GMAT Set Theory – Two sets

Of the 180 students in a school, at least 45% attended the prom night and at least 40% took part in the debating session. What is the maximum number of students who would have neither attended the prom night nor the debating session?
A. 99
B. 90
C. 81
D. 27
E. 153

Correct Answer – Choice E. 99 students attended neither of the events.

Explanatory Answer

Total 180 students.

At least 45% attended the prom night.
So, the number should be a minimum of 45% of 180 = 81 students.

At least 40% took part in the debate.
So, the number should be a minimum of 40% of 180 = 72 students.

We have been asked to find out the maximum number of students who attended neither the prom night nor the debate.

If set A represents those who attended the prom night and set B represents those who attended the debate,
Set (A U B) will give us the set of students who attended at least one of the two events.

And Set (A U B)’ will give the set of students who attended neither of the two events.

If we have to maximize (A U B)’, we have to minimize (A U B).

n(A U B) = n(A) + n(B) – n(A n B)
In this question, n(A U B)  = 81 + 72 – n(A n B) = 153 – n(A n B)

If n(A U B) has to be minimized, we should maximize n(A n B).
The maximum value that n(A n B) can take is the smaller of n(A) and n(B).

So, max n(A n B) = n(B) = 72.

Therefore, min n(A U B) = 153 – 72 = 81.

Hence, max n(A U B)” = 180 – 81 = 99