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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.

What is the difference between the total number of people who like only one game and the number of people who like all three games?

Question

Read the information carefully and answer the question:

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.

What is the difference between the total number of people who like only one game and the number of people who like all three games?

Accepted Answer

**Solution:**

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

2. People who like only chess are 15 less than those who like only volleyball:
   $$	ext{Only Chess} = (x + 10) - 15 = x - 5$$

3. People who like only cricket = 28.

4. Average number of people who like only one game is 21, so:
   $$\frac{(x + 10) + (x - 5) + 28}{3} = 21$$

5. Simplify the equation:
   $$\frac{2x + 33}{3} = 21$$
   $$2x + 33 = 63$$
   $$2x = 30$$
   $$x = 15$$

6. Calculate the number of people who like only volleyball and only chess:
   - Only volleyball = $x + 10 = 15 + 10 = 25$
   - Only chess = $x - 5 = 15 - 5 = 10$

7. Total people who like only one game:
   $$25 + 10 + 28 = 63$$

8. Number of people who like all three games = 50 (given).

9. Find the difference between people who like only one game and those who like all three games:
   $$63 - 50 = 13$$

**Answer:** 13

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