A fort has provisions for 50 days. If after 10 days they are strengthened by 500 men and the food lasts for 35 days longer, the number of men originally in the fort were

### Concept & Inverse Proportion The relationship between the number of men and the number of days food lasts is inversely proportional. The total "man-days" of remaining food must be equal for both the original group and the newly formed group. $$ M_{old} \times D_{remaining} = M_{new} \times D_{new} $$ ### Step-by-Step Solution * **Variables:** Let the original number of men be $M$. * **Remaining provisions:** After 10 days, the food left is enough for the original $M$ men for $50 - 10 = 40$ days. So, remaining man-days $= 40M$. * **New group size:** They are strengthened by 500 men, so the new number of men is $M + 500$. * **New duration:** The food lasts for 35 days for this new group. * **Equation:** Equate the remaining man-days. * $40M = (M + 500) \times 35$ * $40M = 35M + 17500$ * $40M - 35M = 17500$ * $5M = 17500$ * $M = 3500$ ### Exam Strategy & Shortcut Form the equation based on the shifts: The difference in days for the original men ($40 - 35 = 5$ days) must feed the new 500 men for 35 days. Thus, $5 \times M = 500 \times 35 \implies M = 3500$. This mental shift saves significant time. ### Common Pitfall A common mistake is writing the equation as $50M = (M+500) \times 35$, forgetting to subtract the 10 days that have already passed from the original provision duration. ### Final Answer Therefore, the correct answer is **3500**.