In a school, the ratio of boys to girls is 4 : 5. After 100 girls leave, the ratio becomes 6 : 7. How many boys are there in the school initially?

Introduction / Context:
Two consecutive ratio conditions enable solving for actual counts. We set up variables for initial numbers and apply the change to girls to get the second ratio.

Given Data / Assumptions:
Boys : Girls initially = 4 : 5. After 100 girls leave, Boys : Girls = 6 : 7.

Concept / Approach:
Parameterize initial counts as 4x and 5x, update the girls by subtracting 100, and equate to the new ratio 6 : 7.

Step-by-Step Solution:
Initial: Boys = 4x, Girls = 5x. After departure: Boys = 4x, Girls = 5x − 100. Ratio condition: 4x / (5x − 100) = 6 / 7. Cross-multiply: 28x = 30x − 600 ⇒ 2x = 600 ⇒ x = 300. Boys = 4x = 1200.

Verification / Alternative check:
Then girls initially 1500; after 100 leave → 1400; 1200 : 1400 simplifies to 6 : 7. Correct.

Why Other Options Are Wrong:
1800, 1000, 1500, 1400 do not satisfy both ratio conditions when tested.

Common Pitfalls:
Changing the number of boys by mistake or failing to simplify the final ratio for checking.

Final Answer:
1200