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The total number of students who study biology = .4(400) = 160.Among 400 students, 56% study sociology, 44% study mathematics and 40% study biology. If 30% of students study both mathematics and sociology, what is the largest possible number of students who study biology but do not study either mathematics or sociology?
A. 30
B. 90
C. 120
D. 172
E. 188
Eliminate D and E, since the number who study ONLY biology cannot exceed 160.
The greatest remaining answer choice is 120, which is equal to 30% of the 400 students.
Plug (only B) = 30% and the other given percentages into a Venn diagram:
By letting SB = 10% -- so that the total who study biology = 40% -- we can see it's possible that only B = 30%.
The correct answer is C.
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We can let S denote sociology, M denote mathematics, and B denote biology.Among 400 students, 56% study sociology, 44% study mathematics and 40% study biology. If 30% of students study both mathematics and sociology, what is the largest possible number of students who study biology but do not study either mathematics or sociology?
A. 30
B. 90
C. 120
D. 172
E. 188
Since %(S or M) = %(S) + %(M) - %(S and M), we have:
%(S or M) = 56 + 44 - 30 = 70
Thus, 70% of the students study sociology or mathematics or both. If we add this percentage to the 40% who study biology, we have 110%, which is not possible since it's over 100%. So, we must have (at least) 10% of the students who study biology and also study either mathematics or sociology or both. However, we can exclude that 10% of the students and say that the remaining 40% - 10% = 30% of the students study biology only. Thus, the largest possible number of students who study biology but do not study either mathematics or sociology is 0.3 x 400 = 120.
Answer: C
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