There are 200 students in a class playing either badminton, tennis, or both. The number of boys playing only tennis is 1/3 of the total girls playing only one game. The number of girls playing badminton equals the total number of boys. The number of girls playing both games is 40\% more than the number of boys playing only badminton, and the number of girls playing only tennis is twice the number of boys playing only badminton. The number of boys playing both games is half the number of girls playing only badminton.

Compare the following quantities:

- Quantity I: Difference between the number of girls playing only badminton and only tennis.
- Quantity II: 60\% of the number of boys playing both games.
- Quantity III: 40\% of the difference between boys playing only badminton and girls playing only tennis.

Question

There are 200 students in a class playing either badminton, tennis, or both. The number of boys playing only tennis is 1/3 of the total girls playing only one game. The number of girls playing badminton equals the total number of boys. The number of girls playing both games is 40\% more than the number of boys playing only badminton, and the number of girls playing only tennis is twice the number of boys playing only badminton. The number of boys playing both games is half the number of girls playing only badminton.

Compare the following quantities:

- Quantity I: Difference between the number of girls playing only badminton and only tennis.
- Quantity II: 60\% of the number of boys playing both games.
- Quantity III: 40\% of the difference between boys playing only badminton and girls playing only tennis.

Accepted Answer

**Solution:**

1. Let the number of boys be $ B $ and girls be $ G $.

2. Given total students = 200, so $ B + G = 200 $.

3. Girls playing badminton = number of boys $ \Rightarrow G_{badminton} = B $.

4. Boys playing only tennis = $ \frac{1}{3} $ of total girls playing only one game.

5. Let the number of boys playing only badminton be $ b_b $.

6. Girls playing both games = 40\% more than boys playing only badminton:
   $$ G_{both} = b_b + 0.4 b_b = 1.4 b_b $$

7. Girls playing only tennis = twice the boys playing only badminton:
   $$ G_{tennis} = 2 b_b $$

8. Boys playing both games = half of girls playing only badminton:
   $$ B_{both} = \frac{1}{2} G_{badminton\_only} $$

9. Using the above, we find:
   - Girls playing only badminton = $ G_{badminton} - G_{both} = B - 1.4 b_b $

10. Total girls playing only one game = girls playing only badminton + girls playing only tennis:
    $$ (B - 1.4 b_b) + 2 b_b = B + 0.6 b_b $$

11. Boys playing only tennis = $ \frac{1}{3} $ of total girls playing only one game:
    $$ B_{tennis} = \frac{1}{3} (B + 0.6 b_b) $$

12. Total boys = boys playing only badminton + boys playing only tennis + boys playing both:
    $$ B = b_b + B_{tennis} + B_{both} = b_b + \frac{1}{3} (B + 0.6 b_b) + \frac{1}{2} (B - 1.4 b_b) $$

13. Solving the above equation for $ B $ and $ b_b $ gives:
    $$ B = 75, \quad b_b = 25 $$

14. Then:
    - Girls playing only badminton = $ 75 - 1.4 \times 25 = 75 - 35 = 40 $
    - Girls playing only tennis = $ 2 \times 25 = 50 $
    - Boys playing both = $ \frac{1}{2} \times 40 = 20 $

15. Now calculate the quantities:
    - Quantity I: Difference between girls playing only badminton and only tennis:
      $$ |40 - 50| = 10 $$
    - Quantity II: 60\% of boys playing both games:
      $$ 0.6 \times 20 = 12 $$
    - Quantity III: 40\% of difference between boys playing only badminton and girls playing only tennis:
      $$ 0.4 \times (25 - 50) = 0.4 \times (-25) = -10 $$
      Taking absolute value for comparison, $ 10 $.

16. Comparing:
    - Quantity II (12) > Quantity I (10) = Quantity III (10)

**Answer:** Quantity II > Quantity I = Quantity III

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