SOLUTIONS MANUAL
For
EXCURSIONS IN
MODERN MATHEMATICS
TENTH EDITION
By
Peter Tannenbaum
, Table of Contents
Solutions
Chapter 1 1
Chapter 2 21
Chapter 3 44
Chapter 4 70
Chapter 5 91
Chapter 6 104
Chapter 7 120
Chapter 8 130
Chapter 9 148
Chapter 10 160
Chapter 11 173
Chapter 12 191
Chapter 13 211
Chapter 14 224
Chapter 15 233
Chapter 16 247
Chapter 17 261
iii
, Chapter 1
WALKING
1.1. Ballots and Preference Schedules
1. Number of voters 5 3 5 3 2 3
1st choice A A C D D B
2nd choice B D E C C E
3rd choice C B D B B A
4th choice D C A E A C
5th choice E E B A E D
This schedule was constructed by noting, for example, that there were five ballots listing candidate C as the
first preference, candidate E as the second preference, candidate D as the third preference, candidate A as the
fourth preference, and candidate B as the last preference.
2. Number of voters 4 5 6 2
1st choice A B C A
2nd choice D C A C
3rd choice B D D D
4th choice C A B B
3. (a) 5 + 5 + 3 + 3 + 3 + 2 = 21
(b) 11. There are 21 votes all together. A majority is more than half of the votes, or at least 11.
(c) Chavez. Argand has 3 last-place votes, Brandt has 5 last-place votes, Chavez has no last-place votes,
Dietz has 3 last-place votes, and Epstein has 5 + 3 + 2 = 10 last-place votes.
4. (a) 202 + 160 + 153 + 145 + 125 + 110 + 108 + 102 + 55 = 1160
(b) 581; There are 1160 votes all together. A majority is more than half of the votes, or at least 581.
(c) Alicia. She has no last-place votes. Note that Brandy has 125 + 110 + 55 = 290 last-place votes, Cleo
has 202 + 145 + 102 = 449 last-place votes, and Dionne has 160 + 153 + 108 = 421 last-place votes.
5. Number of voters 37 36 24 13 5
1st choice B A B E C
2nd choice E B A B E
3rd choice A D D C A
4th choice C C E A D
5th choice D E C D B
Here Brownstein was listed first by 37 voters. Those same 37 voters listed Easton as their second choice,
Alvarez as their third choice, Clarkson as their fourth choice, and Dax as their last choice.
6. Number of voters 14 10 8 7 4
1st choice B B A D E
2nd choice A D B C B
3rd choice E A E B A
4th choice C E D E C
5th choice D C C A D
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, 2 Chapter 1: The Mathematics of Elections
7. Number of voters 14 10 8 7 4
A 2 3 1 5 3
B 1 1 2 3 2
C 5 5 5 2 4
D 4 2 4 1 5
E 3 4 3 4 1
Here 14 voters had the same preference ballot listing B as their first choice, A as their second choice, E as
their third choice, D as their fourth choice, and C as their fifth and last choice.
8. Number of voters 37 36 24 13 5
A 1 2 5 2 4
B 3 1 2 4 1
C 2 4 3 1 5
D 5 3 1 5 2
E 4 5 4 3 3
9. Number of voters 255 480 765
1st choice L C M
2nd choice M M L
3rd choice C L C
(0.17)(1500) = 255; (0.32)(500) = 480; The remaining voters (51% of 1500 or 1500-255-480=765) prefer M
the most, C the least, so that L is their second choice.
10. Number of voters 450 900 225 675
1st choice A B C C
2nd choice C C B A
3rd choice B A A B
100% - 20% - 40% = 40% of the voters number 225 + 675 = 900. So, if N represents the total number of
voters, then (0.40) N = 900 . This means there are N = 2250 total voters. 20% of 2250 is 450 (these voters
have preference ballots A, C, B). 40% of 2250 is 900 (these voters have preference ballots B, C, A).
1.2. Plurality Method
11. (a) C. A has 15 first-place votes. B has 11 + 8 + 1 = 20 first-place votes. C has 27 first-place votes. D has 9
first-place votes. C has the most first-place votes with 27 and wins the election.
(b) C, B, A, D. Candidates are ranked according to the number of first-place votes they received (27, 20, 15,
and 9 for C, B, A, and D respectively).
12. (a) D. A has 21 first-place votes. B has 18 first-place votes. C has 10 + 1 = 11 first-place votes. D has 29
first-place votes. D has the most first-place votes with 29 and wins the election.
(b) D, A, B, C.
13. (a) C. A has 5 first-place votes. B has 4 + 2 = 6 first-place votes. C has 6 + 2 + 2 + 2 = 12 first-place votes.
D has no first-place votes. C has the most first-place votes with 12 and wins the election.
(b) C, B, A, D. Candidates are ranked according to the number of first-place votes they received (12, 6, 5,
and 0 for C, B, A, and D respectively).
14. (a) B. A has 6 + 3 = 9 first-place votes. B has 6 + 5 + 3 = 14 first-place votes. C has no first-place votes. D
has 4 first-place votes. B has the most first-place votes with 14 and wins the election.
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