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Pianists and Violinists

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Pianists and Violinists

by gmataspirant » Fri Jun 13, 2008 3:31 am
In a group of 24 muscians , 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

How do you solve it ?

OA is B
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by spanlength » Fri Jun 13, 2008 3:40 am
Lets say we hav x pianists and (24-x) violinists. Now the union has
x/2 + 2/3(24-x) members

hence count of members = 16- (x/6).This value needs to be least, hence the value of x/6 must be 16 but here we have the value of x must satisfy the following conditions : 1) should be an integral multiple of 6 and should be less than 24. So value of x is 18.
Hence count = 16- 18/6 = 13.

Any comments !!
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by AleksandrM » Fri Jun 13, 2008 9:27 am
I think the easiest way to approach this problem is to notice that 1/2 is less than 2/3. So, you want to minimize the number of 2/3, therefore, there will be 3 violinists with 2 in the union and 21 pianists. Since you cannot have 10.5 people, you have 11 pianists in the union [the urge may be to round down, but do not do this since you have 10.5, not 10.4].
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by anayak » Fri Jun 13, 2008 9:33 am
As 1/2 of the pianists and 2/3 of musicians are there, think of split as 18 pianists and 6 musicians. (Other combinations could be 12,12; 6,18...etc) but the some of 1/2 pianists and 2/3 musicians would be less when taken the split 18,6.

Therefore the ans : 9+6=13
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by Ian Stewart » Fri Jun 13, 2008 11:21 am
AleksandrM wrote:I think the easiest way to approach this problem is to notice that 1/2 is less than 2/3. So, you want to minimize the number of 2/3, therefore, there will be 3 violinists with 2 in the union and 21 pianists. Since you cannot have 10.5 people, you have 11 pianists in the union [the urge may be to round down, but do not do this since you have 10.5, not 10.4].
Careful! The question says exactly half the pianists are in the union. You can't have 21 pianists, because then 10.5 pianists would be in the union- that doesn't make any sense, of course. If you round up to 11, you have 11/21 of the pianists in the union, not 1/2. Otherwise, your approach is correct- you want to maximize the number of pianists, and the maximum is 18 (unless you admit the possibility that all 24 musicians are pianists).
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by babachal » Fri Jun 13, 2008 1:06 pm
Let's solve this problem, like following;
x-pianists
y-viol
x+y=24
z=total union members
x/2+2y/3=z=(3x+4y)/6

z=(3x+3y+y)/6=(3(x+y)+y)/6=3(x+y)/6 + y/6;
we gonna replace x+y with 24
3*24/6+y/6=12+y/6;
here, the questin is asking minimum, so we have to figure out what we have to find from y/6, it should be equal to 0 or 1, if it is zero y is equal to zero, this is not real answer, if y/6 is 1 then minimum number y members is 6

so, z=12+y/6=12+1=13
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by AleksandrM » Fri Jun 13, 2008 3:08 pm
Good catch, Ian. :)
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