A hostel has 800 boys. 75% boys play hockey and 50% boys play football. If each boy plays hockey or football or both, then how many boys play both sports?

Steps to Solve

1. Define the total number of boys

The total number of boys in the hostel is given as 800.

2. Calculate the number of boys playing hockey

75% of the boys play hockey. We can calculate this as: $$ \text{Number of boys playing hockey} = 0.75 \times 800 = 600 $$

3. Calculate the number of boys playing football

50% of the boys play football. We can calculate this as: $$ \text{Number of boys playing football} = 0.50 \times 800 = 400 $$

4. Use the principle of inclusion-exclusion

Let ( A ) represent the boys playing hockey and ( B ) represent the boys playing football. According to the principle of inclusion-exclusion: $$ |A \cup B| = |A| + |B| - |A \cap B| $$ Where ( |A \cup B| ) is the total number of boys who play hockey or football (or both). Since all boys play at least one sport, we have: $$ |A \cup B| = 800 $$

Substituting the known values: $$ 800 = 600 + 400 - |A \cap B| $$

5. Solve for the number of boys playing both sports

Rearranging the equation to find ( |A \cap B| ): $$ |A \cap B| = 600 + 400 - 800 = 200 $$