Given that fort had provision for 500 persons for 27 days Hence, after 3 days, the remaining food is sufficient for 500 persons for 24 days.
Remaining persons after 3 days = 500 + 300 = 800
Assume that after 10 days,the remaining food is sufficient for 800 persons for x days.

More men, Less days (Indirect Proportion)
⇒ Men 500 : 800 :: x : 24
⇒ 500 × 24 = 800x
⇒ 5 × 24 = 8x
⇒ x = 5 × 3 = 15