# Chapter 13 - Voting and Apportionment - 13.1 Voting Methods - Exercise Set 13.1 - Page 851: 35

(a) Candidate C is declared the new department chair using the Borda count method. (b) Candidate A is declared the new department chair using the Borda count method.

#### Work Step by Step

(a) With the Borda count method, each candidate receives 1 point for each last place vote, 2 points for each second-to-last-place vote, and so on. The candidate who receives the most points is declared the winner. We can find the total points for each candidate. Candidate A: 5(5 + 3) + 4(0) + 3(3) + 2(5 + 2) + 1(3) = 66 points Candidate B: 5(3) + 4(5) + 3(3 + 3 + 2) + 2(0) + 1(5) = 64 points Candidate C: 5(5) + 4(3 + 2) + 3(5) + 2(3 + 3) + 1(0) = 72 points Candidate D: 5(3 + 2) + 4(3) + 3(5) + 2(5) + 1(3) = 65 points Candidate E: 5(0) + 4(5 + 3) + 3(0) + 2(3) + 1(5 + 3 + 2) = 48 points Since Candidate C received the most points, Candidate C is declared the new department chair using the Borda count method. (b) After Candidate E withdraws, the other candidates who were ranked below Candidate E in a given ballot move up one place in that ballot. We can see the update preference table below. We can use the Borda count method to determine who is declared the new department chair. Candidate A: 4(5 + 3) + 3(3) + 2(5) + 1(3+2) = 56 points Candidate B: 4(3) + 3(5) + 2(3 + 3 + 2) + 1(5) = 48 points Candidate C: 4(5) + 3(3 + 2) + 2(5+3) + 1(3) = 54 points Candidate D: 4(3 + 2) + 3(5 + 3) + 2(0) + 1(5 + 3) = 52 points Since Candidate A received the most points, Candidate A is declared the new department chair using the Borda count method.

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