Each of the 2001 students at school studies either Spanish or French and some study both.
The number who study Spanish is between 80% and 85% of the school population and the number who study French is between 30% and 40% Let ‘m’ be the smallest number of students who could study both languages and let ‘M’ be the largest number of students who could study both languages then $\frac{M - m}{149} =$ ______

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answer is 2.

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Detailed Solution

Let $S \to$ Spanish, $F \to$ French, $S \cap F \to$ number of students who study both
$(80 % \times 2001) = 1601 \leq S \leq (85 % \times 2001) = 1700$
$(30 % \times 2001) = 601 \leq F \leq (40 % \times 2001) = 800$ , by the principle of Inclusion-Exclusion
$S + F - S \cap F = S \cup F = 2001$
For $' m' = S \cap F$ to be smallest, S and F must be minimized $\to$ 1601+601-m=2001 $\to$ m=201
For $' M' = S \cap F$ to be largest, S and F must be maximized $\to$ 1700+800-M=2001 $\to$ M=499