In a school, there are a total of $Y$ students (boys and girls combined). 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.

What is the difference between the number of students who like only one sport and the number of students who like all three sports?

Question

In a school, there are a total of $Y$ students (boys and girls combined). 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.

What is the difference between the number of students who like only one sport and the number of students who like all three 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 $Y$ = $0.04Y$
   - Total students who like at least two sports = 312

3. Sum of students who like exactly two sports:
   $$0.16Y + 0.13Y + 0.04Y = 0.33Y$$

4. Students who like all three sports = 114

5. Total students who like at least two sports = students who like exactly two + students who like all three:
   $$0.33Y + 114 = 312$$

6. Solve for $Y$:
   $$0.33Y = 312 - 114 = 198$$
   $$Y = \frac{198}{0.33} = 600$$

7. Now, find the number of students who like exactly two sports:
   $$0.33 	imes 600 = 198$$

8. Total students who like at least two sports = 312, so students who like all three = 114 (given)

9. Let the number of students who like only A, only B, and only C be $2x$, $3x$, and $x$ respectively (ratio 2:3:1).

10. Total students liking only one sport:
    $$2x + 3x + x = 6x$$

11. Total students liking at least one sport = $Y = 600$.

12. Total students liking at least two sports = 312 (given).

13. Students liking only one sport = Total students - students liking at least two sports:
    $$600 - 312 = 288$$

14. So, $6x = 288 \Rightarrow x = 48$

15. Number of students who like only one sport = 288

16. Number of students who like all three sports = 114

17. Difference = $288 - 114 = 174$

**Check:** The options include 174 but the correct answer is 188 according to the options. Let's re-check step 13.

Step 13 re-examined:
- Total students = 600
- Students who like at least two sports = 312
- Students who like only one sport = $600 - 312 = 288$

Step 17:
- Difference = $288 - 114 = 174$

Since 174 is an option, but the question's correct answer is marked as 188, let's verify if the total students liking only one sport is indeed 288.

Alternatively, sum of all students liking sports:
- Only one sport = $6x$
- Exactly two sports = $0.33Y = 198$
- All three sports = 114

Sum:
$$6x + 198 + 114 = 600$$
$$6x + 312 = 600$$
$$6x = 288$$
$$x = 48$$

So, only one sport = 288

Difference = $288 - 114 = 174$

Therefore, the correct difference is 174.

**Answer:** 174

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