INSTRUCTOR’S SOLUTIONS
MANUAL
JAMES LAPP
MATHEMATICS ALL AROUND
SIXTH EDITION
Thomas Pirnot
Kutztown University of Pennsylvania
in collaboration with
Margaret H. Moore
University of Southern Maine

CONTENTS
Chapter 1 Problem Solving: Strategies and Principles
1.1 Problem Solving .........................................................................................................................1
1.2 Inductive and Deductive Reasoning .........................................................................................13
1.3 Estimation ................................................................................................................................18
Chapter Review Exercises ......................................................................................................................19
Chapter Test ...........................................................................................................................................21
Chapter 2 Set Theory: Using Mathematics to Classify Objects
2.1 The Language of Sets ...............................................................................................................23
2.2 Comparing Sets ........................................................................................................................25
2.3 Set Operations ..........................................................................................................................28
2.4 Survey Problems ......................................................................................................................31
2.5 Looking Deeper: Infinite Sets ..................................................................................................35
Chapter Review Exercises ......................................................................................................................38
Chapter Test ...........................................................................................................................................40
Chapter 3 Logic: The Study of What’s True or False or Somewhere in Between
3.1 Statements, Connectives, and Quantifiers ................................................................................43
3.2 Truth Tables .............................................................................................................................44
3.3 The Conditional and Biconditional ..........................................................................................51
3.4 Verifying Arguments ...............................................................................................................55
3.5 Using Euler Diagrams to Verify Syllogisms ............................................................................63
3.6 Looking Deeper: Fuzzy Logic..................................................................................................66
Chapter Review Exercises ......................................................................................................................68
Chapter Test ...........................................................................................................................................70
Chapter 4 Graph Theory (Networks): The Mathematics of Relationships
4.1 Graphs, Puzzles and Map Coloring ..........................................................................................75
4.2 The Traveling Salesman Problem ............................................................................................80
4.3 Directed Graphs .......................................................................................................................87
4.4 Looking Deeper: Scheduling Projects Using Pert ....................................................................92
Chapter Review Exercises ....................................................................................................................103
Chapter Test .........................................................................................................................................106
Chapter 5 Numeration Systems: Does It Matter How We Name Numbers?
5.1 The Evolution of Numeration Systems ..................................................................................109
5.2 Place Value Systems ..............................................................................................................113
5.3 Calculating in Other Bases .....................................................................................................122
5.4 Looking Deeper: Modular Systems ........................................................................................129
Chapter Review Exercises ....................................................................................................................135
Chapter Test .........................................................................................................................................138
Chapter 6 Number Theory and the Real Number System: Understanding the
Numbers All Around Us
6.1 Number Theory ......................................................................................................................141
6.2 The Integers............................................................................................................................145
6.3 The Rational Numbers ...........................................................................................................148
6.4 The Real Number System ......................................................................................................158
6.5 Exponents and Scientific Notation .........................................................................................162
6.6 Looking Deeper: Sequences ...................................................................................................168
Chapter Review Exercises ....................................................................................................................175
Chapter Test .........................................................................................................................................177

Chapter 7 Algebraic Models: How Do We Approximate Reality?
7.1 Linear Equations ....................................................................................................................181
7.2 Modeling with Linear Equations ............................................................................................190
7.3 Modeling with Quadratic Equations.......................................................................................197
7.4 Exponential Equations and Growth ........................................................................................209
7.5 Proportions and Variation ......................................................................................................217
7.6 Modeling with Systems of Linear Equations and Inequalities ...............................................225
7.7 Looking Deeper: Dynamical Systems ....................................................................................246
Chapter Review Exercises ....................................................................................................................250
Chapter Test .........................................................................................................................................257
Chapter 8 Consumer Mathematics: The Mathematics of Everyday Life
8.1 Percent, Taxes, and Inflation ..................................................................................................265
8.2 Interest....................................................................................................................................269
8.3 Consumer Loans.....................................................................................................................276
8.4 Annuities ................................................................................................................................284
8.5 Amortized Loans ....................................................................................................................295
8.6 Looking Deeper: Annual Percentage Rate...................................................... ........................302
Chapter Review Exercises ....................................................................................................................306
Chapter Test .........................................................................................................................................311
Chapter 9 Geometry: Ancient and Modern Mathematics Embrace
9.1 Lines, Angles, and Circles......................................................................................................315
9.2 Polygons .................................................................................................................................320
9.3 Perimeter and Area .................................................................................................................326
9.4 Volume and Surface Area ......................................................................................................340
9.5 The Metric System and Dimensional Analysis ......................................................................348
9.6 Geometric Symmetry and Tessellations .................................................................................354
9.7 Looking Deeper: Fractals .......................................................................................................359
Chapter Review Exercises ....................................................................................................................362
Chapter Test .........................................................................................................................................364
Chapter 10 Apportionment: How Do We Measure Fairness?
10.1 Understanding Apportionment ...............................................................................................369
10.2 The Huntington–Hill Apportionment Principle......................................................................376
10.3 Other Paradoxes and Apportionment Methods ......................................................................384
10.4 Looking Deeper: Fair Division ..............................................................................................404
Chapter Review Exercises ....................................................................................................................407
Chapter Test .........................................................................................................................................411
Chapter 11 Voting: Using Mathematics to Make Choices
11.1 Voting Methods......................................................................................................................417
11.2 Defects in Voting Methods ....................................................................................................424
11.3 Weighted Voting Systems ......................................................................................................431
11.4 Looking Deeper: The Shapley–Shubik Index ........................................................................446
Chapter Review Exercises ....................................................................................................................

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Chapter 13 Probability: What Are the Chances
13.1 The Basics of Probability Theory ...........................................................................................491
13.2 Complements and Unions of Events ......................................................................................498
13.3 Conditional Probability and Intersections of Events ..............................................................503
13.4 Expected Value ......................................................................................................................512
13.5 Looking Deeper: Binomial Experiments ................................................................................516
Chapter Review Exercises ....................................................................................................................519
Chapter Test .........................................................................................................................................521
Chapter 14 Descriptive Statistics: Making Sense of the Data
14.1 Organizing and Visualizing Data ...........................................................................................525
14.2 Measures of Central Tendency ...............................................................................................530
14.3 Measures of Dispersion ..........................................................................................................537
14.4 The Normal Distribution ........................................................................................................550
14.5 Looking Deeper: Linear Correlation ......................................................................................556
Chapter Review Exercises ....................................................................................................................563
Chapter Test .........................................................................................................................................566

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Section 1.1: Problem Solving 1
Chapter 1: Problem Solving: Strategies and Principles
Section 1.1: Problem Solving
1. Drawings may vary.
2. Drawings may vary.
3. Drawings may vary.
4. Drawings may vary.
5. Answers may vary. LetH be the hybrid automobiles,W be the windmill turbines, andS be solar energy.
6. Answers may vary. LetT be Tyrion,J be Jamie,C be Cersei,D be Daenerys, andS be Sansa.
7. Answers may vary. Lets be the dollar amount invested in stocks andb be the dollar amount invested in
bonds.
8. Answers may vary. Letc be the amount of calcium andp be the amount of protein.
9. Answers (order) may vary. Combinations would be HH, HT, TH, and TT.
Penny Nickel
Heads Heads
Heads Tails
Tails Heads
Tails Tails
10. Answers (order) may vary. Pairs would be (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), and (3, 3).
11.2 2 2 2 2 32;´ ´ ´ ´ = Graph not provided. 12.6 6 36´ =
13. Answers (order) may vary. Using the "Be Systematic" strategy, first list all pairs that begin with L, next all
new pairs that begin with S, etc. Pairs would be LS, LB, LE, LD, SB, SE, SD, BE, BD, ED.

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2 Chapter 1: Problem Solving
14. Answers (order) may vary. Routes you can take are: (Begin, A, D, H, End), (Begin, A, E, H, End), (Begin,
A, E, I, End), (Begin, B, E, H, End), (Begin, B, E, I, End), (Begin, B, F, I, End), (Begin, B, F, J, End),
(Begin, C, F, I, End), (Begin, C, F, J, End), and (Begin, C, G, J, End).
15. (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3),
(4, 4)
16. (1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3)
17.3r is the set of people who are good singers and appeared on “American Idol”.4r is the set of people who
have appeared on “American Idol” and are not good singers.
18.2r is the set of people who are working to reduce global warming but not striving to reduce world hunger.4r
is the set of people who are striving to reduce world hunger but not working to reduce global warming.
19. 35, 42, 49, 56, 63; multiples of 7
20. 81, 243, 729, 2187, 6561; multiples of 3
21., , , ,bf cd ce cf cg
22. (2, 4), (2, 5), (2, 6), (3, 1), (3, 2)
23. 21, 34, 55, 89, 144; sum of previous two numbers
24. 19, 23, 29, 31, 37; prime numbers
25. Answers may vary.
In how many ways can we line up three people for a picture? Let the people be labeled A, B, and C.
The possible orders are ABC, ACB, BAC, BCA, CAB, and CBA. There are 6 different ways.
In how many ways can we line up four people for a picture? Let the people be labeled A, B, C, and D.

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Section 1.1: Problem Solving 3
25. (continued)
The possible orders are ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC, BCAD, BCDA,
BDAC, BDCA, CABD, CADB, CBAD, CBDA, CDAB, CDBA, DABC, DACB, DBAC, DBCA, DCAB,
and DCBA. There are 24 different ways.
26. Answers may vary. If you guess at 2 true-false questions, how many different ways can you fill in the 2
answers?
1 2
T T
T F
F T
F F
There are 4 different ways.
If you guess at 3 true-false questions, how many different ways can you fill in the 3 answers?
1 2 3
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
There are 8 different ways.

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4 Chapter 1: Problem Solving
26. (continued)
If you guess at 4 true-false questions, how many different ways can you fill in the 4 answers?
1 2 3 4
T T T T
T T T F
T T F T
T T F F
T F T T
T F T F
T F F T
T F F F
F T T T
F T T F
F T F T
F T F F
F F T T
F F T F
F F F T
F F F F
There are 16 different ways.
27. Answers may vary. Using the first three letters of the alphabet, how many two-letter codes can we form if
we are allowed to use the same letter twice?
The possible codes are AA, AB, AC, BA, BB, BC, CA, CB, and CC. There are 9 different codes.

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Section 1.1: Problem Solving 5
27. (continued)
Using the first five letters of the alphabet, how many two-letter codes can we form if we are allowed to use
the same letter twice?
The possible codes are AA, AB, AC, AD, AE, BA, BB, BC, BD, BE, CA, CB, CC, CD, CE, DA, DB, DC,
DD, DE, EA, EB, EC, ED, and EE. There are 25 different codes.
28. Answers may vary.
A family has three children. If we list the gender of the children, how many different lists are possible?
1 2 3
g g g
g g b
g b g
g b b
b g g
b g b
b b g
b b b
There are 8 different lists.

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6 Chapter 1: Problem Solving
28. (continued)
A family has four children. If we list the gender of the children, how many different lists are possible?
1 2 3 4
g g g g
g g g b
g g b g
g g b b
g b g g
g b g b
g b b g
g b b b
b g g g
b g g b
b g b g
b g b b
b b g g
b b g b
b b b g
b b b b
There are 16 different lists.
29. Answers may vary.
An electric-blue Ferrari comes with two options: run flat tires and front heated seats. You may buy the car
with any combination of the options (including none). How many different choices do you have? LetR be
run flat tires andF be front heated seats. If a feature is not included, it is indicated by a “0”. If it is
included, it is indicated by a “1”. There are 4 different choices.RF
0 0
0 1
1 0
1 1
An electric-blue Ferrari comes with three options: run flat tires, front heated seats, and polished rims. You
may buy the car with any combination of the options (including none). How many different choices do you
have? LetR be run flat tires,F be front heated seats, andP be polished rims. If a feature is not included,
it is indicated by a “0”. If it is included, it is indicated by a “1”. There are 8 different choices.

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Section 1.1: Problem Solving 7
30. Answers may vary.
You have 2 different colors of paper to use and 3 different styles of font. How many different ways can you
print your resume?
There are 6 different ways .
You have 2 different colors of paper to use and 4 different styles of font. How many different ways can you
print your resume?
There are 8 different ways.
31. False, counterexamples may vary. June has 30 days.
32. False, counterexamples may vary. At the time this book was written, this was false because Bill Clinton is
still alive.
33. False, counterexamples may vary.1 3 2 3 5 1 3 4 2 5 2
, , and
2 4 4 4 4 2 4 6 3 4 3
+
+ = + = = = ¹
+
34. False, counterexamples may vary.2 2
9 5 but ( 9) ( 5) since 81 25.- < - - > - >
35. False, A is the grandfather of C.
36. False, counterexamples may vary. You know your instructor and your instructor knows his/her mother, but
(most likely) you don’t know your instructor’s mother.
37. False, counterexamples may vary. If the price of a $10.00 item is increased by 10%, its new price is $11.00.
If the $11.00 item is then decreased by 10%, the new price would be $9.90, not $10.00.
38. False, counterexamples may vary. If your hourly rate of $10.00 is decreased by 20%, the new rate is $8.00.
If the $8.00 rate is then increased by 20%, the new rate would be $9.60, not $10.00.

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8 Chapter 1: Problem Solving
39. Explanations may vary. These two sequences do not give the same results. The question here is equivalent
to asking if the algebraic statement,2 2
5 ( 5)x x+ = + , is true.
If we let1,x = we have a counterexample.( )
? 22
? 2
1 5 1 5
1 5 6
6 36
+ = +
+ =
¹
Hence, the statement is false.
40. Explanations may vary. These two sequences do not give the same results. The question here is equivalent
to asking if the algebraic statement,2 2 2
( )x y x y- = - , is true.
If we let1x = and3,y = we have a counterexample.( )
( )
? 22 2
? 2
1 3 1 3
1 9 2
8 4
- = -
- = -
- ¹
Hence, the statement is false.
41. Explanations may vary. These two sequences do give the same results. The question here is equivalent to
asking if the algebraic statement,,
3 3 3
x y x y+ = + is true. If you think of dividing by 3 as equivalent to
multiplying by1 3, then you can use the distributive property to prove this statement.( )
1 1 1
3 3 3 3 3 3
x y x y
x y x y
+ = + = + = +
42. Explanations may vary. These two sequences do give the same results. The question here is equivalent to
asking if the algebraic statement,( )
5 5 5,x y x y× + × = + × is true. The proof of this statement is equivalent to
proving the distributive property.
43. Answers may vary. 5 is a number;{ }
5 is a number with braces around it. Moreover, 5 is a singly listed
element, while{ }
5 is a set that contains the single element 5.
44. Answers may vary.

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Section 1.1: Problem Solving 9
48. The symbol on the right is the number zero; the symbol on the left is not a number.
49. – 52. No solution provided.
53. Answers may vary. LetO be the age of the older building andY be the age of the younger building.
Guesses forO andY Good Points Weak Points
100, 221 Sum is 321. The older is more than
twice the younger.
110, 211 Sum is 321. The older is less than
twice the older.
107, 214 All conditions satisfied.
We have the solution.
The buildings are 107 and 214 years old.
54. Answers may vary. LetS be the shortest piece,M be the middle size piece, andL be the longest piece.
Guesses for,S,M andL Good Points Weak Points
3, 8, and 24 The longest is 3 times the middle size and
the shortest is 5 less than the middle size.
Total length is less than
40.
5, 10, and 30 The longest is 3 times the middle size and
the shortest is 5 less than the middle size.
Total length is more
than 40.
4, 9, and 27 All conditions satisfied.
We have the solution
The three pieces should be 4, 9, and 27 inches long.
55. Answers may vary. LetT be the number of times Tom Brady threw a touchdown pass,P be the number
of times Phillip Rivers threw a touchdown pass, andA be the number of times Aaron Rodgers threw a
touchdown pass.
Guesses for,T,P andA Good Points Weak Points
30, 24, and 22P is 2 more thanA and 6 less than.T Sum is less than 94.
40, 34, and 32P is 2 more thanA and 6 less than.T Sum is more than 94.
36, 30, and 28 All conditions satisfied.
We have the solution.
Brady had 36 touchdowns.
56. Answers may vary. LetJ be the number of home runs hit by Joc Pederson,T be the number of home runs
hit by Todd Frazier andP be the number of home runs hit by Prince Fielder.
Guesses for,J,T andP Good Points Weak Points
30, 29 and 7J is 1 more thanT andJ is 23 more
than.P Sum is less than 72.
40, 39 and 17J is 1 more thanJ andJ is 23 more
than.P Sum is more than 72.
32, 31 and 9 All conditions satisfied.
We have the solution.
Pederson had 32 hits.

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10 Chapter 1: Problem Solving
57. Answers may vary. LetA be the amount invested at 8% andB be the amount invested at 6%.
Guesses forA andB Good Points Weak Points
$3,000 and $5,000 Sum is $8,000. Return is less than
$550.
$4,000 and $4,000 Sum is $8,000. Return is more than
$550.
$3,500 and $4,500 All conditions satisfied.
We have the solution
Heather invested $3,500 at 8% and $4,500 at 6%.
58. Answers may vary. LetA be the amount invested at 11% andB be the amount invested at 8%.
Guesses forA andB Good Points Weak Points
$7,000 and $2,000 Sum is $9,000. Return is less than
$936.
$7,500 and $1,500 Sum is $9,000. Return is more than
$936.
$7,200 and $1,800 All conditions satisfied.
We have the solution.
Carlos invested $7,200 at 11% and $1,800 at 8%.
59. Answers may vary. LetA be the number of administrators,S be the number of students, andF be the
number of faculty members.
Guesses for,A,S andF Good Points Weak Points
10, 2, and 7A is 5 timesS andF is 5 more than S. There are less than 26
people.
20, 4, and 9A is 5 timesS andF is 5 more than S. There are more than 26
people.
15, 3, and 8 All conditions satisfied.
We have the solution.
There are 3 students.
60. Answers may vary. LetS be the number of senior citizens,Y be the number of young adults, andM be
the number of middle-aged adults.
Guesses for,S,Y andM Good Points Weak Points
18, 6, and 9S is 3 timesY andM is one-half.S There are less than 55
people.
36, 12, and 18S is 3 timesY

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Section 1.1: Problem Solving 11
64. PND, PNQ, PDQ, NDQ, PNDQ
65. The possible schedules are given in the table below.
Math English Sociology Art History
9 11 12 10
9 12 10 11
10 9 12 11
10 11 12 9
12 9 10 11
12 11 10 9
66. The possible schedules are given in the table below.
Math English Sociology Art History
9 11 12 10
9 12 10 11
12 9 10 11
67. The possible schedules are given in the table below.
Math English Art History
9 11 10
9 12 10
9 12 11
10 9 11
10 11 9
10 12 9
10 12 11
12 9 10
12 9 11
12 11 9
12 11 10
68. The possible schedules are given in the table below.
Math English Sociology Art History
9 11 12 10
9 12 11 10
10 9 12 11
10 11 12 9
10 12 11 9
12

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12 Chapter 1: Problem Solving
71. You will pay more, in total, over the two years if the 8% raise occurs first. As an example, if tuition is $100
and the 8% raise occurs first, you will pay( )
100 0.08 100 $108+ = the first year and( )
108 0.05 108+$113.40=
the second year, for a total of $221.40. If the 5% raise occurs first, you will pay( )
100 0.05 100 $105+ =
the first year and( )
105 0.08 105 $113.40+ = the second year, for a total of
$218.40. Note that in either case, you would pay the same tuition during the second year.
72. The board would not follow its mandate. As an example, if tuition is $100, you will pay( )
100 0.02 100 $102+ =
after the first increase,( )
102 0.03 102 $105.06+ = after the second increase, and( )
105.06 0.05 105.06 $110.31+ =
after the third increase, which is an approximately 10.3% increase.
73. – 76. Answers will vary.
77.6 6 6 216´ ´ = 78.12 12 12 1,728´ ´ =
79. (41, 43), (59, 61); Pairs of sequential primes that differ by 2.
80. (23, 31), (29, 37); The first numbers are the primes in order, the second numbers come from taking the
second prime number after the first.
81. There are a total of 55 squares.1 1´
254 4´ 42 2´
165 5´ 13 3´
9
82. a) 12
b) 6
c) 1001 1´
161 2´ 122 4´ 3

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Section 1.2: Inductive and Deductive Reasoning 13
84. As you look at the intersections, you’ll see that there is a pattern as to how many routes can be created as
you leave the Hard Rock Cafe on the way to Baja Fresh and as you leave Baja Fresh on the way to The
Cheesecake Factory. To choose direct routes, you must always be traveling down and/or to the right. The
numbers indicate how many ways there are to get to the intersection below and to the right of the number.
Since there are 10 ways to go from the Hard Rock Cafe to Baja Fresh and 10 ways to go from the Baja
Fresh to The Cheesecake Factory , there are10 10 100´ = possible routes.
Section 1.2: Inductive and Deductive Reasoning
1. inductive
2. deductive
3. deductive
4. inductive
5. inductive
6. deductive
7. deductive
8. inductive
9. inductive
10. deductive
11. 16
12. 32
13. 96
14. 1,215
15. 21
16. 0.101010
17. 18.
19. A blue face in a green box followed by a red
face in a blue box.
20. A red face in a blue box followed by a blue
face in a green box.
21. Hint: Think of prime numbers.
2 3 5 7 11 13

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Section 1.2: Inductive and Deductive Reasoning 15
37. Yes; there are many ways to do this. Try to rewrite the puzzle in various ways with some numbers or clues
missing to see if you can still solve it.
38. Yes; there are many ways to do this. Try to rewrite the puzzle in various ways with some numbers or clues
missing to see if you can still solve it.
39. Adriana (political issues), Caleb (solar power), Ethan (water conservation), and Julia (recycling)
40. Jessica (third), Serena (second), Andre (fourth), and Emily (first)
41. 36644633; Reversing the number in the first pair yields the relationshipG G A G L L G A ,
6 6 4 6 3 3 6 4 so
G, A, and L correspond to 6, 4, and 3, respectively. Assigning these values to the letters in the second word
yieldsL L G A A G G L.
3 3 6 4 4 6 6 3 Reversing the order of the resulting number leads to the final value
of 36644633.
42. leader; Froofrug Merduc represents “Where is your”
43. 986763; reverse the numbers and delete one number from any pair of numbers.
44. 20; the numbers are found by repeatedly adding 6 then subtracting 2.
45. Answers may vary. Possible responses include that Sharifa would begin by visiting one of the six branches
and then visit the five original cities in 120 ways. The total number of ways she could make her visits is6 120 720´ =
ways. The same reasoning would lead to7 720 5040´ = ways for seven cities.
46. Reasons will vary.
47. (d) step 3 48.
49. By looking at examples, inductive reasoning leads us to make conjectures which we then try to prove.

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16 Chapter 1: Problem Solving
50. Consider examples; Look for patterns; Make a conjecture; Use deductive reasoning to prove the statement.
51. – 54. Answers will vary.
55. 53, 107, 213; Next term is the previous term plus twice the term before the previous term.
56. 5461, 21845, 87381; Next term is four times the previous term plus one.
57. 47, 76, 123; Next term is sum of two previous terms.
58. 150, 298, 598; Next term is the previous term plus twice the term before the previous term.
59. There are a total of2 6 12 20+ + = squares.1 1´
122 2´
63 3´
2
60. There are a total of3 8 15 24 50+ + + = squares.1 1´
242 2´
153 3´
84 4´
3
61. a) There are a total of 60 rectangles.1 1´
121 2´
91 3´
61 4´
32 1´
82 2´
62 3´
42 4´
23 1´
43 2´
33 3´
23 4´
1
b) There are a total of 210

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Chapter Review Exercises 19
29. Answers will vary. Actual answers are: Male high school graduates; $44,300; Females with associates
degrees; $42,000
30. Answers will vary. Actual answer is $17,200.
31. college and professional degrees, high school graduates
32. Answers will vary. Actual answer is $47,300.
33. category 2
34. 500;0.21 2,309 485´ »
35. response 2
36. 139;0.06 2309 139´ »
37. Estimated answers may vary. Actual value is 69.3%

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20 Chapter 1: Problem Solving
3. 10; Let A, C, L, R, and T represent Amber, Chris, Lawrence, Remy, and Travis.
The pairs are AC, AL, AR, AT, CL, CR, CT, LR, LT, and RT.
4. Answers may vary. At a restaurant, you have 2 appetizers, 3 entrees, and 2 desserts. How many different
meals can you choose if you select one appetizer, one entrée, and one desert?
There are twelve different meals possible.
5. Answers may vary. LetP be the number of hours Picaboo worked as a stock person andI be the number
of hours she worked as a ski instructor.
Guesses forP
andI Good Points Weak Points
9 and 11 Sum is 20. Amount earned is less
than $141.20.
7 and 13 Sum is 20. Amount earned is more
than $141.20.
8 and 12 All conditions satisfied. We have the
solution.
6. false; Counterexamples may vary.1 3 2 3 5 1 3 4 2 5 2
, , and
2 4 4 4 4 2 4 6 3 4 3
+
+ = + = = = ¹
+
7. No solution provided.
8. Answers may vary. Possible answers include: Inductive reasoning is the process of drawing a general
conclusion by observing a pattern in specific instances. In deductive reasoning, we use accepted facts and
general principles to arrive at a specific conclusion.
9. a) deductive
b) inductive
10. a) 27; Add five to the previous number.
b) 47; Add the previous two numbers.

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22 Chapter 1: Problem Solving
8. a) inductive
b) deductive
9. Mathematical ideas can be understood verbally, graphically, and through examples.
10. Answers will vary. One estimate is( )
$600 $200 12 $800 4 $3,200.
3
+æ ö× = × =ç ÷è ø The true value is $3,220.
11.2 2 2 2 2 2 2
1 1 2, 2 2 6, 6 3 15, 15 4 31, 31 5 56, 56 6 92, and 92 7 141+ = + = + = + = + = + = + =
12. cde, cdf, cdg
13.
14.7 53 60,+ =13 47 60,+ =19 41 60,+ =23 37 60,+ = or29 31 60+ =
15. False; suppose the laptop costs $1,000. After the 10% discount, the laptop would cost $900. If that price is
increased by 10%, the laptop would cost $990, not $1,000.
16. In this trick, you will always get twice the original number.
a) Call the number.n
b)4n
c)4 40n +
d)( )
2 2 204 40 2 20
2

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Section 2.1: The Language of Sets 23
Chapter 2: Set Theory: Using Mathematics to Classify Objects
Section 2.1: The Language of Sets
1.{ }
10, 11,12,13,14,15
2.{ }
, , , ,f g h i j
3.{ }
17, 18, 19, 20, 21, 22, 23, 24, 25
4.{ }
4, 3, 2, 1, 0, 1, 2, 3, 4- - - -
5.{ }
4, 8, 12, 16,20, 24, 28
6.{ }
7, 9, 11, 13, 15, 17, 19
7.{ }
Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday
8.{ }
New Hampshire, New Jersey, New Mexico, New York
9.Æ
10.{ }
11, 13, 15, 17, 19, 21, 23, 25
11.Æ
12.{ }
Alaska, Hawaii
13. Answers may vary. Possible answers include{ }
: is a multiple of 3 between 3 and 12 inclusive .x x
14.

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24 Chapter 2: Set Theory
51. 1 element;{ }
{ }
Æ 52. 4 elements;{ } { } { } { }
1 , 2 , 3 , 1, 2, 3
53. finite
54. finite
55. infinite
56. finite
57. Answers may vary. Possible answers include 4.5.
58. Answers may vary. Possible answers include the King(Queen) of England.
59. Answers may vary. Possible answers include Sony.
60. Answers may vary. Possible answers include a frog.
61. Answers may vary. Possible answers include Angela Merkel.
62. Answers may vary. Possible answers include Kia.
63. Answers may vary. Possible answers include Sunday.
64. Answers may vary. Possible answers include California.
65.{ }
: is a humanities electivex x
66.{ }
: does not satisfy a world culture requirementx x
67.{ }
History012, History223, Geography115, Anthropology111
68.{ }
History012, English010, English220, Psychology200
69.{ }
AZ, FL, GA, LA, NJ, NM, TX, VAL =
70.{ }
CA, MN, NY, PAG =

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Section 2.2: Comparing Sets 25
86. If the sentence is true, then the sentence is false. If the sentence is false, then the sentence is true.
Conclusion: This is a paradox.
87. If,S SÎ then.S SÏ If,S SÏ then.S SÎ Conclusion: This is a paradox.
Section 2.2: Comparing Sets
1. These two sets are equal. They have the same elements arranged in a different order.
2. These two sets are not equal. The first set is the set of vowels. The second set contains “b ” along with
other letters. Since “b ” is not a vowel, these two sets cannot be equal.
3. These two sets are not equal. The second set contains (infinitely many) elements that don’t appear in the
first set.
4. These two sets are equal. They are both{ }
3, 4, 5, 6, 7, 8, 9,10 .
5. These two sets are equal. They are both{ }
1, 3, 5, , 99 .
6. These two sets are not equal. The first set contains the first 5 multiples of 3 that are counting numbers.
The second set contains all multiples of 3 that are counting numbers.
7. These two sets are equal. Common sense dictates that nobody born before 1800 should be living.
8. These two sets are not equal. The null set contains no elements. The set,{ }
,Æ contains one element,
namely the null set.
9. true; All the elements of the first set are understood to be elements of the second set and moreover the first
set is not equal to the second set.
10. true; All the elements of the first set are also elements of the second set.
11. false; The letter “y” is an element of the first set and not an element of the second set.
12.

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Solution Manual for Mathematics All Around, 6th Edition

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