3/8 of all students at a school are in all three of the following clubs: Albanian, Bardic, and Chess. 1/2 of all students are in Albanian, 5/8 are in Bardic, and 3/4 are in Chess  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




We can make the total number of students anything. Let's try 8

Since 3/8 of all students are in all 3 clubs, 3 students are in all 3 clubs
Since 1/2 of all students are in Albanian, 4 students are in Albanian
Since 5/8 of all students are in Bardic, 5 students are in Bardic
Since 3/4 of all students are in Chess, 6 students are in Chess.
From the given information, we know that 0 students are in none 
Letting t = the number of students in exactly two clubs, we can now use the equation:
Total = A + B + C - (exactly two clubs) -2(all three clubs) + none

8 = 4 + 5 + 6 - t - 2(3) + 0
8 = 15 - t - 6
t = 1

So, 1/8 are in exactly two clubs.

A