Let there be N men in the fort initially.
Let one person eat an amount equal to V in a day.
So we have a total amount of food = 42 * V * N
In 10 days, food consumed = 10 * V * V
After 10 days food amount remaining, 32 V N
Number of men now = N + 200.
Food lasts 24 days: ie., food amount = 24 *V * (N + 200)
=> 32 V N = 24 * V * (N+200)
=> 32 N = 24 N + 4800
=> N = 600
There were 600 persons.
food was sufficient for 42 days. After 10 days, food remained for 32 days. 200 people joined. So food finished in 24 days ie., 8 days earlier than planned.
The additional 200 persons had eaten food for 24 days. This eaten food amount was sufficient for 8 more days for the initial group of men. Ie., the food that was to be eaten by initial group on day 25, 26, to day 32.
food eaten by 200 persons in 24 days = food eaten by ? men in 8 days
200 * 24 / 8 = 600 persons.
This is inverse proportions or reciprocal proportions.