In a hostel there was food sufficient for 1000 students for one full month of 30 days. After 10 days, 1000 more students joined the hostel. From that point onward, for how many additional days will the students be able to manage with the remaining food?

Introduction / Context:
This is a typical problem on the concept of “man-days” or in this case “student-days”. The idea is that the total quantity of food is fixed initially and can be expressed as the product of the number of consumers and the number of days that the food lasts. When the number of consumers changes, the duration for which the remaining food will last must be recalculated using the same total food concept.

Given Data / Assumptions:

• Food is initially sufficient for 1000 students for 30 days.
• After 10 days, 1000 more students join the hostel.
• All students consume food at the same constant rate.
• We assume no wastage and no additional supply of food.
• We must find the number of days the remaining food will last after the extra students join.

Concept / Approach:
We use the concept of total student-days of food. Total food initially is equal to 1000 students * 30 days. Some of this food is consumed in the first 10 days by 1000 students. The remaining food is then divided among 2000 students. Using the formula Total food = number of students * number of days, we can compute how many more days the remaining food will last after the change in the number of students.

Step-by-Step Solution:
Total food in terms of student-days = 1000 students * 30 days = 30,000 student-days. In the first 10 days, 1000 students consume = 1000 * 10 = 10,000 student-days of food. Remaining food = 30,000 − 10,000 = 20,000 student-days. After 10 days, total students = 1000 original + 1000 new = 2000 students. Let the remaining number of days be D. Then 2000 * D = 20,000 student-days. So D = 20,000 / 2000 = 10 days.

Verification / Alternative check:
If the additional students had not joined, 1000 students with 20,000 student-days remaining would have managed for 20 more days, giving a total of 30 days as originally planned. Doubling the number of students halves the remaining duration, so the 20 days would become 10 days, which matches our calculation.

Why Other Options Are Wrong:
15 or 20 days ignore the fact that the number of students doubles. 5 days is only one fourth of the original remaining time and would correspond to more than 4000 students, which is not the case. Only 10 days correctly respects the conservation of total student-days of food.

Common Pitfalls:
A frequent mistake is to subtract 10 from 30 and then randomly divide by 2, or to treat the 60 km style reasoning rather than student-days. Always remember that when population changes, the only invariant is total food measured in person-days, and you should form equations around that quantity.

Final Answer:
From the moment the new students join, the remaining food will last for 10 days.