Math 2000 - Solutions 10

Spring 2001

1. Each half-step on a 12-tone scale corresponds to an increase in frequency by a factor of f = 1.05946. If we start at G with a frequency of 390 cps, then one half-step higher (G#) has a frequency of 390 ´ 1.05946 = 415 cps. Continuing in this way, we can produce the following table (rounding to the nearest whole number).

Note	Frequency (cps)	Note	Frequency (cps)
G	390	D	584
G#	413	D#	619
A	438	E	656
A#	464	F	695
B	491	F#	736
C	521	G	780
C#	551

Note that the frequency doubles over an interval of an octave (12 steps).

2. a. A total of 66 votes were cast.

b. A received 8 first place votes, B received 20 first place votes, C received 16 first place votes, and D received 22 first place votes. Therefore, D is the plurality winner (but not by a majority).

c. From part (b), we see that B and D enter the runoff and the votes of A and C are redistributed. Now B receives 20 + 6 = 26 votes and D receives the remainder, or 40, of the votes. Thus D is the winner of the top two runoff.

d. In the sequential runoff we eliminate only the candidate with the fewest first place votes at each stage. From part (b), we see that A is eliminated first. Redistributing A’s votes, D receives A’s 8 first place votes; so at this point B has 20 votes, C has 16 votes, and D has 30 votes. Now C is eliminated and the election is between B and D, as in part (c). The winner by sequential runoff is D.

e. For the Borda Count, we score 4 points for a first place vote, 3 points for a second place vote, 2 points for a third place vote, and 1 point for a fourth place vote. The point totals are as follows:

A: (20 ´ 1) + (15 ´ 3) + (10 ´ 2) + (8 ´ 4) + (7 ´ 3) + (6 ´ 3) = 156.

B: (20 ´ 4) + (15 ´ 1) + (10 ´ 1) + (8 ´ 1) + (7 ´ 2) + (6 ´ 2) = 139.

C: (20 ´ 2) + (15 ´ 2) + (10 ´ 4) + (8 ´ 2) + (7 ´ 1) + (6 ´ 4) = 157.

D: (20 ´ 3) + (15 ´ 4) + (10 ´ 3) + (8 ´ 3) + (7 ´ 4) + (6 ´ 1) = 208.

Note that the point total is 660, as it must be. We see that D is the winner by the Borda count.

f. Here are the results of the 6 pairwise races:

A over B, 46 to 20. C over A, 36 to 30. D over A, 52 to 14. C over B, 39 to 27. D over B, 40 to 26. D over C, 50 to 16.

Thus A scores 1 point, B scores 0 points, C scores 2 points and D scores 3 points. D wins by the pairwise comparison method.

g. As the winner by all five methods, candidate D is clearly the winner of the election.