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

The Correct Option is B

Let $G$ represent the number of girls and $B$ represent the number of boys, where $B = 250 - G$.
Participation in Swimming and Running: 50% of boys and 70% of girls participated in swimming; 80% of boys and 60% of girls participated in running.
Number of students participating in both activities: Let $x$ be the number of boys and $y$ be the number of girls participating in both swimming and running.
Using the principle of inclusion-exclusion:
- The total number of boys participating in swimming and running is:
\[ 0.5B + 0.7B - x = 1.2B - x \]
- The total number of girls participating in swimming and running is:
\[ 0.8G + 0.6G - y = 1.4G - y \]
The total number of students participating in both activities (boys and girls) is the sum of these:
\[ 1.2B - x + 1.4G - y = 1.4G + 1.2B - x - y \]
To find the maximum and minimum values of $x$ and $y$: For the minimum number of students participating in both activities, we assume maximum overlap between boys and girls in swimming and running participation. Therefore, we calculate:
\[ x = 72 \quad \text{and} \quad y = 80 \]
Thus, the maximum number of students participating in both activities is 80.