In a group of 24 musicians, some are pianists and the rest a

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

In a group of 24 musicians, some are pianists and the rest are violinists.
Let x = number of pianists
So, 24-x = number of violinists

Exactly 1/2 of the pianists and exactly 2/3 of the violinists belong to a union.
So, the number of pianists in the union = x/2
And the number of violinists in the union = (2/3)(24-x) = 16 - 2x/3

What is the least possible number of union members in the group?
Number of union members = x/2 + (16 - 2x/3)
Rewrite with common denominators to get: 3x/6 + 16 - 4x/6
Simplify to get: 16 - x/6

Our goal is to MINIMIZE the value of 16 - x/6
To do so, we must MAXIMIZE the value of x/6
Since x must be divisible by 6 and since x must be less than 24, the greatest possible value of x is 18
When x = 18, we get 16 - x/6 = 16 - 18/6 = 16 - 3 = 13

Answer: B

Cheers,
Brent

Hi All,

We're told that in a group of 24 musicians, some are pianists and the rest are violinists and exactly 1/2 of the pianists and exactly 2/3 of the violinists belong to a union. We're asked for the LEAST possible number of union members in the group. This question can be approached in a couple of different ways, including the use of some Number Properties and some simple 'brute force' Arithmetic.

To start, since we can't have a 'fraction' of a musician, we know a couple of things about the number of pianists and the number of violinists:
-The number of pianists MUST be a multiple of 2 (since 1/2 of them are in a union)
-The number of violinists MUST be a multiple of 3 (since 2/3 of them are in a union).

To MINIMIZE the total number of union members, we need there to be MORE pianists (since only 1/2 of pianists are in a union vs. 2/3 of violinists). Thus, we need to add a multiple of 2 to a multiple of 3 and get a total of 24.... with the multiple of 2 being as big as possible. We can list out the first few options until we find a match:

24 total people could be:
3 violinists and 21 pianists --> NOT possible (number of pianists here is NOT a multiple of 2).
6 violinists and 18 pianists --> This matches both of the 'restrictions' in the prompt and gives us the maximum number of pianists

This option gives us 6(2/3) + 18(1/2) = 4 + 9 = 13 union members.

Final Answer: B

GMAT assassins aren't born, they're made,
Rich

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.

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.