## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 13 - Voting and Apportionment - 13.1 Voting Methods - Exercise Set 13.1 - Page 851: 38

#### Answer

The pairwise comparison method leads to a tie. We need to use the Borda count method to determine the winner. Candidate A is declared the new mayor using the Borda count method.

#### Work Step by Step

With the pairwise comparison method, each candidate is compared with every other candidate. For each pair of candidates, if one candidate is ranked higher than the other candidate on a majority of ballots, then the higher-ranked candidate receives 1 point. If the two candidates tie, then they each receive 0.5 points. After all the comparisons have been made, the candidate who receives the most points is declared the winner. We can compare Candidate A and Candidate C. We can find the number of ballots with Candidate A ranked higher than Candidate C, and the number of ballots with Candidate C ranked higher than Candidate A. Candidate A: 60,000 + 20,000 = 80,000 Candidate C: 40,000 + 40,000 + 20,000 = 100,000 Since Candidate C is ranked higher than Candidate A on more ballots, Candidate C receives 1 point. We can compare Candidate A and Candidate B. We can find the number of ballots with Candidate A ranked higher than Candidate B, and the number of ballots with Candidate B ranked higher than Candidate A. Candidate A: 60,000 + 40,000 + 20,000 = 120,000 Candidate B: 40,000 + 20,000 = 60,000 Since Candidate A is ranked higher than Candidate B on more ballots, Candidate A receives 1 point. We can compare Candidate B and Candidate C. We can find the number of ballots with Candidate B ranked higher than Candidate C, and the number of ballots with Candidate C ranked higher than Candidate B. Candidate B: 60,000 + 40,000 = 100,000 Candidate C: 40,000 + 20,000 + 20,000 = 80,000 Since Candidate B is ranked higher than Candidate C on more ballots, Candidate B receives 1 point. After all the pairwise comparisons have been made, we can add up the total number of points for each candidate. Candidate A: 1 point Candidate B: 1 point Candidate C: 1 point The pairwise comparison method leads to a tie. We need to use the Borda count method to determine the winner. With the Borda count method, each candidate receives 1 point for each third-place vote, 2 points for each second-place vote, and 3 points for each first-place vote. The candidate who receives the most points is declared the winner. We can find the total points for each candidate. Candidate A: 3(60,000 + 20,000) + 2(40,000) + 1(40,000 + 20,000) = 380,000 points Candidate B: 3(40,000) + 2(60,000 + 20,000) + 1(40,000 + 20,000) = 340,000 points Candidate C: 3(40,000 + 20,000) + 2(40,000 + 20,000) + 1(60,000) = 360,000 points Since Candidate A received the most points, Candidate A is declared the new mayor using the Borda count method.

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