BTGmoderatorLU wrote:Source: Manhattan Prep

55 people live in an apartment complex with three fitness clubs (A, B, and C). Of the 55 residents, 40 residents are members of exactly one of the three fitness clubs in the complex. Are any of the 55 residents members of both fitness clubs A and C but not members of fitness club B?

1) 2 of the 55 residents are members of all three of the fitness clubs in the apartment complex.
2) 8 of the 55 residents are members of fitness club B and exactly one other fitness club in the apartment complex.

The OA is E

Given that 40 residents are members of exactly one of the three fitness clubs, we have 55 - 40 = 15 residents who are members of (1) A and C clubs (2) A and B clubs (3) B and C clubs and (4) all three clubs (5) no club

Say,

the number of residents who are members of A and B clubs = AB;
the number of residents who are members of A and C clubs = AC;
the number of residents who are members of C and B clubs = BC;
the number of residents who are members of all the three clubs = ABC; and
the number of residents who are members of no club = N;

Thus, we have AB + AC + BC + ABC + N = 15 ---(1).

We have to determine whether AC > 0.

Let's take each statement one by one.

1) 2 of the 55 residents are members of all three of the fitness clubs in the apartment complex.

=> ABC = 2. We can't get the value of AC from equation (1). Insufficient.

2) 8 of the 55 residents are members of fitness club B and exactly one other fitness club in the apartment complex.

=> AB + BC = 8. We can't get the value of AC from equation (1). Insufficient.

(1) and (2) together

We have AB + AC + BC + ABC + N = 15 => 8 + 2 + AC + N = 15 => AC + N = 5. Can't get the value of AC. Insufficient.

The correct answer: E

Hope this helps!

-Jay
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