In a fort there was enough food to feed 200 soldiers for 31 days. After 27 days, 120 soldiers left the fort. For how many extra days will the remaining food last for the soldiers who stay?

Introduction / Context:
This is a question on the concept of “man-days” or “soldier-days” applied to food consumption. We are given that food in a fort is sufficient for a certain number of soldiers for a fixed number of days. Some soldiers leave after a period, and we are asked how long the remaining food will last for the reduced group. This tests understanding of proportionality between total food, number of consumers and time.

Given Data / Assumptions:
- There is enough food for 200 soldiers for 31 days. - All 200 soldiers initially consume food for 27 days. - After 27 days, 120 soldiers leave the fort. - Therefore, 80 soldiers remain in the fort. - The rate of food consumption per soldier per day remains constant.

Concept / Approach:
We treat the total food stock in terms of “soldier-days”, which is analogous to total work in man-days. Total food can be expressed as number of soldiers multiplied by the number of days the food would last. Then we subtract the amount of food already consumed in the first 27 days. The remaining quantity of food, measured in soldier-days, is then divided by the number of soldiers left to find how many more days it will last.

Step-by-Step Solution:
Step 1: Total amount of food in soldier-days = 200 soldiers * 31 days = 6200 soldier-days. Step 2: Food consumed in the first 27 days by 200 soldiers = 200 * 27 = 5400 soldier-days. Step 3: Remaining food = 6200 - 5400 = 800 soldier-days. Step 4: After 27 days, 120 soldiers leave, so remaining soldiers = 200 - 120 = 80 soldiers. Step 5: Time for which remaining food lasts = remaining soldier-days / remaining soldiers. Step 6: Time = 800 soldier-days / 80 soldiers = 10 days.

Verification / Alternative check:
We can check consistency by thinking in reverse. If 80 soldiers consume the food for 10 days, they use 80 * 10 = 800 soldier-days of food, exactly the remaining amount we computed. Adding the first 5400 soldier-days consumed by 200 soldiers in 27 days gives 5400 + 800 = 6200, which matches the original total food. This confirms that 10 extra days is correct.

Why Other Options Are Wrong:
12 days or 8 days: These do not match an integer number of soldier-days when multiplied by 80 soldiers. They would either overuse or underuse the remaining 800 soldier-days of food.
6 days or 4 days: These values are too small and would imply that some food is left over, contradicting the calculation based on total food and consumption.

Common Pitfalls:
A common mistake is to treat the problem as if days scale directly with soldiers without converting to the common unit of soldier-days. Some students may forget to subtract the food consumed over the first 27 days or miscount the remaining soldiers. Remember that total food is fixed and equal to the product of consumers and days; whenever the number of consumers changes, the number of days must adjust in inverse proportion.

Final Answer:
The remaining food will last for 10 extra days for the remaining soldiers.