are
green, and six are yellow. Once a card is drawn, it is not
replaced. Find the probability of each outcome.
1. two blue cards in a row
2. a green card and then a yellow card
B
C
8.
C
B
A
3. Standardized Test Practice Sarita has four $1 bills and three $10 bills in
her wallet. What is the probability that she will reach into her wallet
twice, and pull out a $10 bill each time? Assume she does not replace the
first bill.
1
A 7
2
B 7
6
C 49
12
D 49
3. A
A
7.
©
Glencoe/McGraw-Hill
70
6
B
B
6.
2. 95
C
A
5.
1
4.
Answers: 1. 19
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Experimental Probability (pages 400–403)
You know that because a number cube has six possible outcomes, the
1
probability of tossing a one is 6. This kind of probability is called
theoretical probability. But if you toss a cube a number of times, the
1
fraction of times you get a one may not be exactly . This is known as
6
experimental probability.
Clarise conducted an experiment to find out her probability of making a
free throw during a basketball game. She hit 40 of her 100 free throws.
What is her experimental probability of making a free throw?
experimental probability number of free throws made
number of free throws attempted
So, her experimental probability of making a free throw is
40
100
or
2
.
5
1. If you toss a baseball card, what is the theoretical probability that it will
land with the picture face-up?
2. You have tossed the card 40 times and it lands with the picture face-up 24
times. What is the experimental probability of the card landing face-up?
3. Svetlana and Lenora are playing a game with two
Results of Rolling Two Number Cubes
16
number cubes. Based on the results from the rolls
12
indicated on the graph, what number is Svetlana
Number
of Rolls 8
most likely to roll next?
4
0
2 3 4 5 6 7 8 9 10 11 12
4. Genetics Gregor grows pea plants as a hobby. Some of his pea plants
always produce white flowers. Others always produce red flowers. As an
experiment, Gregor pollinated a white flower with pollen from a red
flower. The cross-pollinated white flower produced 8 seeds.
a. If genetic traits such as flower color are equally likely to occur, how many
of those 8 seeds would you expect to grow into plants with red flowers?
b. If three of the 8 seeds grow into plants with red flowers, what is the
experimental probability of a seed growing into a plant with red flowers?
B
C
C
B
C
8.
B
A
5. Standardized Test Practice Celia has a bag of 10 marbles. Some are blue,
some are yellow. She drew a marble from the bag 100 times, replacing the
marble after each draw. If she drew a blue marble 78 of the 100 times, how
many blue marbles are most likely in the bag?
A 3
B 8
C 7
D 9
5. B
A
7.
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Glencoe/McGraw-Hill
71
3
3. 8 4a. 4 4b. 8
B
6.
3
A
5.
2. 5
4.
1
Answers: 1. 2
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Using Sampling to Predict (pages 406–409)
If you want to make a prediction about a large group of people, you may wish
to use a smaller group, or sample, from the larger group. The large group
from which you gathered your sample is known as the population. To make
sure your information represents the population, the sample must be drawn at
random. A random sample gives everyone the same chance of being selected.
The school math club asked several students at random what they
like to eat during their afternoon snack break. Three students said
they like to eat muffins, five said fruit, and one said bagels.
A What is the size of the sample?
B What percent preferred muffins?
Add the number of people who were asked.
3519
3 out of 9 said that they like to eat muffins.
3
9
C Based on their survey, about how many
of the 1,200 students in the school would
prefer muffins for their afternoon snack?
1
3
1,200 400
So about 400 students would prefer muffins.
1
3
1
or 33 %
3
D Were the students the math club
surveyed an appropriate sample?
The students surveyed by the math club
probably were not an appropriate sample
because there were so few students surveyed
compared to the total number of students in
the school.
1. Brushy Creek Middle School is a new school with 800 students. The
principal asked some students their preference for the new school
mascot. The results were that 22 preferred an eagle, 36 preferred a tiger,
and 42 preferred an armadillo.
a. What is the sample size?
b. What percent wanted the armadillo to be the school mascot?
2. Biology Every month for three years, a biologist has caught 30 fish
from a lake and checked their blood for lead contamination. In the three
years, she has found 270 fish with lead in their blood. If she decides to
check 40 fish next month instead of 30, how many do you predict will
have lead in their blood?
B
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
3. Standardized Test Practice A film company wants to see test-audience
reactions to a new cartoon adventure film before they start advertising.
Which of the following test audiences would make the best sample of the
film’s intended audience?
A college students
B high school students
C senior citizens
D elementary school students
3. D
4.
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72
Answers: 1a. 100 1b. 42% 2. 10
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Chapter 8 Review
Family Photo Opportunity
Eight members of Joaquin’s family (including Joaquin and Irene) are eating
a holiday dinner together. Joaquin has a new camera and wants to take their
pictures in groups of some number (as large as possible) to make an album
for those who could not come. Irene is worried that there will not be
enough film for all those pictures. Help them figure out this problem.
1. How many different groups of two people can they form from those at
the dinner? (Hint: A picture of Uncle Steve with Bill is the same as a
picture of Bill with Uncle Steve.)
2. How many groups of three members each can be formed from the
8 people?
3. How many groups of four members each can be formed from 8 people?
4. How many groups of five members each can be formed from 8 people?
5. How many groups of six members each can be formed from 8 people?
6. Irene and Joaquin have 2 rolls of film with 36 exposures each. What is
the largest size group they can use in their pictures?
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73
NAME ________________________________________ DATE ______________ PERIOD _____
Histograms (pages 420–424)
Statistics involves collecting, organizing, and analyzing data. You can
display data with a type of bar graph called a histogram. A histogram
uses bars to display numerical data organized into equal intervals.
90–
99
60–
69
70–
79
80–
89
The bar for a grade of 90–99 ends halfway between 6 and 8 on the
vertical axis. So 7 students made an A. Categories that have a
frequency of 0 have no bar. Since the category for 50–59 has no bar,
there were 0 students who made a grade of 50–59 percent.
Grades on Science Test
12
10
8
Number of
6
Students
4
2
0
50–
59
In the histogram shown at the right, how many students
made a grade of 90–99 percent on the science test? How many
made a grade of 50–59 percent?
Grade (Percent)
Try These Together
1. Refer to the histogram in Exercise 3.
2. Refer to the histogram in Exercise 3.
How many presidents were between the
How large is each interval?
ages of 40 and 44 when inaugurated?
HINT: Count the number of ages each interval
HINT: What is the height of the bar for the
interval 40–44?
contains.
3. Use the histogram at the right to answer each question.
a. Which interval has the least number of presidents?
40
–
45 44
–
50 49
–
55 54
–
60 59
–
65 64
–6
9
b. Construct a frequency table from the data.
Ages of Presidents
12
Number of 8
Presidents 4
0
Age at Inauguration
4. Genealogy Annabeth surveyed the students in
her grade to find out how many CDs they each
had. The results of her survey are shown in the
table. Make a histogram of the data.
B
C
B
C
B
6.
A
7.
8.
21–10
16
11–20
20
21–30
28
31–40
28
C
A
5.
B
A
5. Standardized Test Practice Refer to the histogram of test grades
in the example above. Which grade interval was earned by the
greatest number of students?
A 90–99 percent
B 80–89 percent
C 70–79 percent
3b. See Answer Key. 4. See Answer Key. 5. B
4.
Students
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74
D 60–69 percent
Answers: 1. 2 2. 5 years 3a. 40–44
3.
CDs per Student
CDs
NAME ________________________________________ DATE ______________ PERIOD _____
Circle Graphs (pages 426–429)
A circle graph compares parts of a set of data to the whole set.
Drawing
a Circle
Graph
• If the data is given in numbers (rather than percents), first find the total number
and find a ratio that compares each category to the total.
• Multiply each ratio or percent by 360 degrees to find the number of degrees for
that section of the graph.
• Use a compass to draw a circle. Draw a radius. Use a protractor to draw any of
the angles. From the new radius, use the protractor to draw the next angle, and
repeat.
• Label each section. Write each ratio as a percent. Title the graph.
How many degrees will you draw in a circle graph to represent 25%?
Write the percent as a decimal: 0.25.
Multiply the decimal by 360 degrees: 0.25 360 90.
90 degrees represents 25% of the circle.
Try This Together
1. Use the table in Exercise 2 to find the number of degrees in the section
of the circle graph that represents dogs in 1-person families.
HINT: Find 13% of 360.
2. Pets The table shows the percent of dogs that lived with
1, 2, 3, and 4-person families in a recent year.
a. Make a circle graph of the data.
b. Which family size owns about one-fifth of the dogs?
Number of
Percent
People in Family of Dogs
1
13%
2
31%
3
21%
4
35%
3. School Suppose that in the United States there are 38,289,000 students in
kindergarten through 8th grade; 16,299,000 students in 9th through 12th
grade; and 16,228,000 students in college. Make a circle graph of this data.
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
4. Standardized Test Practice The circle graph shows the
number of radio and television stations in the U.S. in
1999. About what percent of radio and television stations
were AM radio stations?
A 47%
B 55%
C 39%
D 29%
Number of Radio and
Television Stations in the U.S.
Television
1,599
FM Radio
5,745
AM Radio
4,782
Answers: 1. 47 degrees 2a. See Answer Key. 2b. 3-person families 3. See Answer Key. 4. C
3.
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75
NAME ________________________________________ DATE ______________ PERIOD _____
Choosing an Appropriate Display (pages 430–433)
When you choose what type of graph to use, ask yourself these questions:
What type of information is this? What do I want my graph to show?
Display
Bar Graph
Choosing
a Display
Use
shows the number of items in specific categories in the
data using bars
Circle Graph
compares parts of the data to the whole
Histogram
shows the frequency of data that has been organized
into equal intervals
Line Graph
shows change over a period of time
Line Plot
shows how many times each number occurs in the data
Pictograph
shows the number of items in specific categories using
symbols to represent a quantity
Stem-and-Leaf Plot lists all individual numerical data in a condensed form
Table
may list all the data individually or by groups
Choose an appropriate type of display to compare people’s annual salary to
their number of years of education.
You could plot the salaries and years of education on a line graph.
You could examine the line graph to see if salaries increase with more education.
Try These Together
1. Choose an appropriate type of
display for the populations of five
different cities in 2000.
2. Choose an appropriate type of
display for the votes received by four
candidates in an election.
HINT: There are five categories which are
not numerical.
HINT: Election results are often reported as
percents of the whole.
Choose an appropriate type of display for each situation.
3. the numbers of students in your math class whose heights are 55 to
59 inches, 60 to 64 inches, 65 to 69 inches, and 70 to 74 inches
4. students’ grades on a math test and the numbers of hours they studied
5. numbers of Americans who own 0, 1, 2, 3, or 4 or more cars
B
3.
C
C
A
B
5.
C
B
8.
B
A
6. Standardized Test Practice What type of a display would you use to
show the number of states with different numbers of national parks?
A histogram
B line plot
C circle graph
D line graph
5. circle graph or bar graph
A
7.
©
Glencoe/McGraw-Hill
3. histogram 4. line graph
6.
Answers: 1–7. Sample answers are given. 1. bar graph 2. circle graph
6. A
4.
76
NAME ________________________________________ DATE ______________ PERIOD _____
Measures of Central Tendency (pages 435–438)
Measures of central tendency use one number to describe a set of data.
Measures
of Central
Tendency
• The mean is the sum of the data divided by the number of items in the data set.
• The median is the number in the middle when you order the data from least
to greatest. When there are two middle numbers, the median is the mean of
those two.
• The mode is the number or numbers that occur most often.
Find the mean, median, and mode of the data. 11, 23, 47, 11, 25, 54
171
Find the total. Then divide by 6. 28.5
6
The mean is 28.5.
To find the median, arrange the data in order. 11, 11, 23, 25, 47, 54
There are two middle numbers, 23 and 25. The mean of 23 and 25 is
23 25
2
or 24. The median of the data is 24.
The number that appears most often is 11, which appears twice.
The mode is 11.
Try These Together
1. Find the mean, median, and mode of
the data. 17, 15, 15, 12, 16
2. Find the mean, median, and mode of
the data. 3, 2, 3, 2, 3, 9, 5, 6, 4, 5, 2
HINT: Find the total and divide by 5 to find
the mean. Arrange in order to find the median.
HINT: There are two modes.
Find the mean, median, and mode of each set of data. Round to
the nearest tenth if necessary.
3. 58, 63, 57, 52, 58, 52, 52, 64
4. 110, 150, 142, 120, 113, 110, 123
5. 35, 35, 36, 32, 34, 33, 32, 31
6. 500, 1,000, 700, 1,000, 1,000, 1,200
7. Employment Kezia conducted a study to find out what the average
wage was for high school students who were employed. The data she
gathered is shown below. Find the mean, median, and mode of her data.
Round to the nearest cent.
$5.50 $6.75 $5.25 $5.75 $6.25 $5.75 $6.75 $5.50 $5.25 $5.25
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
8. Standardized Test Practice The high temperatures in New York, NY, for
one week in the summer were 80°F, 78°F, 80°F, 81°F, 85°F, 82°F, and
79°F. What was the median high temperature?
A 79°F
B 80°F
C 81°F
D 85°F
Answers: 1. 15; 15; 15 2. 4; 3; 2 and 3 3. 57; 57.5; 52 4. 124; 120; 110 5. 33.5; 33.5; 32 and 35 6. 900; 1,000; 1,000
7. $5.80; $5.63; $5.25 8. B
3.
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77
NAME ________________________________________ DATE ______________ PERIOD _____
Measures of Variation (pages 442–445)
The spread of data is called the variation. One way to measure it is with
the range, the difference between the greatest and least numbers in the set.
With large sets of data, it is often helpful to separate the data into four
equal parts called quartiles.
Find the range, median, upper and lower quartiles, and interquartile range
for this set of data.
12, 12, 16, 14, 13, 13, 11, 15, 13, 15
Arrange the data in order and divide it into halves.
11, 12, 12, 13, 13,
13, 14, 15, 15, 16
The range is the difference between the greatest and least values.
16 11 5
The range is 5.
There are 2 middle numbers, 13 and 13, so the median is 13.
The median of the upper half of the data is 15, so 15 is the upper quartile.
The median of the lower half of the data is 12, so 12 is the lower quartile.
To find the interquartile range, subtract the lower quartile from the upper
quartile. The difference is 15 12, or 3. The interquartile range is 3.
Try These Together
1. Find the range, median, and upper and
lower quartiles for this set of data.
0, 5, 3, 3, 2, 5, 6, 4, 6, 9, 6
2. Find the interquartile range for the set
of data in Exercise 1.
HINT: Subtract the quartiles.
HINT: First arrange the data in order.
Find the range, median, upper and lower quartiles, and
interquartile range for each set of data.
3. 9, 2, 3, 8, 6, 1, 4, 6
4. 41, 45, 42, 42, 45, 46, 41, 43, 43
5. 75, 85, 75, 75, 85, 95, 96, 130, 78
B
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
7. Standardized Test Practice What is the interquartile range for a set of
data whose upper quartile is 5.5 and whose lower quartile is 1.8?
A 7.3
B 9.9
C 3.7
D 1.9
2. 3 3. 8; 5; 7, 2.5; 4.5 4. 5; 43; 45, 41.5; 3.5 5. 55; 85; 95.5, 75; 20.5 6. 20; 25; 29, 19; 10 7. C
4.
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78
Answers: 1. 9; 5; 6, 3
3.
6. 32, 16, 12, 21, 29, 19, 30, 25, 25, 26
NAME ________________________________________ DATE ______________ PERIOD _____
Box-and-Whisker Plots (pages 446–449)
A box-and-whisker plot uses a number line to show the distribution of a
set of data. The box is drawn around the quartile values, and the whiskers
extend from each quartile to the extreme data points that are not outliers.
1. Draw a number line that includes the least and greatest number in the data.
Drawing a
2. Mark the extremes, the median, and the upper and lower quartile above
Box-and-Whisker
the number line. If the data has an outlier, mark the greatest value that is
Plot
not an outlier.
3. Draw the box and the whiskers.
Draw a box-and-whisker plot for this data: 18, 19, 16, 23, 25, 9, 10, 16
Arrange the data in order from least to greatest
(9, 10, 16, 16, 18, 19, 23, 25). Draw a number line
that includes the least and greatest numbers (9 and 25).
13 17
9
Mark the extremes (9 and 25), the median (17),
the upper quartile (21), and the lower quartile (13)
above the number line.
6
21
25
8 10 12 14 16 18 20 22 24 26
Draw the box and the whiskers.
Draw a box-and-whisker plot for each set of data.
1. 283, 251, 225, 281, 290, 273, 204, 267
2. 102, 105, 80, 15, 90, 95, 106, 87, 80, 80, 105, 87, 85, 86
3. 27, 40, 30, 14, 19, 25, 27, 35, 31, 36, 39, 18, 30, 30, 35, 14
For Exercises 4–7, use the following box-and-whisker plot.
4. Which set of data is more spread out?
31
73 82
51
32
41
62
83
98
97
5. What is the interquartile range of class A’s test scores?
6. Twenty-five percent of the students in class B scored
below what average?
A
B
20 30 40 50 60 70 80 90 100
7. In general, which class scored higher on the test?
B
3.
C
C
A
B
5.
C
B
A
7.
8.
B
A
8. Standardized Test Practice Use the box-and-whisker plot
at the right. Fifty-percent of the data are found between
what two values?
A 25 and 36
B 28 and 36
C 11 and 28
6. 41% 7. A 8. A
6.
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79
11
17 25 28
36
10 15 20 25 30 35 40 45
D 17 and 36
Answers: 1–3. See Answer Key. 4. B 5. 31
4.
NAME ________________________________________ DATE ______________ PERIOD _____
Misleading Graphs and Statistics (pages 450–453)
When dealing with statistics, be careful to identify when statistics are
presented in a misleading way. Recall that there are three different measures
of central tendency or types of averages. They are mean, median, and mode.
These different values can be used to show different points of view.
The prices for sandwiches at a fast-food restaurant are $0.99, $1.29, $3.39,
$0.99, $0.99, $3.19, $2.49, $0.99, $3.19, $1.49, $2.79, $2.49, and $1.49.
A Find the mean, median, and mode of the prices.
$25.77
$1.98
13
Mean
sum of values
number of values
Median
Mode
Ordering the prices from least to greatest, the middle price is $1.49.
$0.99
B Which average would the restaurant use to encourage people who want to save money
to eat at their restaurant? Explain.
People who want to save money would like more inexpensive sandwiches. Therefore, the restaurant
would use the mode since it is the least of the averages.
C Which average or averages would be more representative of the data?
Since the mode is the least value of the thirteen values, it is not as representative as the mean or
median.
For Exercises 1–5, use the list of times, in minutes, it took two different
groups of students to complete a homework assignment.
Group 1: 60, 45, 40, 30, 25, 22, 20, 20, 20, 15
Group 2: 45, 40, 32, 30, 25, 22, 18, 18, 15, 15
1. What is the mean, median, and mode of the times for Group 1?
2. If the two groups are competing to see who finished faster, what
average is most favorable for Group 1?
3. What is the mean, median, and mode of the times for Group 2?
4. Which group was faster in completing the assignment? Explain.
C
5. Standardized Test Practice In the following set of data, which value is
smallest?
Miles traveled per day on a trip: 420, 125, 375, 283, 198, 420, 632, 480
A mean
B median
C mode
D none of these
C
A
B
5.
C
B
6.
A
7.
B
A
8.
3. 13; 23.5; 18 4. Group 2; the mean time was much lower for Group 2. 5. C
©
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80
2. mode
4.
Answers: 1 29.7; 23.5; 20
B
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Matrices (pages 454–457)
One way to organize information is by using a matrix. A matrix is a
rectangular arrangement of numbers in rows and columns. Each number in
a matrix is called an element of the matrix.
Adding and
Subtracting
Matrices
• You can add or subtract matrices that have the same number of rows and
the same number of columns.
• Add or subtract matrices by adding or subtracting the corresponding
elements.
2 0 –1
1 5 1
A Add 1 –2 3 –1 3 4 .

 

2 1
3 1 0
B Add 3 0 2 0 –1 .


5 6
Both matrices have 2 rows and 3 columns, so
you can add them by adding the corresponding
elements.
0 5 1 1  3 5 0
2 1
1 (1) 2 3 3 4  0 1 7
The first matrix has 3 rows and 2 columns, but
the second matrix has 2 rows and 3 columns.
It is impossible to add these matrices.
Try These Together
1. Add. If there is no sum, write
impossible.
5 2 4 1 0 1
1 6 3 8 2 1
2. Subtract. If there is no difference, write
impossible.
5 3 3 1
8 2 7 0
4 6 0 5
HINT: 5 1 6, 2 0 2, and so on.
HINT: Do these matrices have matching
numbers of rows and columns?
Add or subtract. If there is no sum or difference, write impossible.
1 5
2 8
3. –8 –1 –3 2
4. 9 11 [3 7 15]


  2 –4

5. Population The table shows the populations of
Montana and Idaho in 1980, 1990, and 2000. Write
a matrix for the data.
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
8 –3 5
6. Standardized Test Practice Find the sum of 0 –1 2 and


 3 6 –7 .
–8 5 3
11 3 –2
A –8 4 5


5 –9 12
B 8 –6 –1


–11 –3 2
C  18 6 1


–5 9 –12
D –18 4
5

Answers: 1–3. See Answer Key. 4. impossible 5. See Answer Key. 6. A
B
3.
Population (thousands)
Year Montana Idaho
1980
787
944
1990
799
1,007
2000
902
1,294
©
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81
NAME ________________________________________ DATE ______________ PERIOD _____
Chapter 9 Review
Family Album Math
1. Ask your parent or other family member to help you collect data from
your family. Make a table of the names and ages of at least ten people in
your family.
2. Find the mean and mode of the data in your table.
3. Find the range, median, upper and lower quartiles, and the interquartile
range for your data.
4. What do you think is an appropriate display for your data: table,
histogram, bar graph, circle graph, line plot, or line graph? Explain the
reasons for your choice.
5. Display the data for your family using the graphic method you chose in
the previous question.
©
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82
NAME ________________________________________ DATE ______________ PERIOD _____
Simplifying Algebraic Expressions (pages 469–473)
The expressions 3(x 4) and 3x 12 are equivalent expressions,
because no matter what x is, these expressions have the same value.
Simplifying
Algebraic
Expressions
When a plus sign separates an algebraic expression into parts, each part is
called a term. The numerical part of a term that contains a variable is called
the coefficient of the variable. Like terms are terms that contain the same
variables, such as 4x and 5x. A term without a variable is called a constant.
Constant terms are also like terms.
An algebraic expression is in simplest form if it has no like terms and no
parentheses. You can use the Distributive Property to combine like terms.
This is called simplifying the expression.
A Use the Distributive Property to rewrite the expression 8(x 5).
8(x 5) 8(x) 8(5)
8x 40
Simplify.
B Identify the terms, like terms, coefficients, and constants in the expression 5y 4 6y.
terms: 5y, 4, 6y
like terms: 5y and 6y
coefficients: 5, 4 and 6
constants: 4
C Simplify 3t 11 4t.
3t and 4t are like terms.
3t 11 4t 3t 4t 11
[3 (4)]t 11
7t 11
Use the Distributive Property to rewrite each expression.
1. 2( y 11)
2. 3(2b 3)
3. 6(10r 3)
Identify the terms, like terms, coefficients, and constants in each expression.
4. 4 3r r 2
5. 2t 3 11 4t
6. 16y 5 2y y
Simplify each expression.
7. 6x 2x
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
9. 6r – 2r + 1
B
A
10. Standardized Test Practice Which expression represents
the perimeter of the figure at the right?
A 5a 2
B a6
C 9a 6
D 9a 2
3a 1
a 1
5a 4
Answers: 1. 2y 22 2. 6b 9 3. 60r 18 4. terms: 4, 3r, r, 2; like terms: 4, 2 and 3r, r; coefficients: 4, 3, 1, 2;
constant: 4, 2 5. terms: 2t, 3, 11, 4t; like terms: 2t, 4t and 3, 11; coefficients: 2, 3, 11, 4; constant: 3, 11
6. terms: 16y, 5, 2y, y; like terms: 16y, 2y, y; coefficients: 16, 5, 2, 1; constant: 5 7. 4x 8. 16y 7 9. 4r 1 10. D
3.
8. 4y + 7 + 12y
©
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83
NAME ________________________________________ DATE ______________ PERIOD _____
Solving Two-Step Equations (pages 474–477)
In some algebraic equations, two operations, such as addition and
multiplication, are performed on a variable. An example is 2x 1 5.
Such equations are known as two-step equations.
Solve 2x 1 5.
2x 1 1 5 1
2x 4
Solving
Two-Step
Equations
2x
2
4
2
x2
First, use inverse operations to “undo” any addition or
subtraction operations. Then use inverse operations to
“undo” any multiplication or division operations. Notice
that this is in the opposite order from the order of
operations.
Solve 8 3b 26.
8 3b 26
8 8 3b 26 8
3b 18
3b
3
18
3
b 6
Subtract 8 from each side.
The solution is 6. Be sure to check your answer.
Try These Together
1. 2d 10 20
2. 3f 15 12
3. 9 4t 25
HINT: Remember to “undo” operations.
4. 30 5p 25
5. 2x 3 9
7. 17 12r 41
8. 64 4s 16
n
10. 3 8 11
6. 8g 24 8
9. 50 6z 10
m
11. 20
15
12. 5.8 3a 14.8
13. Entertainment At an amusement park, admission for the first 5 people in
Bob’s family cost $20 per person, or $100 total. The remaining people in the
group got in at a lower rate. If Bob’s family is a group of 8, and the total cost
was $145, how much was the admission, per person, for the other three people?
B
C
C
8.
C
B
A
14. Standardized Test Practice Find n if 4n 16 36.
A 14
B 12
C 13
14. C
A
7.
©
12. 3 13. $15 each
B
B
6.
Glencoe/McGraw-Hill
10. 9 11. 30
A
5.
9. 10
4.
84
D 15
Answers: 1. 5 2. 9 3. 4 4. 1 5. 6 6. 4 7. 2 8. 12
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Writing Two-Step Equations (pages 478–481)
Some verbal sentences translate to two-step equations. There are many reallife situations in which you start with a given amount and then increase it at a
certain rate. These situations can be represented by two-step equations.
Translate and solve the equation.
Seven less than twice a number is 15. Find the number.
Words
Variables
Equation
Seven less than twice a number is 15.
Let n = the number.
2n – 7 15
Write the equation.
2n – 7 + 7 15 + 7
Add 7 to each side.
2n 22
Simplify.
2n
2
22
2
n 11
Therefore, the number is 11.
Simplify.
Translate each sentence into an equation. Then find each
number.
1. Eight less than six times a number is equal to 2.
2. The quotient of a number and 4, plus 2, is equal to 10.
3. The difference between four times a number and thirteen is 15.
4. If 11 is increased by three times a number, the result is 2.
5. Six times a number minus three times the number plus 1 is 5.
Solve each problem by writing and solving an equation.
6. Kyle wants to save for a new pair of shoes. The shoes cost $109.99. He already
has $85 in his savings account. How much more does he need to save?
7. Kate has two sisters. Kate is twice as old as one of her sisters and five
years older than her other sister. If the sum of their ages is 35, how old
is each sister?
B
3.
C
C
A
B
5.
C
B
6.
A
7.
8. Standardized Test Practice Brad spent $143.10 dollars at a sporting
goods store. If the sales tax was 6%, which of the following equations
can be used to find the amount (b) before the sales tax?
B
A
8.
A b 0.06b 143.10
B b 6b 143.10
C 143.10 b(0.06) b
D b 0.06 143.10
Answers: 1. 6n 8 2; n 1 2. 4 2 10; n 32 3. 4n 13 15; n 7 4. 11 3n 2; n 3 5. 6n 3n 1 5;
1
n 2 6. let n what Kyle needs to save; 85 n 109.99; n 24.99; Kyle needs to save $24.99 7. let x Kate’s age; 2x (x) (x 5) 35; x=16; Kate is 16 and her sisters are 8 and 11. 8. A
n
4.
©
Glencoe/McGraw-Hill
85
NAME ________________________________________ DATE ______________ PERIOD _____
Solving Equations with Variables
on Each Side (pages 483–487)
Some equations have variables on each side of the equals sign. To solve
these equations, use the Addition or Subtraction Property of Equality to
write an equivalent equation with the variables on one side of the equals
sign. Then solve the equation.
Solve 24 – 2y = 4y. Check your solution.
24 2y 4y
24 2y 2y 4y 2y
24 6y
4y
To check your solution, replace
24 2y 4y
?
24 2(4) 4(4)
16 16
The solution is 4.
Check
Write the equation.
Add 2y to each side.
Simplify.
Mentally divide each side by 6.
y with 4 in the original equation.
Write the equation.
Replace y with 4.
The sentence is true.
11. 6x 4 7x
12.
13. 2p p 21
14.
15. 6 5j 2j 8
16.
17. 16.4 d 3d
18.
19. 5m 26 7m 34
10.
11. 9y 1.2 16.8 21y
12.
3
4
1
4
13. k 6 k 1
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
1
6
1
3
14. 2 m m 7
Define a variable and write an equation to find each number.
Then solve.
15. Three times a number is 21 more than six times a number. What is the
number?
16. Nine less than twice a number equals three times the number plus six.
What is the number?
17. Standardized Test Practice Rental car company A charges $36 a day
plus $0.25 per mile. Rental car company B charges $21 a day plus
$0.35 per mile. Which equation can be used to find the number of miles
for which the companies’ plans cost the same?
A 36 0.25m 21 0.35m
B 36 0.35m 21 0.25m
C 36m 0.25 21m 0.35
D (36 0.25)m (21 0.35)m
2
Answers: 1. x 4 2. k 3 3. p 21 4. r 1 5. j 2 6. s 5 7. d 4.1 8. = 7.5 9. m 3 10. c 0.5
11. y 0.6 12. x 1.2 13. k 14 14. m 10 15. let n number; 3n 21 6n; n 7 16. let n number;
2n 9 3n 6; n 15 17. A
B
3.
13k 12 9k
8 3r 5r
s 2 3s 8
6.1 24 9.3
7 3c 4 3c
1 4x 6x 13
©
Glencoe/McGraw-Hill
86
NAME ________________________________________ DATE ______________ PERIOD _____
Inequalities (pages 492–495)
A mathematical sentence that contains or is called an inequality. When
used to compare a variable and a number, inequalities can describe a range
of values. Some inequalities use the symbols or . The symbol is read is
less than or equal to, while the symbol is read is greater than or equal to.
Common Phrases and Corresponding Inequalities
• is less than
• is fewer than
• is greater than
• is more than
• exceeds
• is less than or
equal to
• is no more than
• is at most
A Write an inequality for the sentence.
Then graph the inequality on a number
line.
• is greater than
or equal to
• is no less than
• is at least
B For the given value, state whether the
inequality is true or false.
13 x 6, x 4
Write the inequality.
13 x 6
13 4 6
Replace x with 4.
96
Simplify.
Since 9 is greater than 6, 13 x 6 is true.
Children 5 years of age and under are
admitted free.
Let c child’s age
3 4 5 6 7
c5
To graph the inequality, place a closed circle at
5. Then draw a line and an arrow to the left.
Try These Together
1. Write an inequality for the sentence.
More than 20 students must sign up in
order to go on the field trip.
2. For the given value, state whether the
inequality is true or false.
t 5 11, t 8
Write an inequality for each sentence.
13. You must sell at least 25 candy bars to qualify for a prize.
14. No more than 4 students at each activity.
For the given value, state whether each inequality is true or false.
15. 7d 28, d 4
16. 15 y 3, y 6
17. 9 a 1, a 12
Graph each inequality on a number line.
18. m 8
9. h 22
10. b 1
B
C
C
B
C
8.
B
A
11. Standardized Test Practice Which inequality represents a number is no
more than 34.
A x 34
B x 34
C x 34
D x 34
8-10. See Answer Key. 11. C
A
7.
©
Glencoe/McGraw-Hill
6. false 7. true
B
6.
87
4. s 4 5. true
A
5.
2. false 3. c 25
4.
Answers: 1. s 20
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Solving Inequalities by Adding or
Subtracting (pages 496–499)
An inequality is a mathematical sentence that compares quantities using
symbols like and instead of an equals sign. Inequalities may have
many solutions, which can be written as a set of numbers or graphed on a
number line.
Addition and Subtraction Properties of Inequality
Words
When you add or subtract the same number from each side of an inequality, the
inequality remains true.
Symbols
For all numbers a, b, and c,
1. if a b, then a c b c and a c b c.
2. if a b, then a c b c and a c b c.
23
38
2 5 3 5
3484
72
1 4
These properties are also true for a ≥ b and a ≤ b.
Examples
Solve n 10 12 and graph the solution on a number line.
n 10 12
n 10 10 12 10
Subtract 10 from each side.
n2
Simplify.
All values of x that are less than or equal to 2 are solutions to the inequality.
This is indicated by a closed circle on the number line at 2, and an arrow
going to the left.
0
1
2
3
4
Try These Together
Solve each inequality and check your solution. Then graph the solution on
a number line.
1. y 5 3
2. 14 9 x
3. f 8 10
HINT: When graphing, use a closed circle for or and an open circle for or .
Solve each inequality and check your solution. Then graph the
solution on a number line.
4. 4 g 3
5. h 1 2
6. 6 q 16
7. 5 k 11
8. m 8 1
9. a 9 12
B
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
10. Standardized Test Practice Solve the inequality x 4 7.
A x 28
7. k 6 8. m 9
4.
©
Glencoe/McGraw-Hill
B x 11
C x3
D x9
Answers: 1–9. See Answer Key for graphs. 1. y 8 2. x 5 3. f 2 4. g 7 5. h 3 6. q 10
9. a 3 10. C
3.
88
NAME ________________________________________ DATE ______________ PERIOD _____
Solving Inequalities by Multiplying
or Dividing (pages 500–504)
You can solve inequalities that have rational numbers in them the same way
you solved inequalities with integers.
Solving
Inequalities
Use the same steps to solve an inequality as you use to solve an equation, with
this one exception.
• When you multiply or divide each side of an inequality by a negative
number, the direction of the inequality symbol must be reversed for the
inequality to remain true.
A Solve 3x 12.
y
B Solve 8 0.
2
3x 12
3x
3
12
3
y
2
x 4
Since you divided each side by 3, the
direction of the inequality symbol must be
reversed. The solution is x 4.
y
2
80
8 8 0 8 Subtract 8 from each side.
2
y
2
y
2
8
2(8)
Multiply each side by 2.
y 16
The solution to the inequality is y 16.
Try These Together
1. Solve 7c 21.
2. Solve j 0.06 4.5.
HINT: Will the solution have a sign or
a sign?
HINT: Solve by subtracting 0.06 from each
side.
Solve each inequality.
k
9
6. 5 9
1
9. 16a 19 17 3
s
12. Standardized Test Practice Solve 3 8 4.
A s 36
B s 36
3
8. v 9 5
2
9. a 48
5
Glencoe/McGraw-Hill
7. m 27 4
©
6. k 36
C
B
A
C s 36
1
8.
5. q 16
A
7.
3n
11. 9
2
10. 2z 6 4
89
4. x 75
C
B
B
6.
8. 5 5v 52
D s 36
1
C
A
5.
m
1
7. 9 4
3
Answers: 1. c 3 2. j 4.44 3. p 2
4.
1
5. 8q 2
10. z 1 11. n 6 12. A
B
3.
4. 15 5
x
3. 6p 3
NAME ________________________________________ DATE ______________ PERIOD _____
Chapter 10 Review
Algebra at the Zoo
Substitute the values in the box into each problem below and solve.
Write your solution in the blank to the left of the problem.
5
3
1.
2
x
4
2.
3.
4.
Draw a square with a side of length
2
inches.
5. Find the area of the square.
6. Find the perimeter of the square.
©
Glencoe/McGraw-Hill
90
NAME ________________________________________ DATE ______________ PERIOD _____
Sequences (pages 511–515)
A sequence is a list of numbers in a certain order. Each number is called a
term of the sequence. In an arithmetic sequence, the difference between any
two consecutive terms is the same. This difference is called the common
difference. In a geometric sequence, the consecutive terms of a sequence
are formed by multiplying by a constant factor called the common ratio.
A Identify the pattern in 22, 19, 16, 13,
10, … and write the next five terms.
B Identify the pattern in 20, 10, 5, 2.5,
1.25, … and write the next three terms.
Try 19 22 3. If you add 3 to 19, do you
get the next term, 16? Yes, and this pattern
continues, so this is an arithmetic sequence with
a common difference of 3. The next five terms
are 7, 4, 1, 2, and 5.
There is no common difference. What can you
multiply 20 by to get 10? 0.5. Does this
common ratio continue? Yes, so this
geometric sequence has a common ratio of
0.5. The next three terms are 0.625, 0.3125,
and 0.15625.
Try These Together
1
1
2. State whether 7, 6 , 6, 52 , … is
2
arithmetic, geometric, or neither. Then
find the next three terms.
1. State whether 0, 3, 6, 9, … is
arithmetic, geometric, or neither.
Then find the next three terms.
HINT: What can you add to each term to give
you the next term?
HINT: What can you add to each term to give
you the next term?
State whether each sequence is arithmetic, geometric, or neither.
If it is arithmetic or geometric, state the common difference or
common ratio. Write the next three terms of each sequence.
1
1
3. 3, 1, , 9 , …
3
4. 3, 2, 0, 3, …
6. 80, 40, 20, 10, …
7. 4, 3, 10, 17, …
5. 88, 93, 99, 106, …
2
1
8. 8, 8 3 , 9 3 , 10, …
9. Fitness Hank wants to increase the number of push-ups he does each
day by 3. If on the first day he does 2, how many will he try to do on
the 10th day?
B
C
C
10. Standardized Test Practice What is the next term in the sequence
1.3, 1.7, 2.1, 2.5, …?
A 3.3
B 3.1
C 2.9
D 2.7
4 3. geometric;
91
Answers: 1. arithmetic; 12, 15, 18 2. arithmetic; 5,
1
;
3
1
,
27
1
,
81
2
1
243
2
1
4. neither; 7, 12, 18
Glencoe/McGraw-Hill
1
4 ,
2
8. arithmetic; ; 10 , 11 , 12
3
3
3
©
7. arithmetic; 7; 24, 31, 38
C
B
A
1
8.
1
A
7.
1
B
B
6.
5. neither; 114, 123, 133 6. geometric; ; 5, 2 , 1
2
2
4
A
5.
10. C
4.
9. 29
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Functions (pages 517–520)
A relationship where one thing depends upon another is called a function.
In a function, one or more operations are performed on one number to get
another number. So, the second number depends on, or is a function of, the
first number. The value of f(x) (which you say as “function of x” or “f of x”)
depends on the value of x.
Finding
Values for
Functions
You can organize the input (original number), rule (the operations performed on the
input), and the output (the value of the function) into a function table like this one.
Input or domain
Rule Output or range
2x 1
x
f(x)
The domain contains all the values of x, and the range contains all the values of f(x).
Complete the function table at the right.
Input
Rule
Output
x
2x 1
f(x)
1 2(1) 1 1
0
2(0) 1
1
1
2(1) 1
3
2
2(2) 1
5
Replace x in the rule with each input value.
The rule, 2x 1, is 2(0) 1 or 1 for an input of 0.
Put the simplified value for f(x) in the output column.
Repeat these same steps for the input values of 1, 1, and 2.
1. Complete this function table.
C
B
C
B
6.
A
7.
8.
B
A
Find each function value.
2. f (6) if f (x) x 3
3. f (0.5) if f (x) 0.5x 1
4. f (3.2) if f (x) x2 2
5. f (12) if f (x) x 3
6. f (4) if f (x) x 5
7. f (0) if f (x) x 5
8. Standardized Test Practice If f (x) 2x2 20, find f (3).
A 2
B 38
C 56
6. 1 7. 5 8. B
C
A
5.
f(x)
©
Glencoe/McGraw-Hill
2. 3 3. 1.25 4. 8.24 5. 15
4.
3x
92
D 236
Answers: 1. 6, 0, 1.5, 6, 12
B
3.
x
2
0
0.5
2
4
NAME ________________________________________ DATE ______________ PERIOD _____
Graphing Linear Functions (pages 522–525)
A function for which the graphs of the solutions form a straight line is
called a linear function.
Graphing
a Linear
Function
To graph a linear function, begin by making a function table. List at least three
values for x. Graph each ordered pair. Connect the points with a straight line.
Add arrows to the ends of the line to show that the line continues indefinitely.
Graph the function y 3x 2.
Choose some values for x, and find the matching
values for y. Make a table to show the ordered pairs.
x
3x 2
1 3(1) 2
0
3(0) 2
1
3(1) 2
2
3(2) 2
y
5
2
1
4
y
(2, 4)
(1, 1)
(x, y)
(1, 5)
(0, 2)
(1, 1)
(2, 4)
x
O
(0, –2)
(–1, –5)
Then graph the ordered pairs from your table.
Draw the line that joins these points. This line
is the graph of y 3x 2.
Try These Together
1. Graph the function y 3x.
2. Graph the function y 6 x.
HINT: Make a function table for the x-values
of 1, 0, 1, 2.
HINT: Make a function table for the x-values
of 1, 0, 2, 6.
Graph each function.
B
4.
C
B
C
B
A
7.
8.
5. y x
1
6. y 2 x 4
7. y 2x 3
8. y 5 2x
C
A
5.
6.
4. y x 10
B
A
9. Standardized Test Practice If it costs 25 cents to manufacture an eraser,
how much would it cost to manufacture 10? Find the ordered pair that
would represent this on a linear graph.
A (10, $2.50)
B (10, $5)
C ($2.5, 8)
D (2, $25)
Answers: 1–8. See Answer Key. 9. A
3.
x
3. y 2 3
©
Glencoe/McGraw-Hill
93
NAME ________________________________________ DATE ______________ PERIOD _____
The Slope Formula (pages 526–529)
You can find the slope of a line by using the coordinates of any two points on the line. The
slope m of a line passing through points (x1, y1) and (x2, y2) is the ratio of the difference
in
y2 y1
the y-coordinates to the corresponding difference in the x-coordinates or m ,
x2 x1
where x1 x2.
Find the slope of the line that passes through L(3, 4) and M(2, 1).
y
L (–3, 4)
y y
2
1
m x2 x1
Definition of slope
m
2 (3)
(x1, y1) (3, 4)
(x2, y2) (2, 1)
m=
Simplify.
14
3
5
M (2, 1)
x
O
Find the slope of the line that passes through each pair of points.
11. P (2, 2), Q (3, 3)
12. R (8, 9), S (2, 1)
13. X (4, 5), Y (8, 2)
14. M (3, 7), N (9, 7)
15. G (0, 0), H (7, 6)
16. V (13, 11), W (2, 21)
1
1
1 7
7. P 5, 8 , Q 35, 8
3 1
3
1
9. J (4.5, 2.5), K (6.5, 1.5)
8. R 4, 4 , S 14, 34
For Exercises 16 and 17, use the following information.
Caroline sells shirts for the pep club. After 3 shirts were sold, she had $45. After 6 shirts
were sold, she had $90. After 7 shirts were sold, she had $105.
10. Graph the information with the number of shirts on the horizontal axis and the profit in
dollars on the vertical axis. Draw a line through the points.
11. What is the slope of the graph?
12. What does the slope of the graph represent?
B
C
C
1
C
13. Standardized Test Practice Which graph has a slope of 2?
A
B
y
O
x
C
y
x
O
D
y
y
x
O
O
1
©
Glencoe/McGraw-Hill
1
B
A
94
32
8.
6
A
7.
3
B
B
6.
3. 4 4. 0 5. 7 6. 1
7. 3 8. 3 9. 1
10. See Answer Key. 11. 15 12. price per shirt 13. B
5
1
A
5.
x
4
4.
Answers: 1. 1 2. 5
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Slope-Intercept Form (pages 533–536)
An equation of a line can be written in the form y = mx + b. This is called
the slope-intercept form, where m is the slope of the line and b is the
y-intercept of the line. For example, in the equation y = 3x + (2),
the slope is –3 and the y-intercept is –2.
State the slope and the y-intercept of the graph of each equation.
A
y
12 3x
y
12 3x
y 3x
B
+ 12
y 4x 1
Write the original
equation.
y 4x 1
Write the original
equation.
Write the equation in the
form y mx b.
y 4x (1)
Write the equation in the
form y mx b.
1
The slope of the line is 4 and the y-intercept is 1.
The slope of the line is 3 and the y-intercept is 2.
Try These Together
1. y x 2
2. y 2x 6
3. y 4x 1
1
1
1
14. y 3x 12
15. y 2x 7
16. y 1\5x 5
17. y x 4
18. y 3x 1
19. 4x y 3
Graph each equation using the slope and the y-intercept.
13. y 6x 2.5
14. 3x y 1
15. y x 1
y
C
16. Standardized Test Practice What is the equation of the
graph at the right?
1
A y = 3x – 2
1
C y 3x 2
(2, 4)
B y 3x 2
x
O
D y 3x – 2
(0, –2)
1
5
1
1
5
7. 1; 4
8. 3; 1 9. 4; 3
Glencoe/McGraw-Hill
5. 2; 7 6. ; ©
95
1
3
B
A
2. 2; 6 3. 4; 1 4. ; 12
B
8.
12. y x 5
C
B
A
7.
6
C
A
5.
6.
11. y 5x 2
10–15. See Answer Key. 16. D
4.
1
Answers: 1. 1; 2
B
3.
10. y 4x 3
NAME ________________________________________ DATE ______________ PERIOD _____
Scatter Plots
(pages 539–542)
A graph of two sets of data as ordered pairs is a scatter plot. Scatter plots
can suggest whether two sets of data are related.
Determining the
Relationship
To determine whether two sets of data are related, imagine a line drawn so
that half of the points are above the line and half are below it.
• A line that slopes upward to the right shows a positive relationship.
• A line that slopes downward to the right shows a negative relationship.
• When the points are very spread out instead of clustering along a line,
the scatter plot shows that there is no relationship between the data sets.
Determine whether a scatter plot of the data for age and weight of people
younger than 21 would show a positive, negative, or no relationship.
In children and young people, as the age increases, so does the weight in most cases.
A scatter plot of this data would show a positive relationship.
Try These Together
1. Determine whether a scatter plot of the
data for bank balance and money spent
would show a positive, negative, or no
relationship. Assume everyone
considered has the same income.
2. Determine whether a scatter plot of the
data for hours of sleep per night and
height would show a positive, negative,
or no relationship.
HINT: Do hours of sleep per night and height
have any influence on each other?
HINT: Does the bank balance rise or fall as
money spent increases?
Determine whether a scatter plot of the data for the following
might show a positive, negative, or no relationship.
3. temperature and hours of sunlight
4. age past 70 and number of health problems
5. age of a computer and its value
6. hours of battery use and remaining battery life
7. number of seats in a car and the last digit in its license plate number
B
C
C
B
C
A
7.
8.
B
A
8. Standardized Test Practice What kind of relationship does
the scatter plot at the right show?
A positive
B negative
C no
D inverse
6
5
4
3
2
1
0
7. no relationship 8. B
B
6.
©
Glencoe/McGraw-Hill
6. negative
A
5.
2. no relationship 3. positive 4. positive 5. negative
4.
96
1 2 3 4 5 6 7 8
Answers: 1. negative
3.
NAME ________________________________________ DATE ______________ PERIOD _____
Graphing Systems of Equations (pages 544–547)
A set of two or more equations is called a system of equations. When you
find an ordered pair that is a solution of all of the equations in the system,
you have solved the system.
Solving
Systems of
Two Equations
by Graphing
The ordered pair that names the point where the two lines intersect (or cross
each other) is the solution of the system of equations. The coordinates of
this ordered pair make the equations of each of the lines true. Check your
solution in both equations.
Solve this system of equations by graphing.
y 3x 2 and y 2 x
First make a function table for each equation.
x
1
0
1
2
3x 2
3(1) 2
3(0) 2
3(1) 2
3(2) 2
x
1
0
1
2
2x
2 (1)
20
21
22
y
y=2–x
y = 3x – 2
y
(x, y)
5 (1, 5)
2 (0, 2)
1
(1, 1)
4
(2, 4)
y
3
2
1
0
(x, y)
(1, 3)
(0, 2)
(1, 1)
(2, 0)
Graph the ordered pairs for each table and
draw each line.
Try These Together
1. Solve the system
y 2x 3 and y x 1 by graphing.
O
Find the coordinates of the point where the lines
cross by looking at the graph. (1, 1)
Check this solution in both equations.
Does 1 3(1) 2? yes
Does 1 2 1? yes
The solution of this system is (1, 1).
2. Solve the system
y x 2 and y 2x 2 by graphing.
HINT: The lines intersect in Quadrant III.
HINT: Choose at least 3 values for x in each
equation.
Solve each system of equations by graphing.
3. y 4x 4
4. x y 9
y 3x 2
y 13 2x
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
5. 2 x y
3x 14 y
B
A
6. Standardized Test Practice You are walking along the path of y 6x 8
and your friend Ramon is walking on the path of y 8x 12. At what
point do your paths cross?
A (0, 8)
B (1, 4)
C (2, 4)
D (1, 14)
Answers: 1–5. See Answer Key for graphs. 1. (2, 1) 2. (0, 2) 3. (2, 4) 4. (4, 5) 5. (3, 5) 6. C
3.
x
©
Glencoe/McGraw-Hill
97
NAME ________________________________________ DATE ______________ PERIOD _____
Graphing Linear Inequalities
(pages 548–551)
To graph an inequality, first graph the related equation. This is the boundary. If the
inequality contains the symbol or , a solid line is used to indicate that the
boundary is included in the graph. If the inequality contains the symbol or , a
dashed line is used to indicate that the boundary is not included in the graph.
Graphing
Linear
Inequalities
Next, test any point above or below the line to determine which region is the
solution of the inequality.
y
Graph y x 3.
Graph the boundary line y x 3.
Since is used in the inequality, make the boundary line
dashed.
(3, 0)
x
O
(0, –3)
Test a point not on the boundary line, such as (0, 0).
yx3
?
0 0 3
03
y
Replace x with 0 and y with 0.
Simplify.
(3, 0)
x
O
Since (0, 0) is a solution of y x 3, shade the region
that contains (0, 0).
Try These Together
Graph each inequality.
1. y 3x 2
1
2
2. y x 1
(0, –3)
3
3. y 2 x 3
Graph each inequality.
14. y x 6
17. y 1
2x
–2
10. y 6x – 1
B
4.
B
4
11. y – x ≤ 3
12. y 3x 5
C
B
A
8.
19. y ≥ 3x 2
C
A
7.
18. y x 1
C
5.
6.
16. y 2x – 2
B
A
13. Standardized Test Practice Which ordered pair is not a solution of
1
2
y x 1?
A (0, 0)
B (2, 3)
C (3, 1)
D (4, 1)
Answers: 1–12. See Answer Key. 13. B
3.
15. y ≥ 3x – 7
©
Glencoe/McGraw-Hill
98
NAME ________________________________________ DATE ______________ PERIOD _____
Chapter 11 Review
Function Map
Franny’s friends leave her a map so she can find their picnic in the park.
The picnic site is located somewhere on the graph of the function
f (x) 2x 3.
f (x)
L
A
J
C
K
B
x
O
D
I
E
F
G
H
1. Complete the function table for f (x) 2x 3
x
2x 3
f(x)
0
2(0) 3
3
1
2
2. Graph the function on the map above.
3. Which points on the map could possibly be the picnic site?
4. If the picnic site is in Quadrant II on the map, which point is it?
5. There is a swing set that is also on the graph of the function in
Quadrant III. Which point is the swing set?
6. A large pecan tree is on the graph of the function in Quadrant IV.
Which point is the pecan tree?
©
Glencoe/McGraw-Hill
99
NAME ________________________________________ DATE ______________ PERIOD _____
Linear and Nonlinear Functions
(pages 560–563)
Linear functions have graphs that are straight lines. These graphs represent constant rates
of change. Nonlinear functions do not have constant rates of change. Therefore, their
graphs are not straight lines.
y
Identify Functions
Using Graphs
o
The graph is a curve, not a straight line. So
it represents a nonlinear function.
x
y x2 1
Since x is raised to a power, the equation cannot be written in the
form y mx b. So this function is nonlinear.
Identify Functions
Using Equations
Identify Functions
Using Tables
x
5
7
9
11
y
8
12
16
20
As x increases by 2, y increases by 4 each time.
The rate of change is constant, so this function is linear.
Determine whether each graph, equation, or table represents a linear or
nonlinear function. Explain.
11.
12.
y
o
7.
B
3.
3 4
10 11
o
x
15. y x2
5 6
12 13
8.
x
y
y
x
16. x – y 5
3
6
4 1
9
3
12
8
9. x 3 2 1
y
4
9
0
16 25
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
10. Standardized Test Practice Which equation represents a linear function?
A xy4
6
B y x
C xy 3
D y = x3 1
Answers: 1. nonlinear; graph is a curve 2. linear; graph is a straight line 3. nonlinear; graph is a curve 4. linear; can be written as
y 0x 2 5. nonlinear; power of x is greater than one 6. linear; can be written as y x 5 7. linear; rate of change is
constant, as x increases by 1, y increases by 1 8. nonlinear; rate of change is not constant, as x increases by 3, y increases by a
greater amount each time 9. nonlinear; rate of change is not constant, as x increases by 1, y increases by a greater amount each
time 10. A
4.
x
y
o
x
14. y 2
3.
y
©
Glencoe/McGraw-Hill
100
NAME ________________________________________ DATE ______________ PERIOD _____
Graphing Quadratic Functions (pages 565–568)
In a quadratic function, the greatest power of the input variable (usually x) is
2. For example, y x2, A s2, and y 3x2 5 are all quadratic functions.
Graphing
Quadratic
Functions
You graph a quadratic function with the same steps you used for graphing a
linear function, but the graph of a quadratic function is a curve, not a straight line.
The graphs of the quadratic functions in this lesson are all curves, called
parabolas, shaped a little like the letter U.
Graph the quadratic function y 2x2 1.
Graph the (x, y) points in the last column of your
table. Draw a smooth curve to join the points.
Choose some values for x and make a table.
x
2 x2 1
2 2 (2)2 1 7
1 2 (1)2 1 1
0
2(0)2 1 1
1
2(1)2 1 1
2
2(2)2 1 7
y
y
(x, y)
7 (2, 7)
1 (1, 1)
1
(0, 1)
1
(1, 1)
7
(2, 7)
x
O
y = –2x 2 + 1
Because the graph is a curve, plot more points
than you would for a straight line, so that you can
see the shape of the curve.
Try These Together
1. Complete the function table and then
graph the function y 2x 2.
x
2
1
0
1
2
2x2
y
2. Complete the function table and then
1
graph the function f(x) 2 x 2.
( x, y)
x
1
x2
2
f(x)
( x, f( x))
4
2
0
2
4
HINT: The y-values repeat.
HINT: Treat the f(x) like y.
3. Graph f(x) 2x2 5.
4. Graph y 12 x2.
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
5. Standardized Test Practice Determine which ordered pair is a solution
of y x2 x 3.
A (6, 9)
B (2, 1)
C (4, 17)
D (3, 15)
Answers: 1–4. See Answer Key for graphs. 1. (2, 8), (1, 2), (0, 0), (1, 2), (2, 8) 2. (4, 8), (2, 2), (0, 0), (2, 2), (4, 8) 5. C
3.
©
Glencoe/McGraw-Hill
101
NAME ________________________________________ DATE ______________ PERIOD _____
Simplifying Polynomials (pages 570–573)
Each monomial in a polynomial is called a term. Monomials with the same
variable to the same power, such as 2x and 3x, are called like terms. You
can simplify polynomials that have like terms. An expression that has no
like terms is in simplest form.
A Simplify 2x 3x.
x
x
x
x
B Simplify 2x2 x2 3.
x2
x
x2
–x2
1
1
With tiles you can see that there are 5 x-tiles.
On paper, you add the like terms.
So 2x 3x 5x.
1
With the tiles, you can see that there are
2 positive x 2-tiles and one negative x 2-tile.
Two positives plus one negative equals one
positive. Or, on paper, 2x 2 x 2 x 2. So the
polynomial in simplest form is x 2 3.
Try These Together
Simplify each polynomial. If the polynomial cannot be simplified, write
simplest form.
1. 3 2q 2 3 q 2
2. 4r 2 2r 2 r
3. 3z 2y 5x 2
HINT: Monomials with the same variable and power are like terms. All numbers without
variables are like terms.
Simplify each polynomial. If the polynomial cannot be simplified,
write simplest form.
4. 5a2 2a 3
5. 6d 2r 3d
6. c2 4c 3
7. m4 m m2 m
8. 1 x 4 x2 x 5
9. t3 t3 t3
10. y 3 y3 y 2 3y3
11. w2 4w 1
12. 5g 2h g 3h
2
2
13. 2b 3 4b 2
14. x 2x 3x 4
15. 2r 2 4r 3r r 2 r
16. a b 3b 1
17. 2y 2y 2 2y 2 y
18. 3a3 2a2 a
19. Money Matters Cesár put his $50 cash birthday gift in a savings
account. He also received $50 last year and also put it in the account.
Adding the interest x he made from his account, write an expression in
simplest form that represents the amount of money in his account.
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
20. Standardized Test Practice Simplify the polynomial x2 x 2x2 3.
A x2 2x 3
B 4x2 2x 3
C 3x2 x 3
D 2x2 x 3
Answers: 1. 3q2 2. 2r 2 r 3. simplest form 4. simplest form 5. 3d 2r 6. simplest form 7. m4 m2 2m 8. x 4 x 2 x 4 9. t3 10. 3y3 y2 11. simplest form 12. 6g 5h 13. 6b 5 14. 4x2 2x 4 15. 3r 2 8r 16. a 2b 1 17. 3y
18. simplest form 19. 100 x 20. C
3.
©
Glencoe/McGraw-Hill
102
NAME ________________________________________ DATE ______________ PERIOD _____
Adding Polynomials (pages 574–577)
To add polynomials, add the like terms in each polynomial. You can use
algebra tiles or pencil and paper to add polynomials.
Find each sum.
A (x 2 2x 1) (x 2 5x 3)
B (2x 2 x 2) (x 2 3x 2)
Use algebra tiles to represent each polynomial.
x2
x
Align the like terms in columns, then add.
2x2 x 2
( x 2) 3x 2
x
–1
x2
x
x
x
x2 2x 4
x
x
1
1
1
Using the tiles, add like terms to find the sum,
2x2 7x 2.
Try These Together
Add.
1.
y2 2y 1
y2 3y 2
2.
HINT: 2y (3y) y
Add.
4.
7x2 6x 2
5x2 3x 4
3x2 y 3
2x2 3y 4
3.
HINT: y (3y) 2y
5.
10q2 7q 1
8q2 2q 6
4m2 2m 5
3m2 m 4
HINT: Like terms are in columns.
6.
4a2 4a 4
(3a2) 3a 3
Add. Then evaluate each sum if x 3 and y 2.
7. (3x 2y) (2 3y)
8. (4x y) (2x 2y)
9. (2x 3y) (3x 4y)
10. (4x 3y) (x y)
11. (5x 3y) (4x 3y)
12. (x y) ( y x)
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
13. Standardized Test Practice What is the sum of t 2 2t 1 and t 2 3t 2?
A t2 t 3
B 2t 2 5t 3
C 2t 2 5t 2 3
D t 2 5t 3
Answers: 1. 2y2 y 1 2. 5x2 2y 7 3. 7m2 3m 1 4. 12x2 3x 6 5. 18q2 9q 5 6. a2 a 1
7. 3x 5y 2; 21 8. 2x 3y; 12 9. x y; 1 10. 5x 4y; 23 11. 9x 6y; 39 12. 2x 2y; 10 13. B
3.
©
Glencoe/McGraw-Hill
103
NAME ________________________________________ DATE ______________ PERIOD _____
Subtracting Polynomials (pages 580–583)
Subtracting polynomials is very similar to adding polynomials. You can use
algebra tiles to subtract polynomials. You can also use paper and pencil.
Since subtracting is the same as adding the opposite, use this procedure to
subtract polynomials with paper and pencil.
Find each difference.
A (3x 2 5x 4) (x 2 2x 3)
B (2x2 4x 3) (x2 3x 2)
Subtracting x2 3x 2 is the same as
adding the additive inverse. To find the additive
inverse, find the opposite of the term,
or x2 3x 2.
2x2 4x 3
x2 3x 2
Use algebra tiles to represent the first
polynomial.
x2
x2
x2
x
x
x
x
x
1
1
1
1
3x2 x 1
MAC3-13-10-C-823594
To
subtract, remove the tiles representing the
second polynomial. The remaining tiles
represent the difference, 2x 2 3x 1.
Try These Together
Subtract.
1.
4x 4
(2x 6)
2.
HINT: The additive inverse of
2x 6 is 2x 6.
Subtract.
4.
7y 2
(4y 3)
3x 5
(x 1)
3.
HINT: The additive inverse of
x 1 is x 1.
5.
8r 2 5a 5
(6r 2 3a 2)
7. (4b2 4b 4) (b2 b 1)
10x 5
(5x 1)
HINT: Add the additive inverse.
6.
7a2 4a 4
(5a2 2a 2)
8. (3b2 3b 3) (2b2 2b 2)
Subtract. Then evaluate if x 3 and y 4.
9. (6x 3y) (3x 2y)
10. (5x 5y) (4x 4y)
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
11. Standardized Test Practice Subtract (5x 3y) (2x 4y) then
evaluate if x 2 and y 5.
A 13
B 29
C 6
D 11
Answers: 1. 2x 2 2. 2x 6 3. 5x 4 4. 3y 1 5. 2r 2 2a 3 6. 2a2 2a 2 7. 5b2 3b 5 8. b2 5b 1
9. 3x y; 5 10. x y; 1 11. D
3.
©
Glencoe/McGraw-Hill
104
NAME ________________________________________ DATE ______________ PERIOD _____
Multiplying and Dividing Monomials
(pages 584–587)
In order to multiply and divide monomials, you will multiply and divide
powers that have the same base.
Product
of Powers
You can multiply powers that have the same base by adding their exponents.
So, for any number a and integers m and n, am an am n.
Quotient
of Powers
You can divide powers that have the same base by subtracting their exponents
am
a
So, for any number a and integers m and n, n am n, where a 0.
Multiply or divide. Express using exponents.
A x3 x5
B
x 3 x 5 x 3 5 or x 8
d6
2
d
d6
2
d
d 6 2 or d 4
Try These Together
Multiply or divide. Express using exponents.
x5
3
3. 32 32
x
HINT: When you multiply powers, use the same base and use a new exponent that is
the sum of the original ones. When you divide powers, the new exponent is the
difference of the original ones. Bases with no exponent written have an understood
exponent of 1.
1. b b4
2.
Multiply. Express using exponents.
4. r 3 r 3
5. 2r 2 r 2
7. 2c c4
8. x5 x10
6. 3a a5
9. 47 49
Divide. Express using exponents.
b12
b
11. 3
12y5
3y
14. 10. 7
13. 4
B
4.
C
B
C
B
A
7.
8.
98
9
64
6
15. 9
f 14
f
C
A
5.
6.
12. 2
B
A
16. Standardized Test Practice Find the product 2x6 x10.
A 2x16
B x16
C 2x4
D 2x60
Answers: 1. b5 2. x2 3. 34 4. r 6 5. 2r 4 6. 3a6 7. 2c5 8. x15 9. 416 10. b5 11. 4m4 12. 96 13. 4y 14. 63 15. f 5
16. A
3.
8m7
2m
©
Glencoe/McGraw-Hill
105
NAME ________________________________________ DATE ______________ PERIOD _____
Multiplying Monomials and Polynomials
(pages 590–592)
You can multiply monomials and polynomials by using the Distributive Property. Often, the
definition of exponents and the Product of Powers rule are also needed to simplify the
product of a monomial and a polynomial.
A Find 2b(b 6).
B Find g3(g 2).
2b(b 6) 2b(b) 2b(6) Distributive Property
2b2 12b
b b b2
Try These Together
Multiply.
1. 4y(y 2)
g3(g 2) g3[ g (2)]
Rewrite g 2 as
g (2).
g3(g) g3(2) Distributive Property
g 4 (2g 3) g3(g) g 3 1 or g 4
g 4 2g 3
Definition of
subtraction
2. n(3n2 n 8)
HINT: Use the Distributive Property, and add exponents when multiplying
powers with the same base.
Multiply.
13. (x 2)(4x)
14. a3(a 3)
15. y 4(y4 6)
6. 5m3(m2 1)
17. y(y2 4y 3)
18. x2(x3 2)
9. 2q2(2q 1)
10. a(a 4)
11. n(3n2 4n 7)
13. (w2 6)(5w)
14. 3q2(q2 2)
12. r3(r5 r3 5)
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
15. Standardized Test Practice What is the product of 2z2 and 4z2 2z 8?
A 8z4 4z2 2z 8
B 8z4 4z3 16z2
C 8z2 4z 16
D 8z4 4z3 2z2 16
Answers: 1. 4y2 8y 2. 3n3 n2 8n 3. 4x2 8x 4. a4 3a3 5. y 8 6y 4 6. 5m5 5m3 7. y 3 4y 2 3y
8. x5 2x2 9. 4q3 2q2 10. a2 4a 11. 3n3 4n2 7n 12. r 8 r 6 5r3 13. 5w3 30w 14. 3q4 6q2 15. B
3.
©
Glencoe/McGraw-Hill
106
NAME ________________________________________ DATE ______________ PERIOD _____
Chapter 12 Review
Match’Em
First, simplify the expressions in each column. Each expression in the left
column matches exactly one expression in the right column. Write the
correct letter in the blank next to each expression in the left column.
45x9
3x
________
1. 2x 1 2x 2
A. 2
________
2. (4x)2
B. 6x4 (4x)
________
3. 6x(x 2)
C. 4(4x2)
________
4. 3
D. x(3x2 6x 12)
________
5. (2x2 x 1) (3x2 x 1)
E. (7x2 x) (4x 1)
________
6. 3x5 (5x2)
F. 2(x 5)
________
7. 6x3 6x4
G. 3x2 4x 5x 3 2x2
________
8. (7x2 3x) (2x2 2x)
H. (5x2 2x 1) (4x2 3)
________
9. 3x(x2 2x 4)
I. 2
412
4
________ 10. 6x2 3x x2 1
J. 9x(4x6)
________ 11. 9x x2 3
K. 3x x 1
20x3
4x
________ 12. 2
L. x(5x2)
________ 13. –8x3(3x2)
M. (43)3
6x3
3x
________ 14. 4
N. 5x2 13x x x2
________ 15. (13x2 2x 10) (13x2 4x)
O. (4x2 5x) (5x2 4x)
©
Glencoe/McGraw-Hill
107
Answer Key
Lesson 1-3
9.
10.
11.
12.
5.
y
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9
(4, 6)
(0, 3)
(–4, 0)
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9
O
x
(–8, –3)
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9
Chapter 4 Review
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9
similar
13.
14.
–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6
Lesson 5-6
3. 120 0.12n
5. n 0.82(30)
7. n 0.4(37)
9. 61 0.5n
11. n 0.12(1.75)
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9
Chapter 1 Review
1. 12; 28 2. 5; 33 3. 17; 50
4. 16; 34
Team A did not score a touchdown. They
are 34 yards short of the goal line.
Chapter 5 Review
1–15. Answers will vary. 16. $3.75
17. 120 18. 25%
Chapter 2 Review
24
Lesson 6-6
19
1.
From least to greatest: 5.3, ,
120
8
7
2
4. 21 0.42n
6. 24 n(96)
8. 13 n(104)
10. n 0.75(98)
12. 8.22 0.15n
2.03 101, , , 4
, 4.7, 11 9
3
4
Chapter 3 Review
1. 45° 2. 4.2 3. 4.5 4. (2, 3)
5. (1, 3) 6. 5.4 units
4.
6.
Lesson 4-3
y
4.
(–1, 5)
(0, 3)
(1, 1)
O
x
(2, –1)
© Glencoe/McGraw-Hill
108
Answer Key
Lesson 6-7
y
J
4.
4b.
5.
y
B
y
C
O
O
x
L
K
Q
T
K
x
Q
T
R R
S
A
O
S
J
x
A y
Chapter 6 Review
perpendicular line
C
D
Chapter 7 Review
C
D
O
1. 88 in2 2. 152 in2 3. 154 in2
4. The rectangular prism in Exercise 3
x
A
Chapter 8 Review
1. 28 2. 56 3. 70 4. 56
6. They can use any size.
Lesson 6-8
F
E
B
A
x
C
A
D
Lesson 9-1
3b. Ages of Presidents
C
E
OD
B y
x
D O
Age
40–44
45–49
50–54
55–59
60–64
65–69
D
3.
y
B
C
E
B
D
C
E
D
x
O
4.
4.
y
H
H
x
O
K
J
I
I
J
CDs per Student
1–1
K
30
25
20
Number of
15
Students
10
5
0
Number
2
6
12
12
7
3
40
F
30
2.
y
5. 28
31–
1.
20
B
0
B
11–
6.
21–
L
CDs
Lesson 6-9
y
4.
4a.
O
y
B
x
A
A
C
O
C
x
B
109
Answer Key
Lesson 9-2
2a.
Dogs Owned by Families
Lesson 10-5
18.
6
4-person
35%
1-person
13%
7
8
9
10
9.
20 21 22 23 24
3-person
21%
3.
10.
2-person
31%
–1
Students in School
College
23%
2
3
1.
0 1 2 3 4 5 6 7 8 9 10
2.
9–12
23%
0 1 2 3 4 5 6 7 8 9 10
3.
Lesson 9-6
0 1 2 3 4 5 6 7 8 9 10
238
270 282
290
4.
200 220 240 260 280 300
5.
204
2.
1
Lesson 10-6
K–8
54%
1.
0
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
80 87 102
15
106
6.
0 1 2 3 4 5 6 7 8 9 10
0
3.
20 40 60 80 100 120
14
22
30
35
7.
40
0 1 2 3 4 5 6 7 8 9 10
8.
0 1 2 3 4 5 6 7 8 9 10
10 15 20 25 30 35 40 45
9.
Lesson 9-8
0 1 2 3 4 5 6 7 8 9 10
2 2
1. 6 2 5 2. 1 2
9 8 4
4 1
787
3.  4 3 5. 799
–10 3
902
Chapter 10 Review
1. 8 2. 1
6. 12 in.
944
1,007
1,294
3. 0.5
4. x 2
5. 9 in2
Chapter 9 Review
1–5. Answers will vary.
110
Answer Key
Lesson 11-3
1.
7.
y
y
x
O
x
O
8.
2.
y
y
x
O
x
O
Lesson 11-4
3.
y
10.
4.
4
–4 O
Profit (dollars)
x
O
Shirt Sale
y
120
105
90
75
60
45
30
15
0
(7, 105)
(6, 90)
(3, 45)
1
2
3
4
5
6
7
8
Number of Shirts Sold
4
8
x
12
–4
Lesson 11-5
–8
10.
y
–12
5.
x
O
y
(4, –2)
(0, –3)
x
O
6.
11.
y
(5, 8)
y
(0, 2)
O
O
x
x
111
Answer Key
12.
2.
y
y
x
O
(0, 2)
x
O
(0, –5)
(1, –6)
13.
3.
y
y
(0, 2.5)
x
O
x
O
(–2, –4)
(1, –3.5)
14.
y
4.
y
(0, 1)
(4, 5)
x
O
(1, –2)
x
O
15.
y
5.
y
(–3, 5)
O
x
(0, –1)
(1, –2)
x
O
Lesson 11-7
1.
Lesson 11-8
y
1.
(–2, –1)
O
y
x
O
112
x
Answer Key
2.
18.
y
y
O
3.
19.
y
y
x
O
4.
x
O
x
x
O
y
10.
y
x
O
x
O
5.
y
11.
x
O
y
x
O
6.
y
12.
y
O
x
x
O
17.
y
Chapter 11 Review
O
1.
x
113
x
0
1
2
2x 3
2(0) 3
2(1) 3
2(2) 3
f(x)
3
1
1
Answer Key
2.
f(n)
L
A
C
J
K
B
n
O
D
E
F
I
G
H
3. A, D, or G 4. A 5. D 6. G
Lesson 12-2
1.
O
3.
2.
y
x
O
O
4.
f (x)
f (x)
x
y
x
O
x
Chapter 12 Review
1. K; 4x 1 2. C; 16x2 3. N; 6x2 12x
4. M; 49 5. H; x2 2x 2 6. A; 15x7
7. J; 36x7 8. O; 9x2 x 9. D;
3x3 6x2 12x 10. E; 7x2 3x 1
11. G; x2 9x 3 12. L; 5x3 13. B;
24x5 14. I; 2 15. F; 2x 10
114

Parent and Student Study Guide Workbook

Transcription

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