Clubs

3/8 of all students at Social High are in all three of the following clubs: Albanian, Bardic, and Checkmate. 1/2 of all students are in Albanian, 5/8 are in Bardic, and 3/4 are in Checkmate. If every student is in at least one club, what fraction of the student body is in exactly 2 clubs?

(A) 1/8

(B) 1/4

(C) 3/8

(D) 1/2

(E) 5/8

Here is the formula for 3 overlapping groups:

Total = Group 1 + Group 2 + Group 3 - (those in 2 of the groups) - 2(those in all 3 groups)

The big idea with overlapping group problems is to SUBTRACT THE OVERLAPS.
When we add together everyone in club A, everyone in club B, and everyone in club C:
Those in exactly 2 of the clubs are counted twice, so they need to be subtracted from the total ONCE.
Those in all 3 clubs are counted 3 times, so they need to be subtracted from the total TWICE.
By subtracting the overlaps, we ensure that no one is overcounted.

In the problem above:
Let Total = 8.
Group 1 = Albanian = (1/2)8 = 4.
Group 2 = Bardic = (5/8)8 = 5.
Group 3 = Checkmate = (3/4)8 = 6.
Those in exactly 2 clubs = x.
Those in all 3 clubs = (3/8)8 = 3.

Plugging these values into the formula, we get:
8 = 4 + 5 + 6 - x - 2(3)
x = 1.

Thus:
(Those in exactly 2 of the clubs)/total = 1/8.

The correct answer is A.