Read the information and answer the following questions. There are total Y students (boys + girls) in a school. Each student like at least one sport out of three sports (A, B & C). The number of students who like all the three sports is 114. 16% of the total students like A and B but not C. 13% of the total students like B and C but not A. 4% of the total students like A and C but not B. The ratio of number of students who like only A, only B and only C is 2: 3: 1 respectively. Total number of students who like at least two sports are 312.

What is the ratio of girls to boys in the school?

Question

Read the information and answer the following questions. There are total Y students (boys + girls) in a school. Each student like at least one sport out of three sports (A, B & C). The number of students who like all the three sports is 114. 16% of the total students like A and B but not C. 13% of the total students like B and C but not A. 4% of the total students like A and C but not B. The ratio of number of students who like only A, only B and only C is 2: 3: 1 respectively. Total number of students who like at least two sports are 312.

What is the ratio of girls to boys in the school?

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 and all three sports = 312

So,
   $$0.16Y + 0.13Y + 0.04Y + 114 = 312$$

$$0.33Y + 114 = 312$$

$$0.33Y = 312 - 114 = 198$$

$$Y = \frac{198}{0.33} = 600$$

4. Now, the number of students who like exactly two sports is:

$$0.16Y + 0.13Y + 0.04Y = 0.33 	imes 600 = 198$$

5. The number of students who like all three sports is 114.

6. Total students who like at least one sport is $Y = 600$.

7. The ratio of students who like only A, only B, and only C is 2:3:1.

Let the common ratio factor be $x$.

So, number of students who like only A = $2x$

Number of students who like only B = $3x$

Number of students who like only C = $x$

8. Total students liking at least one sport is sum of:

- Only A, only B, only C
   - Exactly two sports
   - All three sports

So,

$$2x + 3x + x + 198 + 114 = 600$$

$$6x + 312 = 600$$

$$6x = 288$$

$$x = 48$$

9. Number of students who like only A = $2 	imes 48 = 96$

Number of students who like only B = $3 	imes 48 = 144$

Number of students who like only C = $48$

10. The problem does not provide any information about the number of boys or girls or their distribution among these categories.

11. Therefore, it is impossible to determine the ratio of girls to boys with the given data.

**Answer:** Can’t determine

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