A fort has provisions of food sufficient for 150 men for 45 days. After 10 days, 25 men leave the fort. For how many additional days will the remaining food last for the men who remain?

Introduction / Context:
This is a classic fort and food problem involving the concept of man-days of food. It tests your ability to account for consumption over time and adjust remaining supplies when the number of consumers changes during the period.

Given Data / Assumptions:
- Initial provisions are enough for 150 men for 45 days.
- After 10 days, 25 men leave, so 125 men remain.
- We must find how many more days the remaining food will last.
- All men consume food at the same constant rate.

Concept / Approach:
The key idea is that total food can be measured in man-days: one man-day is the amount of food consumed by one man in one day. We first compute the total food available in man-days, subtract what is consumed in the first 10 days, and then divide the remaining man-days by the new number of men to find how long the food will last thereafter.

Step-by-Step Solution:
Step 1: Compute total food in man-days: 150 men * 45 days = 6750 man-days.Step 2: In the first 10 days, the fort still has 150 men.Step 3: Food consumed in first 10 days = 150 * 10 = 1500 man-days.Step 4: Remaining food in man-days = 6750 - 1500 = 5250 man-days.Step 5: After 10 days, 25 men leave, so remaining men = 150 - 25 = 125 men.Step 6: Let D be the number of days the remaining food will last for 125 men.Step 7: Then 125 * D = 5250 man-days.Step 8: Solve for D: D = 5250 / 125.Step 9: Compute: 125 * 40 = 5000, remainder 250, and 125 * 2 = 250, so D = 42 days.

Verification / Alternative check:
Total elapsed time from the start until the food ends would be 10 days (initial phase) + 42 days (after men leave) = 52 days. Check by total consumption: 150 * 10 + 125 * 42 = 1500 + 5250 = 6750 man-days, which matches the original total available, confirming the calculation.

Why Other Options Are Wrong:
Values like 29, 32 or 37 days underestimate the remaining time and would correspond to total consumption less than 6750 man-days. 54 days greatly overshoots the remaining man-days and would require more food than initially stored. Only 42 days balances the man-days exactly.

Common Pitfalls:
Common errors include forgetting to subtract the food consumed in the first 10 days, or using 150 men instead of 125 men in the second phase. Some students also miscalculate 5250 / 125. A step-by-step man-days approach keeps the logic clear and prevents such mistakes.

Final Answer:
The remaining food will last for 42 more days.