True/False question from GRE Quant about no. of students taking French and Geography.

Question

Today when I was solving problems from GRE Manhattan I ran into a strange word problem.

The first line of the problem already seems weird since $\frac{3}{8}$ of $420$ is not an integer, namely it is equal to $\frac{3}{8}\cdot 420=157,5$. Am I right?

Answer 1

You're right. It works out that $105$ students took both, $52.5$ took French but not Geography, $63$ took Geography but not French, and $199.5$ took neither. So the answer intended is presumably that only C is true, but in fact the situation described is impossible.

Answer 2

Yes, you are absolutely right. A typo has crept in or something like that. It happens both in actual tests and in various textbooks.

Students who took French: $3/8\cdot420=157.5$

Students who took geography: $2/5\cdot 420=168$

Students who took both: $1/4\cdot 420=105$

Students who took geography but not French: $168-105=63$

Students who took French but not geography: $157.5-105=52.5$

Students who took neither: $420-63-52.5-105=199.5$

Therefore answer C should be correct. A Venn diagram is the best way to illustrate the solution.

Please note that unfortunately this is normal for a test. There might be $1\mbox{-}2\%$ of questions, which are either badly prepared, or allow for some ambiguity, or as in your case something like a fractional number of persons pops up, or etc. The verbal section allows ten times as much ambiguity which is not good but not a deal breaker. I mean a student can still pass his/her test with flying colors. It’s just that mathematical questions are much more precise and have no ambiguity and no mistakes with some rare exceptions as was your question. My advice is not to waste time on such things. Such aberrations will pop up very, very rarely in math. So, there’s nothing wrong with Manhattan, Princeton, Barron’s, Kaplan or the actual exam. They are all good books in my opinion.