bryan88 wrote: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:
T = G1 + G2 + G3 - (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 T = 8.
G1 = Albanian = (1/2)8 = 4.
G2 = Bardic = (5/8)8 = 5.
G3 = 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.
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doing this, u get all three only once (as needed).
