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 !!

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].

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

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

Good catch, Ian.