BTGmoderatorDC wrote:In a group of 24 musicians, some are pianists and the rest are violinists. Exactly 1/2 of the pianists and exactly 2/3 of the violinists belong to a union. What is the least possible number of union members in the group?
A. 12
B. 13
C. 14
D. 15
E. 16
OA B
Source: Princeton Review
We can let p = the number of pianists, v = the number of violinists, and u = the number of union members. We see that p is a multiple of 2, and v is a multiple of 3. Also, the number of violinists can't be odd; otherwise, the number of pianists would be odd. Thus, v can only be 6, 12, or 18.
If v = 6, then p = 18 and u = ½(18) + ⅔(6) = 9 + 4 = 13.
If v = 12, then p = 12 and u = ½(12) + ⅔(12) = 6 + 8 = 14.
If v = 18, then p = 6 and u = ½(6) + ⅔(18) = 3 + 12 = 15.
Of the 3 possible values of u that we have calculated (13, 14, 15), the least possible number of union members in the group is 13.
Alternate Solution:
We see that the number of violinists must be a multiple of 3, and the number of pianists must be an even number. We want to minimize the number of violinists, since a greater percentage of violinists are in the union.
We could start with 3 violinists and 21 pianists, but this won't work. But with 6 violinists, we have 18 pianists. Thus, we would have (2/3)(6) = 4 violinists and 18/2 = 9 pianists in the union, for a total of 13 musicians in the union.
Answer: B.
