pgmat wrote:
At a certain school with 200 students, all children must take at least one of three language classes: German, French, and Spanish. If 100 students take German and none of the students who take French also take Spanish, then how many students take exactly two of the three language classes?
(1) 80 of the students study only German.
(2) 120 students study French or Spanish
Veritas Prep Official Explanation
To deal with this type of three-set Venn Diagram problem, first create a diagram that inserts all the information given in the question stem itself. The obvious information given is the fact that there are 200 total students, of which 100 take German. Also given is the fact that there are no students taking French and Spanish together. Less obvious is the fact that there are no students taking neither of the classes (“all students must take at least one”), and there are no students taking all three languages. (If there are no students taking both French and Spanish, then there cannot be any students taking all three.)
Once this is mapped out (which is the hardest part of the problem and is shown below), it is clear that statement (1) is sufficient, because if 80 take only German, then there must be 20 students total who take only German and Spanish or only German and French. Since the question wants exactly that total, then the answer is 20, and statement (1) is sufficient. In statement (2), you learn that the total value of the French and Spanish circles is 120. Since you know that there are 200 total students and 100 students who take German, the overlap between the sets must be 20 (200 = 100 + 120 – overlap ), and this is also sufficient to answer the question.
The correct answer choice is D.Below is the information given in the question stem. This allows you to see how plugging in each statement in will allow you to solve for the sum of exactly two regions:
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