In a school, there are a total of $Y$ students (boys + girls). Each student likes at least one of the three sports: A, B, and C. The number of students who like all three sports is 114. Additionally, 16\% of the total students like sports A and B but not C, 13\% like sports B and C but not A, and 4\% like sports A and C but not B. The ratio of students who like only A, only B, and only C is 2:3:1 respectively. The total number of students who like at least two sports is 312. Among the students who like exactly two sports, one-third are girls.

What is the number of boys who like exactly two sports?

Question

In a school, there are a total of $Y$ students (boys + girls). Each student likes at least one of the three sports: A, B, and C. The number of students who like all three sports is 114. Additionally, 16\% of the total students like sports A and B but not C, 13\% like sports B and C but not A, and 4\% like sports A and C but not B. The ratio of students who like only A, only B, and only C is 2:3:1 respectively. The total number of students who like at least two sports is 312. Among the students who like exactly two sports, one-third are girls.

What is the number of boys who like exactly two sports?

Accepted Answer

**Solution:**

1. Let the total number of students be $Y$.

2. Given:
   - Number of students who like all three sports (A, B, C) = 114
   - Students who like A and B but not C = 16\% of $Y$ = $0.16Y$
   - Students who like B and C but not A = 13\% of $Y$ = $0.13Y$
   - Students who like A and C but not B = 4\% of $Y$ = $0.04Y$

3. Total students who like at least two sports = 312. This includes:
   - Students who like exactly two sports = sum of those who like A and B but not C, B and C but not A, A and C but not B
   - Students who like all three sports = 114

Therefore, sum of students who like exactly two sports = $312 - 114 = 198$

4. Students who like exactly two sports together = 198.

5. Among these 198 students, one-third are girls. So,
   - Number of girls who like exactly two sports = $\frac{1}{3} 	imes 198 = 66$
   - Number of boys who like exactly two sports = $198 - 66 = 132$

6. However, the options do not have 132 as the correct answer. Let's verify the calculations carefully.

7. Re-examining the problem statement: "Total number of students who like at least two sports are 312. The students who like only two sports together, one-third are girls."

- "Only two sports together" means students who like exactly two sports, excluding those who like all three.
   - So, exactly two sports = 312 - 114 = 198 (correct)

- One-third of these 198 are girls, so girls = 66, boys = 132.

8. The answer 132 is present in the options as (d).

**Answer:** 132

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