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

• A. 75 and 90, respectively
• B. 72 and 80, respectively
• C. 72 and 88, respectively
• D. 75 and 96, respectively

Answer 1

Answer: (B)

72 and 80, respectively

Answer 2

Let the number of girls be $G$, and the number of boys $B = 250 - G$.
Swimming and Running Participation: 50-70, 80-60
Number of students who opted for both swimming and running: Let $x$ be the number of boys who opted for both swimming and running, and $y$ be the number of girls who opted for both swimming and running.
From the principle of inclusion and exclusion, we have:
- The total number of boys who opted for swimming and running is:
$0.5B + 0.7B - x = 1.2B - x$
- The total number of girls who opted for swimming and running is:
$0.8G + 0.6G - y = 1.4G - y$
The total number of students who opted for both swimming and running (boys and girls) is the sum of these:
$1.2B - x + 1.4G - y = 1.4G + 1.2B - x - y$
Maximum and Minimum Values of $x$ and $y$: For the minimum number of students who opted for both swimming and running, we assume maximum overlap of boys and girls in swimming and running. Therefore, we calculate:
$x = 72 \quad \text{and} \quad y = 80$
Thus, the maximum number of students who opted for both swimming and running is 80.