Question

In a group of 250 students, the percentage of girls was at least 44% and at most 60%. The rest of the students were boys. Each student opted for either swimming or running or both. If 50% of the boys and 80% of the girls opted for swimming while 70% of the boys and 60% of the girls opted for running, then the minimum and maximum possible number of students who opted for both swimming and running, are

Accepted Answer

A. Let total students $= 250$. Number of girls $G$ satisfies $0.44 	imes 250 \le G \le 0.60 	imes 250 \implies 110 \le G \le 150$. Number of boys $B = 250 - G$. For boys: $S_B = 0.5B$, $R_B = 0.7B$. Since every student opted for at least one activity, $(S \cap R)_B = S_B + R_B - B = 0.5B + 0.7B - B = 0.2B$. For girls: $S_G = 0.8G$, $R_G = 0.6G$. $(S \cap R)_G = 0.8G + 0.6G - G = 0.4G$. Total students opting for both: $$	ext{Both} = 0.2B + 0.4G = 0.2(250 - G) + 0.4G = 50 + 0.2G$$ - Minimum value (when $G = 110$): $50 + 0.2(110) = 72$. - Maximum value (when $G = 150$): $50 + 0.2(150) = 80$. Thus, the minimum and maximum numbers are 72 and 80, respectively.

Students opting for both activities

Explanation

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