Problem Solving

BTGmoderatorLU wrote:Source: Magoosh

Highmont private school polled the graduating class of 125 for the musical genres they listen to. 88% indicated at least one of the following genres: indie rock, classical, country, and electronica. Of these students, 40% responded they listen to only one of the four genres. The average number of students within each distinct group of those who listen to exactly two genres is greater than the total number of those who listen to exactly three genres. If the only lack of overlap in musical tastes was in country and electronica, what is the greatest number of students who could listen to classical, country, and indie rock?

A. 5
B. 6
C. 9
D. 10
E. 13

We have 0.88(125) = 110 students who listen to at least one of the four genres. We also have 0.4(110) = 44 students who listen to exactly one of the four genres. This leaves us 110 - 44 = 66 students who listen to more than one genre. However, since there is no overlap in country and electronica, we don't have any students who listen to all four genres. In other words, the 66 students either listen to exactly two genres or exactly three genres.

The two-genre groups are: indie rock & classical, indie rock & country, indie rock & electronica, classical & country, and classical & electronica.

The three-genre groups are: indie rock & classical & country and indie rock & classical & electronica.

Let's say the average number of students in the two-genre groups is n and the average number of students in the three-genre groups is m, we have:

4n + 2m = 66

2n + m = 33

We are also given that n > 2m. Therefore, 2n > 4m. So 2n + m > 4m + m = 5m. That is, we have:

33 > 5m

6.6 > m

However, since 2n + m = 33 and n must be an integer, we see that m must be odd. In that case, the largest value of m is 5. Since m is the average number of students in the 2 three-genre groups, the total number of students of the 2 three-genre groups is 2 x 5 = 10. Since each of these 2 groups has at least 1 student, we can assume that the "indie rock & classical & electronica" group has exactly 1 student, so that the number of students in the "indie rock & classical & country" group will be 9, which is the maximum number of students it can have.

Answer: C