## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 13 - Voting and Apportionment - 13.1 Voting Methods - Exercise Set 13.1 - Page 851: 30

#### Answer

Soup B is declared the winner using the pairwise comparison 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 Soup A and Soup C. We can find the number of ballots with Soup A ranked higher than Soup C, and the number of ballots with Soup C ranked higher than Soup A. Soup A: 34 Soup C: 30 + 6 + 2 = 38 Since Soup C is ranked higher than Soup A on more ballots, Soup C receives 1 point. We can compare Soup A and Soup B. We can find the number of ballots with Soup A ranked higher than Soup B, and the number of ballots with Soup B ranked higher than Soup A. Soup A: 34 Soup B: 30 + 6 + 2 = 38 Since Soup B is ranked higher than Soup A on more ballots, Soup B receives 1 point. We can compare Soup B and Soup C. We can find the number of ballots with Soup B ranked higher than Soup C, and the number of ballots with Soup C ranked higher than Soup B. Soup B: 34 + 30 + 2 = 66 Soup C: 6 Since Soup B is ranked higher than Soup C on more ballots, Soup B receives 1 point. We can compare Soup A and Soup D. We can find the number of ballots with Soup A ranked higher than Soup D, and the number of ballots with Soup D ranked higher than Soup A. Soup A: 34 Soup D: 30 + 6 + 2 = 38 Since Soup D is ranked higher than Soup A on more ballots, Soup D receives 1 point. We can compare Soup D and Soup C. We can find the number of ballots with Soup D ranked higher than Soup C, and the number of ballots with Soup C ranked higher than Soup D. Soup D: 2 Soup C: 34 + 30 + 6 = 70 Since Soup C is ranked higher than Soup D on more ballots, Soup C receives 1 point. We can compare Soup B and Soup D. We can find the number of ballots with Soup B ranked higher than Soup D, and the number of ballots with Soup D ranked higher than Soup B. Soup B: 34 + 30 = 64 Soup D: 6 + 2 = 8 Since Soup B is ranked higher than Soup D on more ballots, Soup B receives 1 point. After all the pairwise comparisons have been made, we can add up the total number of points for each candidate. Soup A: 0 points Soup B: 1 + 1 + 1 = 3 points Soup C: 1 + 1 = 2 points Soup D: 1 point Since Soup B received the most points, Soup B is declared the winner. Soup B is declared the winner using the pairwise comparison method.

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