In a village of 800 persons, each plays at least one sport among Cricket, Football, and Hockey. The percentages of persons playing each sport are: Cricket 43\%, Football 52.5\%, and Hockey 55\%. The number of persons playing both Cricket and Football but not Hockey is 96, those playing both Cricket and Hockey but not Football is 100, those playing both Hockey and Football but not Cricket is 88, and those playing all three sports is 60.

What is the ratio of the number of persons playing both Cricket and Football to the number of persons playing both Football and Hockey?

Question

In a village of 800 persons, each plays at least one sport among Cricket, Football, and Hockey. The percentages of persons playing each sport are: Cricket 43\%, Football 52.5\%, and Hockey 55\%. The number of persons playing both Cricket and Football but not Hockey is 96, those playing both Cricket and Hockey but not Football is 100, those playing both Hockey and Football but not Cricket is 88, and those playing all three sports is 60.

What is the ratio of the number of persons playing both Cricket and Football to the number of persons playing both Football and Hockey?

Accepted Answer

**Solution:**

1. Total persons = 800

2. Number playing Cricket (C) = 43\% of 800 = $0.43 \times 800 = 344$

3. Number playing Football (F) = 52.5\% of 800 = $0.525 \times 800 = 420$

4. Number playing Hockey (H) = 55\% of 800 = $0.55 \times 800 = 440$

5. Given:
   - Persons playing both Cricket and Football but not Hockey = 96
   - Persons playing both Cricket and Hockey but not Football = 100
   - Persons playing both Hockey and Football but not Cricket = 88
   - Persons playing all three sports = 60

6. Number playing both Cricket and Football (including those who play all three) = 96 + 60 = 156

7. Number playing both Football and Hockey (including those who play all three) = 88 + 60 = 148

8. Required ratio = $\frac{156}{148} = \frac{12}{11}$

**Answer:** 12 : 11

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