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Contents
of Student Resources
Skills Review Handbook
pages 975–1009
Operations with Positive and
Negative Numbers
Perimeter and Area
991
975
Circumference and Area of a Circle
992
Fractions, Decimals, and Percents
976
Surface Area and Volume
993
Calculating with Percents
977
Angle Relationships
994
Factors and Multiples
978
Triangle Relationships
995
Ratios and Proportions
980
Congruent and Similar Figures
996
Converting Units of Measurements
981
More Problem Solving Strategies
998
Scientific Notation
982
Logical Argument
1000
Significant Digits
983
Writing Algebraic Expressions
984
Conditional Statements and
Counterexamples
1002
Binomial Products
985
Venn Diagrams
1004
LCDs of Rational Expressions
986
Mean, Median, Mode, and Range
1005
The Coordinate Plane
987
Graphing Statistical Data
1006
Transformations
988
Organizing Statistical Data
1008
Line Symmetry
990
Extra Practice for Chapters 1–14
pages 1010–1023
Tables
pages 1024–1034
Symbols
Measures
Formulas
Properties
1024
1025
1026
1033
English-Spanish Glossary
pages 1035
1035–
–1084
Index
pages 1085
1085–
–1104
Credits
pages 1105
1105–
–1106
Worked-Out Solutions
Selected Answers
974
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Sk ills Re
Rev
v iew Handboo
Handbook
k
To add positive and negative
numbers, you can use a
number line.
To subtract any number,
add its opposite.
EXAMPLE
To add a positive number, move to the right.
To add a negative number, move to the left.
26
25
24
22
0
21
1
2
3
4
5
6
Add or subtract.
a. 1 1 (25)
End
23
SKILLS REVIEW HANDBOOK
Operations with Positive and Negative Numbers
b. 22 2 (25) 5 22 1 5
Move 5 units to the left.
25 24 23 22 21
0
Start
Start
1
Move 5 units to the right.
23 22 21
2
c 1 1 (25) 5 24
The opposite of 25 is 5.
0
1
2
End
3
4
c 22 2 (25) 5 3
To multiply or divide positive and negative numbers, use the following rules.
• The product or quotient of two numbers with the same sign is positive.
• The product or quotient of two numbers with different signs is negative.
EXAMPLE
Multiply or divide.
a. 3 p 7 5 21
b. 23(27) 5 21
c. 18 4 2 5 9
d. 218 4 (22) 5 9
e. 23(7) 5 221
f. 3(27) 5 221
g. 218 4 2 5 29
h. 18 4 (22) 5 29
PRACTICE
Perform the indicated operation.
1. 2 1 (28)
2. 5 2 12
3. 26(10)
4. 230 4 (22)
6. 7(25)
7. 18 2 10
8. 27 1 (212)
9. 11(4)
5. 24 1 6
10. 81 4 (29)
11. 212 4 3
12. 29(28)
13. 21 1 13
14. 45 4 (29)
15. 26(12)
16. 14 2 (29)
17. 232 4 16
18. 223 1 (25)
19. 28 2 (25)
20. 17 2 (218)
21. 29(21)
22. 23 2 (211)
23. 218 4 (23)
24. 14 1 (27)
25. 5(23)
26. 21 1 (28)
27. 22 2 10
28. 29 1 26
29. 220 4 (24)
30. 22 4 (22)
31. 27(26)
32. 1 2 24
33. 215 2 2
34. 0 1 (24)
35. 16 4 8
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SKILLS REVIEW HANDBOOK
Fractions, Decimals, and Percents
A percent is a ratio with a denominator of 100. The word percent means
“per hundred,” or “out of one hundred.” The symbol for percent is %.
In the model at the right, 71 of the 100 squares are shaded. You can write
the shaded part of the model as a fraction, a decimal, or a percent.
71
Fraction: seventy-one divided by one hundred, or }}
100
Decimal: seventy-one hundredths, or 0.71
Percent: seventy-one percent, or 71%
EXAMPLE
Write as a fraction.
94 5 47
a. 94% 5 }}
}}
100
50
EXAMPLE
3
c. 0.3 5 three tenths 5 }}
10
Write as a decimal.
15 5 0.15
a. 15% 5 }}
100
EXAMPLE
20 5 1
b. 20% 5 }}
}
5
100
106 5 1.06
b. 106% 5 }}
100
5 5 5 4 8 5 0.625
c. }
8
Write as a percent.
41 5 41%
a. 0.41 5 }}
100
8 5 80 5 80%
b. 0.8 5 }}
}}
10
100
5 5 5 p 25 5 125 5 125%
c. }
}}}
}}
4
4 p 25
100
PRACTICE
Write as a fraction.
1. 0.65
2. 0.08
3. 1.5
4. 0.13
5. 0.7
6. 50%
7. 26%
8. 3%
9. 95%
10. 110%
Write as a decimal.
1
11. }
4
9
12. }
10
30
13. }
25
2
14. }
5
3
15. }
8
16. 16%
17. 142%
18. 1%
19. 30%
20. 6.5%
21. 0.6
22. 0.24
23. 1.3
24. 0.07
25. 0.45
1
26. }
10
4
27. }
5
17
28. }
20
5
29. }
2
3
30. }
16
Write as a percent.
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Calculating with Percents
EXAMPLE
Word
what
of
is
n
3
5
Symbol
Answer the question.
a. What is 15% of 20?
b. What percent of 8 is 6?
n 5 0.15 3 20
n53
c. 80% of what number is 4?
n3856
0.8 3 n 5 4
n 5 6 4 8 5 0.75 5 75%
3 is 15% of 20.
n 5 4 4 0.8 5 5
75% of 8 is 6.
SKILLS REVIEW HANDBOOK
You can use equations to calculate with percents. Replace
words with symbols as shown in the table at the right. Below
are three types of questions you can answer with percents.
80% of 5 is 4.
Amount of increase or decrease .
To find a percent of change, calculate }}}}}}}}}}}}}}
Original amount
EXAMPLE
Find the percent of change.
a. A class increases from 21 students to 25 students.
25 2 21
4
}}}} 5 }} ø 0.19 5 19% increase
21
21
b. A price decreases from $12 to $9.
12 2 9
3
}}} 5 }} 5 0.25 5 25% decrease
12
12
PRACTICE
Answer the question.
1. What is 98% of 200?
2. What is 25% of 8?
3. What is 30% of 128?
4. What is 5% of 700?
5. What is 100% of 17?
6. What is 150% of 14?
7. What is 0.2% of 500?
8. What is 6.5% of 3000?
9. What percent of 100 is 54?
10. What percent of 18 is 9?
11. What percent of 80 is 8?
12. What percent of 15 is 20?
13. What percent of 30 is 6?
14. What percent of 5 is 8?
15. What percent of 50 is 1?
16. 50% of what number is 6?
17. 55% of what number is 44?
18. 10% of what number is 6?
19. 75% of what number is 45?
20. 1% of what number is 2?
21. 90% of what number is 63?
22. 12% of what number is 60?
23. 200% of what number is 16?
Find the percent of change. Round to the nearest percent if necessary.
24. A class increases from 20 to 28 students.
25. Time decreases from 60 to 45 minutes.
26. A price is reduced from $200 to $180.
27. Votes increase from 200 to 300.
28. A test is shortened from 40 to 32 items.
29. Membership increases from 820 to 1605.
30. A wage rises from $8.75 to $10.00.
31. The temperature drops from 248F to 58F.
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SKILLS REVIEW HANDBOOK
Factors and Multiples
Factors are numbers or expressions that are multiplied together.
A prime number is a whole number greater than 1 that has exactly
two whole number factors, 1 and itself. The table shows all the prime
numbers less than 100. A composite number is a whole number
greater than 1 that has more than two whole number factors.
Prime Numbers
Less Than 100
2, 3, 5, 7, 11, 13, 17, 19, 23,
29, 31, 37, 41, 43, 47, 53, 59,
61, 67, 71, 73, 79, 83, 89, 97
When you write a composite number as a product of prime numbers,
you are writing its prime factorization.
EXAMPLE
Write the prime factorization of 60.
Use a factor tree. Write 60 at the top. Then draw two branches
and write 60 as the product of two factors. Continue to draw
branches until all the factors are prime numbers. Two factor
trees for 60 are given at the right. Both show 60 5 2 p 2 p 3 p 5.
60
60
2 p 30
3 p 20
2 p 15
2
c The prime factorization of 60 is 2 p 3 p 5.
3 p 5
4 p 5
2 p 2
A whole number that is a factor of two or more nonzero whole numbers is a
common factor of the numbers. The largest of the common factors is the
greatest common factor (GCF).
EXAMPLE
Find the greatest common factor (GCF) of 18 and 45.
Method 1 List factors.
Method 2 Use prime factorization.
Factors of 18: 1, 2, 3, 6, 9, 18
Prime factorization of 18: 2 p 3 p 3
Factors of 45: 1, 3, 5, 9, 15, 45
Prime factorization of 45: 3 p 3 p 5
The GCF is 9, the greatest of the
common factors.
The GCF is the product of the common
prime factors: 3 p 3 5 9.
A multiple of a whole number is the product of the number and any nonzero
whole number. A common multiple of two or more numbers is a multiple of all of
the numbers. The least common multiple (LCM) is the smallest of the common
multiples.
EXAMPLE
978
n2pe-9020.indd 978
Find the least common multiple (LCM) of 12 and 15.
Method 1 List multiples.
Method 2 Use prime factorization.
Multiples of 12: 12, 24, 36, 48, 60, . . .
Prime factorization of 12: 22 p 3
Multiples of 15: 15, 30, 45, 60, . . .
Prime factorization of 15: 3 p 5
The LCM is 60, the least of the
common multiples.
Form the LCM of the numbers by writing
each prime factor to the highest power it
occurs in either number: 22 p 3 p 5 5 60.
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EXAMPLE
3
10
SKILLS REVIEW HANDBOOK
The least common denominator (LCD) of two fractions is the least common
multiple of the denominators. Use the LCD to add or subtract fractions with
different denominators.
5
8
Add: }} 1 }
The least common multiple of the denominators, 10 and 8, is 40.
So, the least common denominator (LCD) of the fractions is 40.
3 5 3 p 4 5 12 and 5 5 5 p 5 5 25
Rewrite the fractions using the LCD of 40: }}
}}}
}}
}
}}}
}}
10
10 p 4
40
8
8p5
40
3 1 5 5 12 1 25 5 37
Add the numerators and keep the same denominator: }}
}
}}
}}
}}
10
8
40
40
40
PRACTICE
Write the prime factorization of the number. If the number is prime,
write prime.
1. 42
2. 104
3. 75
4. 23
5. 70
6. 27
7. 72
8. 180
9. 47
10. 100
11. 88
12. 49
14. 142
15. 32
13. 83
Find the greatest common factor (GCF) of the numbers.
16. 4, 6
17. 24, 40
18. 10, 25
19. 55, 44
20. 28, 35
21. 8, 20
22. 5, 8
23. 15, 12
24. 16, 32
25. 70, 90
26. 2, 18
27. 9, 21
28. 36, 42, 54
29. 7, 12, 17
30. 45, 63, 81
Find the least common multiple (LCM) of the numbers.
31. 4, 16
32. 2, 14
33. 5, 6
34. 16, 24
35. 6, 8
36. 12, 20
37. 3, 6
38. 18, 8
39. 9, 12
40. 9, 5
41. 10, 15
42. 7, 9
43. 40, 4, 5
44. 25, 30, 3
45. 27, 81, 33
Perform the indicated operation(s). Simplify the result.
113
46. }
}
2
8
32 5
47. }
}}
4
16
7 23
48. }}
}
5
10
111
49. }
}
2
3
5 11
50. }}
}
3
12
411
51. }
}
5
8
1 13
52. }}
}
4
10
521
53. }
}
6
2
7 2 11
54. }
}}
8
16
9 21
55. }}
}
3
10
221
56. }
}
3
6
2
1 1}
57. }
5
4
41 1 25
58. }
}}
}
5
12
6
32 3 23
59. }
}} }
4
2
10
9 2121
60. }}
}
}
5
10
2
71 3 21
61. }
}}
}
4
16
8
8122 7
62. }
}
}}
9
3
12
4 11
1 1 }}
63. }
}
15
3
6
11211
64. }
}
}
4
2
3
15 2 7 1 1
65. }}
}}
}
16
10
2
5 211 7
66. }}
}
}}
24
6
12
321
1 1}
67. }
}
5
4
2
5232 2
68. }
}
}}
5
6
15
4132 7
69. }
}
}}
4
9
12
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SKILLS REVIEW HANDBOOK
Ratios and Proportions
A ratio uses division to compare two quantities.
Three Ways to Write the Ratio of a to b
You can write a ratio of two quantities a and b,
where b is not equal to 0, in three ways.
a to b
a
b
a:b
}
You should write ratios in simplest form.
EXAMPLE
Write the ratio of 12 boys to 16 girls in three ways.
Boys
Girls
12 5 12 4 4 5 3
First write the ratio as a fraction in simplest form: }}} 5 }}
}}}
}
16
16 4 4
4
3.
c Three ways to write the ratio of boys to girls are 3 to 4, 3 : 4, and }
4
A proportion is an equation stating that
two ratios are equal.
You can use cross multiplication to solve
a proportion.
EXAMPLE
9
a
b
c
d
b.
}} 5 }
If } 5 }, where b ? 0 and d ? 0, then ad 5 bc.
Solve the proportion.
n
54
5
} 5 }}
a.
Using Cross Multiplication to Solve Proportions
5 p 54 5 9 p n
270 5 9n
30 5 n
Cross multiply.
Simplify.
Solve for n.
x
40
3
8
x p 8 5 40 p 3
8x 5 120
x 5 15
Cross multiply.
Simplify.
Solve for x.
PRACTICE
Write the ratio in simplest form. Express the answer in three ways.
1. 3 to 9
2. 16 to 24
3. 10 to 8
4. 6 to 2
5. 25 to 30
6. 60 to 10
7. 4 to 4
8. 8 to 20
9. 32 to 72
10. 42 to 15
11. 14 to 2
12. 12 to 15
x 5 12
13. }}
}}
14
24
8 5 d
14. }}
}}
24
36
15 5 3
15. }}
}
4
n
9 5 5
16. }}
}
45
h
a5 4
17. }
}}
6
12
13 5 91
18. }}
}}
t
7
75 5 r
19. }}
}
120
8
b 52
20. }}
}
90
3
4 5 n
21. }}
}}
11
110
5 5 150
22. }
}}
90
z
95x
23. }
}
8
6
72 5 24
24. }}
}}
105
m
17 5 51
25. }}
}}
33
a
20 5 24
26. }}
}}
125
n
16 5 8
27. }}
}
144
x
96 5 t
28. }}
}
6
3
Solve the proportion.
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Converting Units of Measurement
SKILLS REVIEW HANDBOOK
The table of measures on page 1025 gives many statements of equivalent
measures. Using each statement, you can write two different conversion factors.
Statement of Equivalent Measures
Conversion Factors
100 cm 5 1 m
}}}} 5 1 and }}}} 5 1
100 cm
1m
1m
100 cm
To convert from one unit of measurement to another, multiply by a conversion
factor. Use the one that will eliminate the starting unit and keep the desired unit.
EXAMPLE
Copy and complete.
a. 3.5 m 5 ? cm
b. 620 cm 5 ? m
100 cm 5 (3.5 3 100) cm 5 350 cm
3.5 m 3 }}}}
1 m 5 620 m 5 6.2 m
620 cm 3 }}}}
}}
c So, 3.5 m 5 350 cm.
c So, 620 cm 5 6.2 m.
1m
100 cm
100
Sometimes you need to use more than one conversion factor.
EXAMPLE
Copy and complete: 7 days 5 ? sec
Find the appropriate statements of equivalent measures.
24 h 5 1 day, 60 min 5 1 h, and 60 sec 5 1 min
24 h , 60 min , and 60 sec
Write conversion factors: }}}
}}}}
}}}
1 day
1h
1 min
Multiply by conversion factors to eliminate days and keep seconds.
24 h 3 60 min 3 60 sec 5 (7 3 24 3 60 3 60) sec 5 604,800 sec
7 days 3 }}}
}}}}
}}}
1 day
1 min
1h
c So, 7 days 5 604,800 sec.
PRACTICE
Copy and complete.
1. 6 L 5 ? mL
2. 2 mi 5 ? ft
3. 80 oz 5 ? lb
4. 4 days 5 ? h
5. 77 mm 5 ? cm
6. 5 gal 5 ? qt
7. 48 ft 5 ? yd
8. 1500 mL 5 ? L
10. 125 lb 5 ? oz
11. 800 g 5 ? kg
12. 900 sec 5 ? min
13. 72 in. 5 ? ft
14. 2.5 ton 5 ? lb
15. 90 min 5 ? h
16. 65,000 mg 5 ? g
17. 100 yd 5 ? in.
18. 3.5 kg 5 ? g
19. 6 pt 5 ? qt
20. 1 week 5 ? min
21. 2 oz 5 ? lb
22. 1 km 5 ? mm
23. 1 mi 5 ? in.
24. 5 gal 5 ? c
25. 288 in.2 5 ? ft 2
26. 24 pt 5 ? gal
27. 4 kg 5 ? g
28. 7 hr 5 ? sec
9. 40 m 5 ? cm
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SKILLS REVIEW HANDBOOK
Scientific Notation
Scientific notation is a way to write numbers using powers of 10. A number is
written in scientific notation if it has the form c 3 10n where 1 ≤ c < 10 and n is
an integer. The table shows some powers of ten in order from least to greatest.
Power of Ten
1023
1022
1021
100
101
102
103
Value
0.001
0.01
0.1
1
10
100
1000
EXAMPLE
a. 12,800,000
Write the number in scientific notation.
Standard form
b. 0.0000039
Standard form
12,800,000
Move the decimal point
7 places to the left.
0.0000039
Move the decimal point
6 places to the right.
1.28 3 107
Use 7 as an exponent of 10.
3.9 3 1026
Use 26 as an exponent of 10.
EXAMPLE
a. 6.1 3 104
Write the number in standard form.
Scientific notation
b. 5.74 3 1025
Scientific notation
6.1 3 104
The exponent of 10 is 4.
5.74 3 1025
The exponent of 10 is 25.
61,000
Move the decimal point
4 places to the right.
0.0000574
Move the decimal point
5 places to the left.
61,000
Standard form
0.0000574
Standard form
PRACTICE
Write the number in scientific notation.
1. 0.6
2. 25,000,000
3. 0.08
4. 0.00542
5. 40.8
6. 7
7. 0.000385
8. 8,145,000
9. 41,236
10. 0.0000016
11. 486,000
12. 0.000000009
13. 0.01002
14. 1,000,000,000
15. 7050.5
16. 0.37
17. 9850
18. 0.0000206
19. 805
20. 0.0005
Write the number in standard form.
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21. 5 3 103
22. 4 3 1022
23. 8.2 3 1021
24. 6.93 3 102
25. 3.2 3 1023
26. 9.01 3 1025
27. 7.345 3 105
28. 2.38 3 1022
29. 1.814 3 100
30. 2.7 3 108
31. 1 3 106
32. 4.9 3 1024
33. 8 3 1026
34. 5.6 3 104
35. 1.87 3 109
36. 7 3 1024
37. 6.08 3 106
38. 9.009 3 1023
39. 3.401 3 107
40. 5.32 3 101
Student Resources
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Significant Digits
SKILLS REVIEW HANDBOOK
Significant digits indicate how precisely a number is known. Use the following
guidelines to determine the number of significant digits.
• All nonzero digits are significant.
• All zeros that appear between two nonzero digits are significant.
• For a decimal, all zeros that appear after the last nonzero digit are significant.
For a whole number, you cannot tell whether any zeros after the last nonzero
digit are significant, so you should assume that they are not significant.
Sometimes calculations involve measurements that have various numbers of
significant digits. In this case, a general rule is to carry all digits through the
calculation and then round the result to the same number of significant digits
as the measurement with the fewest significant digits. When you calculate with
units that cannot be divided into fractional parts, such as number of people,
consider only the significant digits of the other number(s).
EXAMPLE
a.
12.6
3 0.05
0.63
0.6
Perform the calculation. Write your answer with the
appropriate number of significant digits.
3 significant digits
1 significant digit
b.
840
2 significant digits
1 702
3 significant digits
The product has 2 significant
digits.
1542
The sum has 4 significant
digits.
Round to 1 significant digit.
1500
Round to 2 significant digits.
c. $61.20 restaurant bill 4 6 people
The number of people is exact, so consider only the 4 significant digits of the bill,
$61.20. The answer should have 4 significant digits.
$61.20 4 6 5 $10.20
c Each person pays $10.20.
PRACTICE
Perform the calculation. Write your answer with the appropriate number of
significant digits.
1. 600 1 30
2. 5 2 2.6
3. 12 p 6.75
4. 0.098 1 0.14 1 0.369
5. 3.6053 2 1.720
6. 40 4 3.5
7. 8.0 2 3.1
8. 31.7 p 6.8 p 0.435
9. 30.5 p 6.40
13. 4016 2 3007
10. 3.18 1 2.0005
11. 0.088 4 2.44
12. 8650 1 380 2 49
14. 1.35 1 14.8
15. 320 4 18
16. 38.1 p 3.04 4 0.024
17. $1.45 per notebook p 12 notebooks
18. 10.0 liters of water 2 4.5 liters of water
19. 260 pints of milk 4 106 students
20. 0.5 yard of fabric 1 0.87 yard of fabric
21. 27,973 books 4 11 libraries
22. 12.76 gallons of gas 1 6.08 gallons of gas
23. $6.95 per ticket p 180 tickets
24. 1540 pounds 2 160 pounds 2 85 pounds
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SKILLS REVIEW HANDBOOK
Writing Algebraic Expressions
To solve a problem using algebra, you often need to write a phrase as an
algebraic expression.
EXAMPLE
Write the phrase as an algebraic expression.
a. 6 less than a number
b. The cube of a number
c. Double a number
“Less than” indicates
subtraction.
“Cube” indicates raising
to the third power.
“Double” indicates
multiplication by 2.
cn26
c n3
c 2n
EXAMPLE
Write an algebraic expression to answer the question.
a. Rebecca walks three times as far to school as Meghan does. If Meghan
walks m blocks to school, how many blocks to school does Rebecca walk?
c 3m
b. Kate is 8 inches taller than Noah. If Noah is n inches tall, how tall is Kate?
cn18
PRACTICE
Write the phrase as an algebraic expression.
1. 8 more than a number
2. 10 times a number
3. Twice a number
4. 6 less than a number
5. One fifth of a number
6. 4 greater than a number
7. 5 times a number
8. A number squared
9. 25% of a number
10. Half a number
11. 2 less than a number
12. The square root of a number
Write an algebraic expression to answer the question.
13. Allison is 4 years younger than her sister Camille. If Camille is c years old,
how old is Allison?
14. Ryan bought a movie ticket for x dollars. He paid with a $20 bill. How much
change should Ryan get?
15. Bridget spent $5 more than Tom spent at the mall. If Tom spent x dollars,
how much did Bridget spend?
16. Marc has twice as many baseball cards as hockey cards. If Marc has h hockey
cards, how many baseball cards does he have?
17. Elizabeth’s ballet class is 45 minutes long. If Elizabeth is m minutes late for
ballet class, how many minutes will she spend in class?
18. Steve drove x miles per hour for 5 hours. How many miles did Steve drive?
19. Wendy bought 10 pens priced at x dollars each. How much did she spend?
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Binomial Products
EXAMPLE
Simplify (2x 1 1)(x 1 3).
Draw a rectangle with dimensions 2x 1 1 and x 1 3. Use the
dimensions to divide the rectangle into parts. Then find the
area of each part. The binomial product (2x 1 1)(x 1 3) is
the sum of the areas of all the parts.
2
There are 2 blue parts with area x , 7 green parts with area x,
and 3 yellow parts with area 1.
x
2x 1 1
x
1
x
x2
x2
x
1
1
1
x
x
x
x
x
x
1
1
1
x13
(2x 1 1)(x 1 3) 5 2x2 1 7x 1 3
SKILLS REVIEW HANDBOOK
A monomial is a number, a variable, or the product of a number and one or more
variables. A binomial is the sum of two monomials. In other words, a binomial is
a polynomial with two terms. You can use a geometric model to find the product
of two binomials.
Another way to find the product of two binomials is to use the distributive
property systematically. Multiply the first terms, the outer terms, the inner terms,
and the last terms of the binomials. This is called FOIL for the words First, Outer,
Inner, and Last.
EXAMPLE
Simplify (x 1 2)(4x 2 5).
First
Outer Inner
Last
(x 1 2)(4x 2 5) 5 x(4x) 1 x(25) 1 2(4x) 1 2(25)
2
Use FOIL.
5 4x 2 5x 1 8x 2 10
Multiply.
5 4x2 1 3x 2 10
Combine like terms.
PRACTICE
Simplify.
1. (a 1 5)(a 1 3)
2. (m 1 4)(m 1 11)
3. (t 1 8)(t 1 7)
4. (z 1 1)(z 1 6)
5. (y 1 4)(y 1 2)
6. (x 1 9)(x 1 9)
7. (y 2 2)
2
8. (n 1 6)
2
9. (4 2 z)2
10. (a 1 10)(a 2 10)
11. (y 1 3)(y 2 7)
12. (k 1 1)2
13. (5x 2 4)(5x 1 4)
14. (3 1 n)2
15. (c 1 5)(2c 2 7)
16. (a 1 5)(a 1 5)
17. (7 2 z)(7 1 z)
18. (3x 2 8)(x 2 6)
20. (3 2 g)(2g 1 3)
21. (4 2 x)(8 1 x)
22. (3n 2 1)(n 2 4)
23. (2a 1 9)(a 2 9)
24. (8x 1 1)(x 1 1)
25. (5x 1 2)(2x 2 5)
26. (2d 2 5)(3d 2 1)
27. (24z 1 3)(6z 2 1)
19. (4a 1 3)
2
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SKILLS REVIEW HANDBOOK
LCDs of Rational Expressions
A rational expression is a fraction whose numerator and denominator are
nonzero polynomials. The least common denominator (LCD) of two rational
expressions is the least common multiple of the denominators. To find the LCD,
follow these three steps:
STEP 1
Write each denominator as the product of its factors.
STEP 2 Write the product consisting of the highest power of each factor that
appears in either denominator.
STEP 3 Simplify the product from Step 2 to write the LCD.
EXAMPLE
Find the least common denominator of the rational expressions.
3 and 1
b. }}
}}
12x
8x 2
2 and 2
a. }}
}}3
5xy
y
STEP 1
Factors:
Factors:
5xy 5 5 p x p y
8x2 5 23 p x2
3
y 5y
3
Factors:
2
STEP 2 Product: 5 p x p y 3
STEP 3 LCD: 5xy
x
21 and
c. }}}
}}}}}}
3x 1 6
x2 2 3x 2 10
3
3x 1 6 5 3 p (x 1 2)
2
12x 5 2 p 3 p x
x 2 3x 2 10 5 (x 1 2) p (x 2 5)
Product: 23 p 3 p x2
Product: 3 p (x 1 2) p (x 2 5)
LCD: 24x
2
LCD: 3(x 1 2)(x 2 5)
PRACTICE
Find the least common denominator of the rational expressions.
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1 and 4
1. }}
}}
2ab
a2
5 and 6
2. }}
}}
6k 2
7k 2
2 and 2
3. }}
}}
z3
z2
4 and 23
4. }}
}}
5x
10x
m and 1
5. }}
}}}
14
18m
19 and 3
6. }}}
}}}
20xy
16xy
1 and 1
7. }}
}}
3y
3y 2
24 and 2
8. }}}
}}}
9ab2
21a2b
n and n2
9. }}}
}}}
n12
n22
21 and 3
10. }}}
}}}
x21
x13
28 and 4
11. }}}
}}}
5n 1 5
n11
y
1
12. } and }}}
8
2y 1 8
1
2
13. }}}}
and }}}}
2m 2 6
3m 2 9
a and 2a
14. }}
}}}}
n2
n2 2 6n
1 and
1
15. }}}
}}}}
x24
(x 2 4)2
3
4
16. }}}}
and }}}}
4x 1 12
6x 1 18
29
1 and
17. }}
}}}}}
2n3
10n2 1 8n
10
17b
18. }}}}
and }}}}
15b 2 30
9b 2 18
25 and
3
19. }}}}
}}}}
(k 1 3)4
(k 1 3)2
8
1 and
20. }}}
}}}}
y25
3y 2 15
n2
n
21. }}}}}
and }}}}
10n 1 20
7n 1 14
20
1
22. }}}}
and }}}}
5z 2 40
9z 2 56
2a
2
23. }}}}}}
and }}}
a12
a2 1 4a 1 4
1
21
24. }}}
and }}}}}
2z 2 6
z2 2 z 2 6
3k and
2k
25. }}}
}}}}}}
k23
k 2 2 5k 1 6
x
2x
26. }}}
and }}}}}}
x2 2 9
x 2 1 3x 2 18
m2
25
27. }}}}}}}
and }}}}}}}
2
m 2 11m 1 28
m2 1 5m 2 45
Student Resources
11/21/05 10:26:58 AM
The Coordinate Plane
Each point in a coordinate plane is represented by an ordered
pair. The first number is the x-coordinate, and the second
number is the y-coordinate.
The ordered pair (3, 1) is graphed at the right. The x-coordinate
is 3, and the y-coordinate is 1. So, the point is right 3 units and
up 1 unit from the origin.
EXAMPLE
y-axis
Quadrant II 4
(2, 1)
3
origin 2
(0, 0)
y
Quadrant I
(1, 1)
(3, 1)
1 2 3 4 5 6x
262524232221
21
22
23
(2, 2)
Quadrant III 24
x-axis
(1, 2)
Quadrant IV
SKILLS REVIEW HANDBOOK
A coordinate plane is formed by the intersection of a horizontal
number line called the x-axis and a vertical number line called
the y-axis. The axes meet at a point called the origin and divide
the coordinate plane into four quadrants, numbered I, II, III,
and IV.
Graph the points A(2, 21) and B(24, 0) in a coordinate plane.
A(2, 21) Start at the origin.
The x-coordinate is 2, so move right 2 units.
The y-coordinate is 21, so move down 1 unit.
Draw a point at (2, 21) and label it A.
B(24, 0) Start at the origin.
The x-coordinate is 24, so move left 4 units.
The y-coordinate is 0, so move up 0 units.
Draw a point at (24, 0) and label it B.
4
3
2
1
B(24, 0)
y
3 4 5 6x
1
262524232221
21
22
23
24
A(2, 21)
PRACTICE
Graph the points in a coordinate plane.
1. A(7, 2)
2. B(6, 27)
3. C(2, 23)
4. D(28, 0)
5. E(24, 28)
6. F(1, 3)
7. G(3, 0)
8. H(1, 25)
9. I(0, 22)
10. J(26, 5)
11. K(5, 8)
12. L(8, 22)
13. M(23, 24)
14. N(27, 8)
15. P(25, 1)
16. Q(22, 26)
17. R(0, 6)
18. S(24, 21)
19. T(4, 4)
20. V(23, 7)
Give the coordinates and the quadrant or axis of the point.
21. A
24. D
22. B
25. E
23. C
5
26. F
27. G
28. H
29. J
30. K
31. L
32. M
33. N
34. O
35. P
36. Q
37. R
38. S
39. T
40. U
41. V
42. W
43. X
44. Y
T 4
F
3
N
2
1
D
y
A
M
G
S
W
U
E
L
26
24
22
O
R
V
H
3 4 5 6x
1
K
J
B23 X
Œ
24
Y
P
C
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SKILLS REVIEW HANDBOOK
Transformations
A transformation is a change made to the position or to the size
of a figure. Each point (x, y) of the figure is mapped to a new point,
and the new figure is called an image.
A translation is a transformation in which each point of a figure
moves the same distance in the same direction. A figure and its
translated image are congruent.
EXAMPLE
Translation a Units Horizontally
and b Units Vertically
(x, y) → (x 1 a, y 1 b)
Translate }
FG right 3 units and down 1 unit.
y
13
F
21
F9
To move right 3 units, use a 5 3. To move down 1 unit, use
b 5 21. So, use (x, y) → (x 1 3, y 1 (21)) with each endpoint.
1
F(2, 4) → F9(2 1 3, 4 1 (21)) 5 F9(5, 3)
G(1, 1) → G9(1 1 3, 1 1 (21)) 5 G9(4, 0)
G
G9
1
x
Graph the endpoints (5, 3) and (4, 0). Then draw the image.
A reflection is a transformation in which a figure is
reflected, or flipped, in a line, called the line of reflection.
A figure and its reflected image are congruent.
EXAMPLE
Reflection in x-axis
Reflection in y-axis
(x, y) → (x, 2y)
(x, y) → (2x, y)
Reflect n ABC in the y-axis.
y
A9
A
B9 B
Use (x, y) → (2x, y) with each vertex.
A(4, 3) → A9(24, 3)
B(1, 2) → B9(21, 2)
C(3, 1) → C9(23, 1)
1
C9
Change each
x-coordinate
to its opposite.
C
x
1
Graph the new vertices. Then draw the image.
A rotation is a transformation in which a figure is turned
about a fixed point, called the center of rotation. The
direction can be clockwise or counterclockwise. A figure
and its rotated image are congruent.
EXAMPLE
Rotation About the Origin
1808 either direction
(x, y) → (2x, 2y)
908 clockwise
(x, y) → (y, 2x)
908 counterclockwise
(x, y) → (2y, x)
Rotate RSTV 1808 about the origin.
y
R(2, 2) → R9(22, 22)
S(4, 2) → S9(24, 22)
T(4, 1) → T9(24, 21)
V(1, 0) → V9(21, 0)
Change every
coordinate
to its opposite.
R
2
Use (x, y) → (2x, 2y) with each vertex.
S
V9
S9
T
V
T9
2
x
R9
Graph the new vertices. Then draw the image.
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EXAMPLE
Dilation with Scale Factor k
with Respect to the Origin
SKILLS REVIEW HANDBOOK
A dilation is a transformation in which a figure stretches
or shrinks depending on the dilation’s scale factor. A figure
stretches if k > 1 and shrinks if 0 < k < 1. A figure and its
dilated image are similar.
(x, y) → (kx, ky)
Dilate JKLM using a scale factor of 0.5.
The scale factor is k 5 0.5, so multiply every coordinate by 0.5.
Use (x, y) → (0.5x, 0.5y) with each vertex.
J(4, 4) → J9(0.5 p 4, 0.5 p 4) 5 J9(2, 2)
K(6, 4) → K9(0.5 p 6, 0.5 p 4) 5 K9(3, 2)
L(6, 21) → L9(0.5 p 6, 0.5 p (21)) 5 L9(3, 20.5)
M(4, 21) → M9(0.5 p 4, 0.5 p (21)) 5 M9(2, 20.5)
y
J
J9
K
K9
1
M9
L9 5
M
x
L
Graph the new vertices. Then draw the image.
PRACTICE
Find the coordinates of N(23, 8) after the given transformation. For rotations,
rotate about the origin.
1. Rotate 1808.
2. Reflect in x-axis.
3. Translate up 3 units.
4. Reflect in y-axis.
5. Rotate 908 clockwise.
6. Translate left 5 units.
7. Rotate 908 counterclockwise.
8. Translate right 2 units and down 9 units.
Transform n PST. Graph the result. For rotations, rotate about
the origin.
9. Reflect in x-axis.
y
T
10. Rotate 908 counterclockwise.
11. Rotate 908 clockwise.
12. Translate down 7 units.
13. Reflect in y-axis.
14. Translate left 4 units.
15. Rotate 1808.
16. Translate right 2 units.
x
1
22
P
S
17. Translate right 1 unit and up 4 units.
18. Translate left 6 units and up 2 units.
The coordinates of the vertices of a polygon are given. Draw the polygon. Then
find the coordinates of the vertices of the image after the specified dilation, and
draw the image.
19. (1, 3), (3, 2), (2, 5); dilate using a scale factor of 3
3
20. (2, 8), (2, 4), (6, 8), (6, 4); dilate using a scale factor of }
2
1
21. (3, 3), (6, 3), (3, 23), (6, 23); dilate using a scale factor of }
3
22. (2, 2), (2, 7), (5, 7); dilate using a scale factor of 2
1
23. (2, 22), (6, 22), (4, 26), (0, 26); dilate using a scale factor of }
2
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SKILLS REVIEW HANDBOOK
Line Symmetry
A figure has line symmetry if a line, called a line of symmetry, divides the figure
into two parts that are mirror images of each other. Below are four figures with
their lines of symmetry shown in red.
Trapezoid
No lines of symmetry
EXAMPLE
Isosceles Triangle
1 line of symmetry
Rectangle
2 lines of symmetry
Regular Hexagon
6 lines of symmetry
A line of symmetry for the figure is shown in red.
Find the coordinates of point A.
Point A is the mirror image of the point (3, 26) with respect
to the line of symmetry y 5 22. The x-coordinate of A is 3,
the same as the x-coordinate of (3, 26). Because 26 is the
y-coordinate of (3, 26), and 22 2 (26) 5 4, the point (3, 26)
is down 4 units from the line of symmetry. Therefore, point A
must be up 4 units from the line of symmetry. So, the
y-coordinate of A is 22 1 4 5 2. The coordinates of point A
are (3, 2).
y
A
1
x
2
y 5 22
C
B(3, 26)
PRACTICE
Tell how many lines of symmetry the figure has.
1.
2.
3.
4.
5. A parallelogram
6. A square
7. A rhombus
8. An equilateral triangle
A line of symmetry for the figure is shown in red. Find the coordinates of
point A.
9.
4
(24, 3)
10.
y
11.
y
(0, 4)
A
1x
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y5x
1
y51
A
y
A
x52
1
1
1
x
x
(2, 22)
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Perimeter and Area
SKILLS REVIEW HANDBOOK
The perimeter P of a figure is the distance around it. To find the perimeter of a
figure, add the side lengths.
EXAMPLE
Find the perimeter of the figure.
a.
b.
13 in.
5 in.
18 m
4m
4m
18 m
12 in.
P 5 5 1 12 1 13 5 30 in.
P 5 2(4) 1 2(18) 5 8 1 36 5 44 m
The area A of a figure is the number of square units enclosed by the figure.
Area of a Triangle
Area of a Rectangle
Area of a Parallelogram
Area of a Trapezoid
b1
h
w
h
l
b
1
2
b
A 5 lw
A 5 }bh
EXAMPLE
h
b2
1
2
A 5 }(b1 1 b2)h
A 5 bh
Find the area of the figure.
a.
b.
c.
7 in.
6m
5 ft
3m
15 in.
A 5 (15)(7) 5 105 in.2
A 5 (5)(5) 5 25 ft 2
1 (6)(3) 5 9 m 2
A5}
2
PRACTICE
Find the perimeter and area of the figure.
1.
2.
8 ft
3 cm
17 ft
3.
3 in.
15 ft
4.
4 in.
5 in.
5m
12 in.
6m
2 cm
5.
6.
10 yd
8 yd
7.
8.
12 mm
17 yd
8 in.
2.7 m
3m
9 mm
9 mm
21 yd
4m
12 mm
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SKILLS REVIEW HANDBOOK
Circumference and Area of a Circle
A circle consists of all points in a plane that are the same distance from a fixed
point called the center.
The distance between the center and any point on the circle is the radius. The
distance across the circle through the center is the diameter. The diameter is
twice the radius.
circle
radius
diameter
center
The circumference of a circle is the distance around the circle. For any circle,
the ratio of the circumference to the diameter is π (pi), an irrational number
22 .
that is approximately 3.14 or }}
7
To find the circumference C of a circle with radius r, use the formula C 5 2πr.
To find the area A of a circle with radius r, use the formula A 5 πr 2.
EXAMPLE
Find the circumference and area of a circle with radius 6 cm.
Give an exact answer and an approximate answer for each.
Circumference
Area
C 5 2πr
A 5 πr 2
5 2π(6)
5 π(6)2
5 12π
5 36π
< 12(3.14)
< 36(3.14)
< 37.7
< 113
c The circumference is 12π centimeters,
or about 37.7 centimeters.
6 cm
c The area is 36π square centimeters,
or about 113 square centimeters.
PRACTICE
Find the circumference and area of the circle. Give an exact answer and an
approximate answer for each.
1.
2.
3.
4.
5 in.
10 m
2 cm
5.
6.
4 in.
7.
8.
6 ft
12 ft
16 m
9 cm
9.
10.
2 cm
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11.
14 ft
12.
22 in.
36 cm
Student Resources
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Surface Area and Volume
SKILLS REVIEW HANDBOOK
A solid is a three-dimensional figure that encloses part of space.
The surface area S of a solid is the area of the solid’s outer surface(s).
The volume V of a solid is the amount of space that the solid occupies.
Cylinder
Rectangular Prism
h
S 5 2lw 1 2lh 1 2wh
V 5 lwh
w
l
EXAMPLE
r
2
S 5 2πr 1 2πrh
h
V 5 πr 2h
Find the surface area and volume of the rectangular prism.
Surface area
Volume
S 5 2lw 1 2lh 1 2wh
V 5 lwh
7 ft
5 2(5)(3) 1 2(5)(7) 1 2(3)(7)
5 (5)(3)(7)
5 30 1 70 1 42
5 105 ft 3
5 ft
3 ft
5 142 ft 2
EXAMPLE
Find the surface area and volume of the cylinder.
Surface area
3m
Volume
2
12 m
2
V 5 πr h
S 5 2πr 1 2πrh
2
5 2π(3) 1 2π(3)(12)
5 π(3)2 (12)
5 90π m 2
Exact answer
5 108π m3
Exact answer
< 283 m 2
Approximate answer
< 339 m3
Approximate answer
PRACTICE
Find the surface area and volume of the solid.
1.
2.
6.5 mm
3.
3 in.
3 cm
12 mm
5 in.
3 cm
3 cm
8 in.
4.
2m
4m
5.
6.
14 yd
10 ft
4 yd
10 m
15 ft
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SKILLS REVIEW HANDBOOK
Angle Relationships
An angle bisector is a ray that divides an angle into two congruent angles.
Two angles are complementary angles if the sum of their measures is 908.
Two angles are supplementary angles if the sum of their measures is 1808.
EXAMPLE
Find the value of x.
]›
a. BD bisects ∠ ABC
and m ∠ ABC 5 648.
C
b. ∠ GFJ and ∠ HFJ are
J
D
x8
648
]›
Because BD bisects
∠ ABC, the value of
x is half m ∠ ABC.
64 5 32
x 5 }}
2
D
4x 8
(3x 2 1)8
F
G
A
supplementary.
H
(2x 2 6)8
B
c. ∠ CBD and ∠ ABD are
complementary.
E
C
Because ∠ GFJ and ∠ HFJ
are complementary
angles, their sum is 908.
(x 2 3)8
B
A
Because ∠ CBD and ∠ ABD
are supplementary angles,
their sum is 1808.
(2x 2 6) 1 4x 5 90
(3x 2 1) 1 (x 2 3) 5 180
6x 2 6 5 90
4x 2 4 5 180
x 5 16
x 5 46
PRACTICE
]›
BD is the angle bisector of ∠ ABC. Find the value of x.
1.
2.
A
D
3.
A
D
248
788
x8
C
B
A
(11x 2 19)8
(8x 1 5)8
B
(2x 2 4)8
B
D
C
C
∠ ABD and ∠ DBC are complementary. Find the value of x.
4.
5.
A
6. B
D
A
A
(3x 2 18)8
(3x 2 4)8
(7x 1 5)8
D
(4x 1 10)8
B
D
(5x 2 20)8
C
(5x 1 1)8
C
C
B
∠ ABD and ∠ DBC are supplementary. Find the value of x.
7.
8.
D
(x 2 28)8
(4x 1 17)8 (3x 1 2)8
A
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B
A
C
9.
D
D
3x 8
B
(4x 1 2)8 (2x 1 4)8
C
A
B
C
Student Resources
11/21/05 10:27:08 AM
Triangle Relationships
SKILLS REVIEW HANDBOOK
The sum of the angle measures of any triangle is 1808.
EXAMPLE
Find the value of x.
60 1 35 1 x 5 180
x8
95 1 x 5 180
608
358
x 5 85
The sum of the angle measures is 1808.
Simplify.
Solve for x.
In a right triangle, the hypotenuse is the side opposite the
right angle. The legs are the sides that form the right angle.
The Pythagorean theorem states that the sum of the squares
of the lengths of the legs equals the square of the length of
the hypotenuse.
Pythagorean Theorem
a2 1 b2 5 c 2
c
a
b
EXAMPLE
Find the value of x.
a.
b.
12 cm
x
6 ft
x
13 cm
8 ft
6 2 1 82 5 x 2
2
Simplify.
100 5 x2
Simplify.
36 1 64 5 x
x 5 10 ft
x2 1 122 5 132
Pythagorean theorem
Pythagorean theorem
2
x 1 144 5 169
Simplify.
x2 5 25
Solve for x2 .
x 5 5 cm
Solve for x.
Solve for x.
PRACTICE
Find the value of x.
1.
2.
688
378
3.
x8
4.
x8
348
728
x8
x8
5.
6.
40 cm
x
568
x8
7.
8 in.
8 in.
x
x8
37 ft
35 ft
30 cm
8.
5m
6m
x
x
9. A triangle with angles that measure x8, x8, and 708
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Congruent and Similar Figures
Two figures are congruent if they have the same shape and the same size.
If two figures are congruent, then corresponding angles are congruent
and corresponding sides are congruent. The triangles at the right are
congruent. Matching arcs show congruent angles, and matching tick
marks show congruent sides.
Two figures are similar if they have the same shape but not necessarily the same
size. If two figures are similar, then corresponding angles are congruent and the
ratios of the lengths of corresponding sides are equal.
EXAMPLE
Tell whether the figures are congruent, similar, or neither.
a.
7
3
3
3
3
7
7
b.
10
11
As shown, corresponding
angles are congruent, but
corresponding sides have
different lengths. So, the
figures are not congruent,
but they may be similar.
F 3.75 E
A 3 B
D
As shown, corresponding
angles are congruent and
corresponding sides are
congruent. So, the figures
are congruent.
7
6
7.5
C
G
12.5
13.75
H
The figures are similar if the ratios of the lengths of corresponding sides are equal.
BC
FG
3
3.75
AB
EF
}} 5 }} 5 0.8
6
7.5
}} 5 }} 5 0.8
CD
GH
11
13.75
}} 5 }}} 5 0.8
AD
EH
10
12.5
}} 5 }} 5 0.8
c Because corresponding angles are congruent and the ratios of the lengths
of corresponding sides are equal, ABCD is similar to EFGH.
EXAMPLE
The two polygons are similar. Find the value of x.
a.
608
x8
The angle with measure x°
corresponds to the angle with
measure 60°, so x 5 60.
308
b.
8
12
x
9
The side with length 12 corresponds to
the side with length 8, and the side
with length 9 corresponds to the side
with length x.
12
8
9
x
}} 5 }
12x 5 72
x56
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Write a proportion.
Cross multiply.
Solve for x.
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PRACTICE
SKILLS REVIEW HANDBOOK
Tell whether the figures are congruent, similar, or neither. Explain.
1.
2.
4
2
6
14
11
3.5
2
3.
1
10
7
14
3
4.
5.
7
6.
8
12
9
9
7
7
12
8
8.
6
4.5
9
1.5
12
1.6
3.2
5
1.5
3
9.
2.4
4.8
4
3
3
8
8
5
7.
12
6
4.8
7
3
3.2
7
1.6
3
3
2.4
The two polygons are similar. Find the value of x.
10.
11.
438
7
x21
12.
12
14
20
18
8
4x 1 3
x8
13.
14.
1138
34.5
27
15.
568
568
678
(5x 2 3)8
16.
18
17.
538
568
568 1248
5x 2 7
538
18.
1198
618
1198
36
15
30
x15
(3x 1 4)8
(11x 2 5)8
1198
618
(7x 1 4)8
538
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More Problem Solving Strategies
Problem solving strategies can help you solve mathematical and real-life
problems. Lesson 1.5 shows how to apply the strategies use a formula, look for a
pattern, draw a diagram, and use a verbal model. Below are four more strategies.
Strategy
When to Use
How to Use
Make a list or table
Make a list or table when a problem
requires you to record, generate, or
organize information.
Make a table with columns, rows, and any
given information. Generate a systematic list
that can help you solve the problem.
Work backward
Work backward when a problem
gives you an end result and you
need to find beginning conditions.
Work backward from the given information
until you solve the problem. Work forward
through the problem to check your answer.
Guess, check, and revise
Guess, check, and revise when you
need a place to start or you want to
see how the problem works.
Make a reasonable guess. Check to see if
your guess solves the problem. If it does not,
revise your guess and check again.
Solve a simpler problem
Solve a simpler problem when a
problem can be made easier by
using simpler numbers.
Think of a way to make the problem simpler.
Solve the simpler problem, then use what
you learned to solve the original problem.
EXAMPLE
Lee works as a cashier. In how many different ways can Lee
make $.50 in change using quarters, dimes, and nickels?
Use the strategy make a list or table. Then count the number of different ways.
Quarters
Dimes
Nickels
2
0
0
1
2
1
1
1
3
1
0
5
0
5
0
0
4
2
0
3
4
0
2
6
0
1
8
0
0
10
Start with the greatest number of quarters.
Then list all the possibilities with 1 quarter,
starting with the greatest number of dimes.
Then list all the possibilities with 0 quarters,
starting with the greatest number of dimes.
c Lee can make $.50 in quarters, dimes, and nickels in 10 different ways.
EXAMPLE
In a cafeteria, 3 cookies cost $.50 less than a sandwich. If a
sandwich costs $4.25, how much does one cookie cost?
Use the strategy work backward.
4.25 2 0.50 5 3.75
Cost of 3 cookies
3.75 4 3 5 1.25
Cost of 1 cookie
CHECK 1.25 3 3 5 3.75
Cost of 3 cookies
3.75 1 0.50 5 4.25 Cost of sandwich
c One cookie costs $1.25.
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Nolan’s class has 6 more boys than girls. There are
28 students altogether. How many girls are in Nolan’s class?
Use the strategy guess, check, and revise. Guess a number of girls that is less than
half of 28.
First guess:
12 girls, 12 1 6 5 18 boys, 12 1 18 5 30 students
Too high ✗
Second guess: 10 girls, 10 1 6 5 16 boys, 10 1 16 5 26 students
Too low ✗
Third guess:
Correct ✓
11 girls, 11 1 6 5 17 boys, 11 1 17 5 28 students
c There are 11 girls in Nolan’s class.
EXAMPLE
SKILLS REVIEW HANDBOOK
EXAMPLE
How many diagonals does a regular decagon have?
Use the strategy solve a simpler problem. A decagon has 10 sides, so find the
number of diagonals of polygons with fewer sides and look for a pattern.
3 sides
0 diagonals
4 sides
2 diagonals
5 sides
5 diagonals
6 sides
9 diagonals
7 sides
14 diagonals
Notice that the difference of the numbers of diagonals for consecutive figures
keeps increasing by 1:
22052
52253
92554
14 2 9 5 5
So, an 8-sided polygon has 14 1 6 5 20 diagonals, a 9-sided polygon has
20 1 7 5 27 diagonals, and a 10-sided polygon has 27 1 8 5 35 diagonals.
c A regular decagon (a 10-sided polygon) has 35 diagonals.
PRACTICE
1. Ben has a concert at 7:30 P.M. First he must do 2 hours of homework. Then,
dinner and a shower will take about 45 minutes. Ben wants to allow a half
hour to get to the concert. What time should Ben start his homework?
2. Quinn and Kyle collected 87 aluminum cans to recycle. Quinn collected
twice as many cans as Kyle. How many cans did each person collect?
3. In how many different ways can three sisters form a line at a ticket booth?
4. The 8 3 8 grid at the right has some 1 3 1 squares, some 2 3 2 squares, some
3 3 3 squares, and so on. How many total squares does the grid have?
5. If Kaleigh draws 20 different diameters in a circle, into how many parts will
the circle be divided?
6. Six friends form a tennis league. Each friend will play a match with every
other friend. How many matches will be played?
7. Susan has 13 coins in her pocket with a total value of $1.05. She has only
dimes and nickels. How many of each type of coin does Susan have?
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Logical Argument
A logical argument has two given statements, called premises, and a statement,
called a conclusion, that follows from the premises. Below is an example.
Premise 1
Premise 2
Conclusion
If a triangle has a right angle, then it is a right triangle.
In nABC, ∠B is a right angle.
nABC is a right triangle.
Letters are often used to represent the statements of a logical argument and
to write a pattern for the argument. The table below gives five types of logical
arguments. In the examples, p, q, and r represent the following statements.
p: a figure is a square
Type of Argument
q: a figure is a rectangle
Pattern
r: a figure is a parallelogram
Example
Direct Argument
If p is true, then q is true.
p is true.
Therefore, q is true.
If ABCD is a square, then it is a rectangle.
ABCD is a square.
Therefore, ABCD is a rectangle.
Indirect Argument
If p is true, then q is true.
q is not true.
Therefore, p is not true.
If ABCD is a square, then it is a rectangle.
ABCD is not a rectangle.
Therefore, ABCD is not a square.
Chain Rule
If p is true, then q is true.
If q is true, then r is true.
Therefore, if p, then r.
If ABCD is a square, then it is a rectangle. If ABCD is
a rectangle, then it is a parallelogram. Therefore, if
ABCD is a square, then it is a parallelogram.
Or Rule
p is true or q is true.
p is not true.
Therefore, q is true.
ABCD is a square or a rectangle.
ABCD is not a square.
Therefore, ABCD is a rectangle.
And Rule
p and q are not both true.
q is true.
Therefore, p is not true.
ABCD is not both a square and a rectangle.
ABCD is a rectangle.
Therefore, ABCD is not a square.
An argument that follows one of these patterns correctly has a valid conclusion.
EXAMPLE
State whether the conclusion is valid or invalid. If the
conclusion is valid, name the type of logical argument used.
a. If it is raining at noon, Peter’s family will not have a picnic lunch. Peter’s
family had a picnic lunch. Therefore, it was not raining at noon.
c The conclusion is valid. This is an example of indirect argument.
b. If a triangle is equilateral, then it is an acute triangle. Triangle XYZ is an acute
triangle. Therefore, triangle XYZ is equilateral.
c The conclusion is invalid.
c. If x 5 4, then 2x 2 7 5 1. If 2x 2 7 5 1, then 2x 5 8. x 5 4. Therefore, if x 5 4, then
2x 5 8.
c The conclusion is valid. This is an example of the chain rule.
d. If it is at least 808F outside today, you will go swimming. It is 858F outside today.
Therefore, you will go swimming.
c The conclusion is valid. This is an example of direct argument.
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A compound statement has two or more parts joined by or or and.
• For an or compound statement to be true, at least one part must be true.
EXAMPLE
State whether the compound statement is true or false.
a. 12 < 20 and 212 > 220
True
b. 2 < 4 and 4 < 3
True
True
c True, because each part is true.
c. 10 > 0 or 210 > 0
True
False
c False, because one part is false.
d. 28 > 27 or 27 > 26 or 26 > 25
False
False
c True, because at least one part is true.
False
SKILLS REVIEW HANDBOOK
• For an and compound statement to be true, each part must be true.
False
c False, because every part is false.
PRACTICE
State whether the conclusion is valid or invalid. If the conclusion is valid, name
the type of logical argument used.
1. If Scott goes to the store, then he will buy sugar. If he buys sugar, then he will
bake cookies. Scott goes to the store. Therefore, he will bake cookies.
2. If a triangle has at least two congruent sides, then it is isosceles. Triangle MNP
has sides 5 in., 6 in., and 5 in. long. Therefore, triangle MNP is isosceles.
3. If a horse is an Arabian, then it is less than 16 hands tall. Andrea’s horse is
13 hands tall. Therefore, Andrea’s horse is an Arabian.
4. If a figure is a rhombus, then it has four sides. Figure WXYZ has four sides.
Therefore, WXYZ is a rhombus.
5. Jeff cannot buy both a new coat and new boots. Jeff decides to buy new
boots. Therefore, Jeff cannot buy a new coat.
6. If x 5 0, then y 5 4. If y 5 4, then z 5 7. Therefore, if z 5 7, then x 5 0.
7. Kate will order either tacos or burritos for lunch. Kate does not order tacos for
lunch. Therefore, Kate orders burritos for lunch.
8. If a triangle is equilateral, then it is equiangular. Triangle ABC is not
equiangular. Therefore, triangle ABC is not equilateral.
9. An animal cannot be both a fish and a bird. Courtney’s pet is not a fish.
Therefore, Courtney’s pet must be a bird.
State whether the compound statement is true or false.
10. 27 < 25 and 25 < 26
11. 6 > 2 or 8 < 4
12. 0 ≤ 21 or 5 ≥ 5
13. 4 ≤ 3 or 12 ≥ 13
14. 3 < 5 and 23 < 25
15. 1 5 21 or 1 5 1 or 1 5 0
16. 7 < 8 and 8 < 12
17. 22 < 2 and 3 ≥ 2
18. 3(24) 5 12 or 23(4) 5 12
19. 28 > 8 or 28 5 8 or 28 ≥ 0
20. 140 Þ 145 or 140 > 2145 or 2140 < 2145
21. 28(9) 5 272 and 8(29) 5 272
22. 22 ≤ 23 and 222 < 223 and 23 > 22
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Conditional Statements and
Counterexamples
A conditional statement has two parts, a hypothesis and a conclusion. When
a conditional statement is written in if-then form, the “if” part contains the
hypothesis and the “then” part contains the conclusion. An example of a
conditional statement is shown below.
If a triangle is equiangular, then each angle of the triangle measures 608.
Hypothesis
Conclusion
The converse of a conditional statement is formed by switching the hypothesis
and the conclusion. The converse of the statement above is as follows:
If each angle of a triangle measures 608, then the triangle is equiangular.
EXAMPLE
Rewrite the conditional statement in if-then form. Then write
its converse and tell whether the converse is true or false.
a. Bob will earn $20 by mowing the lawn.
If-then form: If Bob mows the lawn, then he will earn $20.
Converse: If Bob earns $20, then he mowed the lawn. False
b. x 5 8 when 5x 1 1 5 41.
If-then form: If 5x 1 1 5 41, then x 5 8.
Converse: If x 5 8, then 5x 1 1 5 41. True
A biconditional statement is a statement that has the words “if and only if.” You
can write a conditional statement and its converse together as a biconditional
statement.
A triangle is equiangular if and only if each angle of the triangle measures 608.
A biconditional statement is true only when the conditional statement and its
converse are both true.
EXAMPLE
Tell whether the biconditional statement is true or false.
Explain.
a. An angle measures 90° if and only if it is a right angle.
Conditional: If an angle is a right angle, then it measures 908. True
Converse: If an angle measures 908, then it is a right angle. True
c The biconditional statement is true because the conditional and its converse
are both true.
b. Bonnie has $.50 if and only if she has two quarters.
Conditional: If Bonnie has two quarters, then she has $.50. True
Converse: If Bonnie has $.50, then she has two quarters. False
c The biconditional statement is false because the converse is not true.
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A counterexample is an example that shows that a statement is false.
SKILLS REVIEW HANDBOOK
EXAMPLE
Tell whether the statement is true or false. If false,
give a counterexample.
a. If a polygon has four sides and opposite sides are parallel, then
it is a rectangle.
c False. A counterexample is the parallelogram shown.
b. If x2 5 49, then x 5 7.
c False. A counterexample is x 5 27, because (27)2 5 49.
PRACTICE
Rewrite the conditional statement in if-then form. Then write its converse and
tell whether the converse is true or false.
1. The graph of the equation y 5 mx 1 b is a line.
2. You will earn $35 for working 5 hours.
3. Abby can go swimming if she finishes her homework.
4. In a right triangle, the sum of the squares of the lengths of the
legs equals the square of the length of the hypotenuse.
5. x 5 5 when 4x 1 8 5 28.
6. The sum of two even numbers is an even number.
Tell whether the biconditional statement is true or false. Explain.
7. Two lines are perpendicular if and only if they intersect to form a right angle.
8. x 3 5 27 if and only if x 5 3.
9. A vegetable is a carrot if and only if it is orange.
10. A rhombus is a square if and only if it has four right angles.
11. The graph of a function is a parabola if and only if the function is y 5 x2.
12. An integer is odd if and only if it is not even.
Tell whether the statement is true or false. If false, give a counterexample.
13. If an integer is not negative, then it is positive.
14. If you were born in the summer, then you were born in July.
15. If a polygon has exactly 5 congruent sides, then the polygon is a pentagon.
16. If x 5 26, then x2 5 36.
17. If B is 6 inches from A and 8 inches from C, then A is 14 inches from C.
18. If a triangle is isosceles, then it is obtuse.
19. If Charlie has $1.00 in coins, then he has four quarters.
20. If you are in Montana, then you are in the United States.
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Venn Diagrams
A Venn diagram uses shapes to show how sets are related.
EXAMPLE
Draw a Venn diagram of the positive integers less than
13 where set A consists of factors of 12 and set B consists
of even numbers.
Positive integers less than 13:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
Positive integers less than 13
A
Set A (factors of 12): 1, 2, 3, 4, 6, 12
1
Set B (even numbers): 2, 4, 6, 8, 10, 12
Both set A and set B: 2, 4, 6, 12
B
2
4
3
6
8
12
10
5
11
7
9
Neither set A nor set B: 5, 7, 9, 11
EXAMPLE
Use the Venn diagram above to decide if the statement is true
or false. Explain your reasoning.
a. If a positive integer less than 13 is not even, then it is not a factor of 12.
c False. 1 and 3 are not even, but they are factors of 12.
b. All positive integers less than 13 that are even are factors of 12.
c False. 8 and 10 are even, but they are not factors of 12.
PRACTICE
Draw a Venn diagram of the sets described.
1. Of the positive integers less than 11, set A consists of factors of 10 and set B
consists of odd numbers.
2. Of the positive integers less than 10, set A consists of prime numbers and
set B consists of even numbers.
3. Of the positive integers less than 25, set A consists of multiples of 3 and
set B consists of multiples of 4.
Use the Venn diagrams you drew in Exercises 1–3 to decide if the statement is
true or false. Explain your reasoning.
4. The only factors of 10 less than 11 that are not odd are 2 and 10.
5. If a number is neither a multiple of 3 nor a multiple of 4, then it is odd.
6. All prime numbers less than 10 are not even.
7. If a positive odd integer less than 11 is a factor of 10, then it is 5.
8. There are 2 positive integers less than 25 that are both a multiple of 3 and a
multiple of 4.
9. If a positive even integer less than 10 is prime, then it is 2.
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Mean, Median, Mode, and Range
The mean of a data
set is the sum of the
values divided by the
number of values.
The mean is also
called the average.
EXAMPLE
The median of a data set is the
middle value when the values
are written in numerical order. If
a data set has an even number
of values, the median is the
mean of the two middle values.
The mode of a data
set is the value that
occurs most often. A
data set can have no
mode, one mode, or
more than one mode.
The range of a
data set is the
difference between
the greatest value
and the least value.
SKILLS REVIEW HANDBOOK
Mean, median, and mode are measures of central tendency; they measure the
center of data. Range is a measure of dispersion; it measures the spread of data.
Find the mean, median, mode(s), and range of the data.
Daily High Temperatures, Week of June 21–27
Day
Sunday
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
76
74
70
69
70
75
78
Temperature (8F)
Mean
Add the values. Then divide by the number of values.
76 1 74 1 70 1 69 1 70 1 75 1 78 5 512
mean 5 512 4 7 ø 73
The mean of the data is about 738F.
Median Write the values in order from least to greatest. Find the middle value(s).
69, 70, 70, 74, 75, 76, 78
median 5 74
Mode
Find the value that occurs most often.
mode 5 70
Range
The median of the data is 748F.
The mode of the data is 708F.
Subtract the least value from the greatest value.
range 5 78 2 69 5 9
The range of the data is 98F.
PRACTICE
Find the mean, median, mode(s), and range of the data.
1. Apartment rents: $650, $800, $700, $525, $675, $750, $500, $650, $725
2. Ages of new drivers: 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 18, 18
3. Monthly cell-phone minutes: 581, 713, 423, 852, 948, 337, 810, 604, 897
4. Prices of a CD: $12.98, $14.99, $13.49, $12.98, $13.89, $16.98, $11.98
5. Cookies in a batch: 36, 60, 52, 44, 48, 45, 48, 41, 60, 45, 38, 55, 60, 48, 40
6. Ages of family members: 41, 45, 8, 10, 40, 44, 3, 5, 42, 42, 13, 14, 67, 70
7. Hourly rates of pay: $8.80, $6.50, $10.85, $7.90, $9.50, $9, $8.70, $12.35
8. Weekly quiz scores: 8, 9, 8, 10, 10, 7, 9, 8, 9, 9, 10, 7, 8, 6, 10, 9, 9, 8, 8, 10
9. People on a bus: 9, 14, 5, 22, 18, 30, 6, 25, 18, 12, 15, 10, 8, 22, 10, 11, 20
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Graphing Statistical Data
There are many ways to display data. An appropriate
graph can help you analyze data. The table at the
right summarizes how data are shown in some
statistical graphs.
EXAMPLE
Bar Graph
Compares data in categories.
Circle Graph
Compares data as parts of a whole.
Line Graph
Shows data change over time.
Use the bar graph to answer the questions.
a. On which day of the week were the greatest
Cars Parked in Student Lot
number of cars parked in the student lot?
120
b. How many cars were parked in the student
lot on Monday?
80
Cars
c The tallest bar on the graph is for Friday.
So, the answer is Friday.
40
c The bar for Monday shows that about
70 cars were parked in the student lot.
EXAMPLE
0
Tu
W
Th
F
Use the circle graph to answer the questions.
a. Which type of transportation is used almost half the
Transportation to School
time?
Car 45%
Bus 20%
c Almost half of the total area of the circle is labeled
“Car 45%.” So, a car is used almost half the time.
Walk or bike
35%
b. Which type of transportation is used the least often?
c The smallest part of the circle is labeled “Bus 20%.”
So, a bus is used the least often.
EXAMPLE
M
Use the line graph to answer the questions.
a. In which month(s) was Jamie’s balance
Jamie’s Savings Account Balance
$250?
b. Between which two consecutive months
did Jamie’s balance increase the most?
c Of the graph’s line segments that have
positive slope, the graph is steepest from
June to July. So, Jamie’s balance increased
the most between June and July.
400
300
Dollars
c The points on the graph to the right of
$250 show that Jamie’s balance was $250
in May and December.
200
100
0
J F M A M J J A S O N D
Month
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PRACTICE
Friday at Ferraro’s Restaurant
1. At which hour did Ferraro’s have 22 diners?
3. How many diners were at Ferraro’s at 11 P.M.?
Were they gone by midnight?
Diners
30
2. At which hour did Ferraro’s have the most diners?
20
10
0
4. Between which two consecutive hours did the
5
number of diners at Ferraro’s change the most?
6
7
8
9
10
11
Time (hours since noon)
12
5. How many fewer diners were at Ferraro’s at
10 P.M. than at 6 P.M.?
Use the bar graph to answer Exercises 6–8.
SKILLS REVIEW HANDBOOK
Use the line graph to answer Exercises 1–5.
Seasons of Students’ Birthdays
6. In which season were the fewest students born?
12
Students
7. In which season(s) were 7 students born?
8. How many more students were born in spring
than in summer?
8
4
0
Use the circle graph to answer Exercises 9–11.
Fall
Winter
Spring
Summer
Heat Sources for U.S. Homes
9. What is the heat source of more than half the
Natural gas 52%
Electricity 22%
homes in the United States?
10. What percent of homes in the United States are
Fuel oil 10%
heated with electricity?
Other 16%
11. If you randomly selected 500 U.S. homes, about
how many would be heated with fuel oil?
12. The table below shows the high temperatures in degrees Fahrenheit for one
week. Display the data in a line graph.
Mon.
Tues.
Wed.
Thurs.
Fri.
Sat.
Sun.
83
89
79
73
69
67
71
13. A high school conducted a survey to determine the numbers of students
involved in various school activities. Display the survey results
in a bar graph.
Computer
club
Music
club
Yearbook
club
Drama
club
Student
council
Chess
club
34
75
16
57
28
12
14. The table below shows the items sold at a café in one day. Display the data in
a circle graph.
Juice
Soda
Water
Muffin
Cookie
95
180
100
55
40
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SKILLS REVIEW HANDBOOK
Organizing Statistical Data
Because it is difficult to analyze unorganized data, it is helpful to organize data
using a line plot, stem-and-leaf plot, histogram, or box-and-whisker plot.
EXAMPLE
Sydney’s math test scores are 90, 85, 88, 95, 100, 77, 85, 100,
80, 77, and 90.
a. Draw a line plot to display the data.
Make a number line from 75 to 100. Each time a value is listed in the data set,
draw an X above the value on the number line.
75
3
3
3
3
3
3
3
3
3
3
3
77
80
85
88
90
95
100
b. Draw a stem-and-leaf plot to display the data.
First write the leaves next to their stems.
7
7
7
8
5
8
5
9
0
5
0
10
0
Then order the leaves from least to greatest.
7
7
8
0
5
5
9
0
0
5
10
0
0
0
Key: 7 | 7 5 77
0
7
8
Key: 7 | 7 5 77
c. Draw a histogram to display the data.
First make a frequency table. Use equal
intervals.
Then make a histogram.
Sydney’s Math Test Scores
Tally
Frequency
71–80
3
3
81–90
5
5
91–100
3
3
6
Frequency
Score
4
2
0
71–80
81– 90
Score
91–100
d. Draw a box-and-whisker plot to display the data.
Write the data in order from least to greatest. Ordered data are divided into a
lower half and an upper half by the median. The median of the lower half is
the lower quartile, and the median of the upper half is the upper quartile.
77
77
Low
value
80
Lower
quartile
85
85
88
90
90
Median
Plot the median, quartiles, and low
and high values below a number
line. Draw a box between quartiles
with a vertical line through the
median as shown. Draw whiskers
to the low and high values.
95
100
100
Upper
quartile
High
value
Sydney’s Math Test Scores
75
80
77
80
85
90
88
95
100
95
100
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PRACTICE
SKILLS REVIEW HANDBOOK
Use the following list of ticket prices to answer Exercises 1–4: $50, $42, $65,
$54, $70, $65, $59, $30, $67, $49, $54, $30, $73, $47, and $54.
1. Draw a line plot to display the data.
2. How many ticket prices are $50 or less?
3. Draw a stem-and-leaf plot to display the data.
4. What is the range of ticket prices costs?
Use the following list of hourly wages of employees to answer Exercises 5–8:
$8.50, $6, $10, $14.25, $5.75, $7, $6.50, $14, $10, $9, $6.50, $8.25, $8.50,
$11.25, $7, $16, $12, $6, $6.75.
5. Draw a histogram to display the data. Begin with the interval $5.00 to $6.99.
6. Copy and complete: The greatest number of employees earn from ? to ?
per hour.
7. Draw a box-and-whisker plot to display the data.
8. Copy and complete: About half of the employees have an hourly wage of ?
or less.
Use the line plot, which shows the results of a survey
asking people the average number of e-mails they
receive daily, to answer Exercises 9 and 10.
9. Copy and complete: Most people surveyed receive
an average of ? e-mails per day.
3
3 3
3 3
4
5
3
3
3
3
3
3
3
7
10
12
15
17
10. How many people receive an average of more than 10 e-mails per day?
Use the stem-and-leaf plot, which shows the weights
(in pounds) of dogs at an animal shelter, to answer
Exercises 11–13.
11. How many dogs were at the shelter?
12. Find the median of the data.
2
2
5 5 9
3
1
3 5 8
4
0
0 1 2 2 5 6 7
5
0
3 5 8 9
6
4
5
Key: 2 | 2 5 22
13. Find the range of the data.
Use the histogram to answer Exercises 14–16.
Baseball Game Attendance
9
9
–6
60
–5
9
50
–4
40
–3
9
30
20
oldest group?
–2
9
0
–1
9
16. Which age group had the same attendance as the
20
9
the baseball game?
40
10
15. How many children up to the age of 9 years attended
0–
baseball game? Which had the least?
People
14. Which age group had the greatest attendance at the
Age (years)
Use the box-and-whisker plot to answer Exercises 17–19.
17. What is the median number of songs on Sam’s CDs?
Number of Songs on Sam’s CDs
10
12
14
10 11 12
14
16
18
18. What is the upper quartile of songs on Sam’s CDs?
19. What is the least number of songs on one of Sam’s
CDs? What is the greatest number?
18
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