Read the information carefully and answer the question below:

- The number of people who like **only volleyball** is $(x + 10)$.
- The number of people who like **only chess** is 15 less than those who like only volleyball.
- The number of people who like **only cricket** is 28.
- The average number of people who like **only one game** is 21.
- The number of people who like **all three games** is 50.
- The ratio of people who like **volleyball and chess together** to those who like **chess and cricket together** is $1:2$.
- The total number of people who like **chess** is 96.

**Question:** What is the ratio of the number of people who like **only cricket and chess together** to the number of people who like **only volleyball**?

Question

Read the information carefully and answer the question below:

- The number of people who like **only volleyball** is $(x + 10)$.
- The number of people who like **only chess** is 15 less than those who like only volleyball.
- The number of people who like **only cricket** is 28.
- The average number of people who like **only one game** is 21.
- The number of people who like **all three games** is 50.
- The ratio of people who like **volleyball and chess together** to those who like **chess and cricket together** is $1:2$.
- The total number of people who like **chess** is 96.

**Question:** What is the ratio of the number of people who like **only cricket and chess together** to the number of people who like **only volleyball**?

Accepted Answer

**Solution:**

1. Let the number of people who like only volleyball be $a = x + 10$.

2. The number of people who like only chess is $c = a - 15 = x + 10 - 15 = x - 5$.

3. The number of people who like only cricket is given as $g = 28$.

4. The average number of people who like only one game is 21, so:
   $$\frac{a + c + g}{3} = 21$$
   Substitute values:
   $$\frac{(x + 10) + (x - 5) + 28}{3} = 21$$

5. Simplify the numerator:
   $$\frac{2x + 33}{3} = 21$$

6. Multiply both sides by 3:
   $$2x + 33 = 63$$

7. Solve for $x$:
   $$2x = 63 - 33 = 30 \implies x = 15$$

8. Now, calculate $a$ and $c$:
   - $a = x + 10 = 15 + 10 = 25$
   - $c = x - 5 = 15 - 5 = 10$

9. Let the number of people who like volleyball and chess together be $b = n$.

10. The number of people who like chess and cricket together is $f = 2n$ (given ratio $1:2$).

11. The number of people who like all three games is $e = 50$.

12. Total people who like chess is 96, so:
    $$c + b + f + e = 96$$
    Substitute values:
    $$10 + n + 2n + 50 = 96$$

13. Simplify:
    $$3n + 60 = 96 \implies 3n = 36 \implies n = 12$$

14. Therefore:
    - People who like volleyball and chess together = $b = 12$
    - People who like chess and cricket together = $f = 24$

15. The question asks for the ratio of people who like **only cricket and chess together** (i.e., $g + f = 28 + 24 = 52$) to people who like **only volleyball** ($a = 25$):

$$	ext{Ratio} = 52 : 25 = 24 : 25 \quad 	ext{(after simplifying by dividing both sides by 2)}$$

**Answer:** $24 : 25$

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