M15-16

Official Solution:

Of the 200 surveyed students, 20% of the students who read book A also read book B, and 25% of the students who read book B also read book A. If each student read at least one of the two books, what is the positive difference between the number of students who read only book A and the number of students who read only book B?

A. 20
B. 25
C. 30
D. 35
E. 40


Let the number of students who read book A be A and the number of students who read book B be B.

We know that 20% of those who read book A also read book B, hence the number of students who read both books is 0.2A. We also know that 25% of those who read book B also read book A, hence the number of students who read both books is 0.25B. Therefore, 0.2A = 0.25B, which gives A = 1.25B.

Since we're given that each student read at least one of the books, then {None} = 0 and we have {Total} = {A} + {B} - {Both}. Substituting A = 1.25B and Total = 200, we get 200 = 1.25B + B - 0.25B, which gives B = 100, therefore, A = 1.25B = 125, and Both = 0.25B = 25.

Thus, the number of students who read only book A is A - Both = 125 - 25 = 100, and the number of students who read only book B is B - Both = 100 - 25 = 75. The positive difference between these two numbers is 100 - 75 = 25.


Answer: B