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

Question

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

A. 12
B. 13
C. 14
D. 15
E. 16

aayush.works · Accepted Answer

number of pianists = p
number of violinists = 24-p

number of union members = p/2 + 2/3(24-p)
=> p(1/2-2/3) + 16 => 16 +p( 3-4/6 ) => 16 - p/6
=> now for minimum value of above expression, p has to get max value.
p cannot be 24 as we are given there are "some" violinists as well.. So next highest value p can take is 18.

Substituting value of p=18, in the above expression for number of union members we get :-
=> 16 - 3 = 13 Ans.

Archit3110 · Answer

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

A. 12
B. 13
C. 14
D. 15
E. 16

As the question stem says "some are pianists and the rest are violinists" so the ratio is definately not 50:50 .
The no of pianists and violinist have to such that we get an integer value for union so pairs can be ( 13,11), (14,10), (15,9),(16,8),(18,6)... so on
One thing to observe is that only pair (18,6) gives integer value for union

Let pianist be 18 and 6 be violinist : union ( 6*2/3+18*1/2)= 4+9=13 option b

dave13 · Answer

Since 2/3 are violonists, we need to minimize # of violonists. Let total # of violonists be 4 Let total # of pianists be 20 20*\frac{1}{2} = 10 pianists 4*\frac{2}{3} = 2.6 (round it) and get 3 violonists 10+3 =13

my fav number

souvikgmat1990 · Answer

The answer is B 13.
Please find the solution below:

P+V=24 ---(i)
Now taking Union into the equation we get,
(1/2)P+(2/3)V=U (Union) ---(ii)

Few points can be deduced from the above two eqn.s:
1. From the eqn (ii) we see that V needs to be minimized since it gets multiplied by 2. Also, V needs to be a multiple of 3.
2. From eqn (ii) we also see P needs to be an even number since it is getting divided by 2
3. Thereby V also needs to be an even number since e+e=e since 24 is an even number

So based on the constraints on V the smallest number we can come up is 6 (Even and a multiple of 3).
Plugging in 6 in eqn (i) we get P=18

Now plugging in the value of P and V in eqn (ii) we get (1/2)*18+(2/3)*6=9+4=13

ScottTargetTestPrep · Answer

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 is also 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.

So 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

priyanknema · Answer

Ok ! lets just say 13 is the min no. of persons in a group -- but I'm confuse because we dont have integer value for 1/2 of 13 or 2/3 of 13

aayush.works · Answer

1. Since the question asked us to minimize the number of union members we need to maximize (p/6) in the expression 16 - p/6.
2. We assumed the number of pianists to be p; and number of violinists to be 24-p. From point number 1. as stated above we know that we need to maximize (p/6) and the only integer values (we need integer values since we are counting people) that p can take are multiples of 6 : {6,12,18,24}. p cannot be 24 since that will defeat the question stem which states that there are "some" violinists as well. So the next largest integer value is "18".
3. Finally, to answer your question: 
we know from points 1 and 2 above that p=18 and 
number of pianists = p = 18 (integer value for 18/2 = 9) 
number of violinists to be 24-p = 6 (integer value for 6 * 2/3 = 4)

priyanknema · Answer

Understood....thanks..

Sent from my iPhone using GMAT Club Forum

BrentGMATPrepNow · Answer

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

EMPOWERgmatRichC · Answer

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

Fdambro294 · Answer

Use Number Properties to Solve:

Let P = No. of Pianists in the group

Let V = No. of Violinists in the group

P + V = 24 = Total No. of People ----- (equation 1)

(1/2)P + (2/3)V = No. of People in the UNION ----- (equation 2)

The Goal of the Question is to MINIMIZE the No. of People in the Union:

Looking at equation 2:

We know that people either have to be V or P - there is no other option. So it comes down to how best we can allocate the people so that we have the LEAST Number of Union members

Case 1: if more people were Violinists > and less people were Pianists

Since (2/3)rds of V are in the Union and a lower % of P (1/2) are in the Union,
we would end up with MORE PEOPLE in the Union

than IF -----

Case 2: LESS people were Violinists < and more people were Pianists

In other words, in order to Minimize:

(1/2)P + 2/3(V)

Given that every person has to be part of at least 1 of the Sets: P or V

we are Best off making as many people part of Set P as we can and as FEW people part of Set V as we can

However, since we are dealing with an Integer Constraint (we can NOT have Fractional People) AND since EXACTLY (2/3)rds of the Violinists are in the Union, it must be True that:

No. of Violinists = V = Multiple of 3

(1) Let V = 3

P + V = 24 people total

P = 24 - 3 = 21

However, we are told that (1/2) of the Pianists are in the Union. We can NOT have 1/2 of 21 People be in the Union because it Fails the Integer Requirement

(2) Let V = 6

P + V = 24

P = 24 - 6 = 18

18 IS Divisible by the DEN of (1/2) and there can be an Integer No. of Pianists in the Union

Summary:
Minimize the No. of People in the Union when -

V = 6 and P = 18 (P + V = 24 ------ 18 + 6 = 24)

LEAST No. of People we can possibly have in the Union = (1/2)P + (2/3)V = (1/2)18 + (2/3)6 = 9 + 4 =

13 People

-B-

Prowess · Answer

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

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