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Try this GRE Word Problems question [Difficulty level – Medium]

Of the 200 students, 130 are studying Mathematics and 150 are studying Physics. If at least 30 of the students are not studying either Mathematics or Physics, then the number of students studying both Mathematics and Physics could be between

(A) 20 to 50
(B) 40 to 70
(C) 50 to 130
(D) 110 to 130
(E) 110 to 150

Here is the solution:

This is a Venn diagram question. There are two things to consider:

(1) Minimum students studying both Mathematics and Physics
(2) Maximum students studying both Mathematics and Physics.

(1) If at least 30 students are not studying either Mathematics or Physics, then at most 200 – 30 = 170 students are studying in either or both. Since there are 130 + 150 = 280 Physics and Mathematics students, then there are at least 280 – 170 = 110 studying both.

(2) The maximum number of students studying both Mathematics and Physics is 130, since 130 are studying Mathematics.

The total number of students studying Mathematics, or Physics, or both is 130 + 20 = 150. Thus, there are 200 – 150 = 50 students who are neither studying Mathematics nor Physics. This number is consistent with the condition that 30 is the minimum number of students not studying Mathematics and Physics. Thus, the number of students studying both Mathematics and Physics could be any number from a minimum of 110 to a maximum of 130.