Step 1: Understanding the Question:
This is a problem based on the concept of 'man-days' or 'student-days' in work and time equivalence.
The total quantity of provisions remains constant, and we must track how much is consumed during each distinct time block to find out how long the remainder will last.
Step 2: Key Formula or Approach:
Total Provisions = Number of Students \( \times \) Number of Days.
Remaining Provisions = Total Provisions - Consumed Provisions.
Remaining Days = Remaining Provisions / Current Number of Students.
Step 3: Detailed Explanation:
The hostel initially has provisions for 250 students lasting 35 days.
Let's calculate the total food in terms of "student-days". Total provisions = \( 250 \times 35 = 8,750 \) student-days.
For the first 5 days, all 250 students consume the food.
Provisions consumed in the first 5 days = \( 250 \times 5 = 1,250 \) student-days.
The remaining provisions after these 5 days = \( 8,750 - 1,250 = 7,500 \) student-days.
After the 5th day, a fresh batch of 25 students joins, making the new total \( 250 + 25 = 275 \) students.
This new group of 275 students stays for the next 10 days before another change occurs.
Provisions consumed during these 10 days = \( 275 \times 10 = 2,750 \) student-days.
The remaining provisions after this period = \( 7,500 - 2,750 = 4,750 \) student-days.
After these 10 days, a batch of 25 students leaves the hostel.
The number of students remaining in the hostel is now \( 275 - 25 = 250 \) students.
We need to find how many days the remaining 4,750 student-days of provisions will last for these 250 students.
Remaining days = \( \frac{4,750}{250} \).
Simplifying the division: \( 4,750 \div 250 = 19 \).
Thus, the remaining provisions will survive for exactly 19 days.
Step 4: Final Answer:
The remaining provisions will survive for 19 days.