Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 13 - Voting and Apportionment - 13.2 Flaws of Voting Methods - Exercise Set 13.2: 2

Answer

(a) Yogurt A has a majority of first-place votes. (b) Yogurt C is declared the winner using the Borda count method. (c) The majority criterion is not satisfied.

Work Step by Step

(a) The total number of votes in this election is 53. Yogurt A has 27 first-place votes which is more than half of the total votes in this election. Therefore, Yogurt A has a majority of first-place votes. (b) With the Borda count method, each yogurt brand receives 1 point for each third-place vote, 2 points for each second-place vote, and 3 points for each first-place vote. The yogurt brand which receives the most points is declared the winner. We can find the total points for each yogurt brand. Yogurt A: 3(27) + 2(0) + 1(24 + 2) = 107 points Yogurt B: 3(24) + 2(2) + 1(27) = 103 points Yogurt C: 3(2) + 2(27 + 24) + 1(0) = 108 points Since Yogurt C received the most points, Yogurt C is declared the winner using the Borda count method. (c) The majority criterion is not satisfied. Yogurt A has a majority of the first-place votes but Yogurt A was not declared the winner of the election.
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