In a mens hostel there are initially 95 students. When 30 more students join, the average expenditure per student decreases by Rs. 5, yet the total expenditure on the hostel increases by Rs. 500. What is the total expenditure of the hostel now?

Introduction / Context:
This problem combines the concept of average expenditure per person with changes in group size. It illustrates how total expenditure and per person expenditure are related when the number of people changes in a hostel or similar shared cost situation.

Given Data / Assumptions:
- Initial number of students = 95.
- Additional students = 30, so final number = 125.
- Average expenditure per student decreases by Rs. 5 after the increase in students.
- Total expenditure increases by Rs. 500 overall.
- We assume expenditure is calculated for the same time period before and after.

Concept / Approach:
Let the initial average expenditure per student be x rupees. Then the initial total expenditure is:
Initial total = 95 * x
After 30 new students join, the new average becomes (x - 5) and the new total expenditure is:
New total = 125 * (x - 5)
The new total is also given to be Rs. 500 more than the old total, which gives an equation to solve for x and then for the new total.

Step-by-Step Solution:
Step 1: Let initial average expenditure per student be x.Step 2: Initial total expenditure = 95x.Step 3: New number of students = 95 + 30 = 125.Step 4: New average expenditure per student = x - 5.Step 5: New total expenditure = 125 * (x - 5).Step 6: Given that the new total is Rs. 500 more: 125 * (x - 5) = 95x + 500.Step 7: Expand and simplify: 125x - 625 = 95x + 500.Step 8: Rearrange: 125x - 95x = 500 + 625, so 30x = 1125.Step 9: Solve for x: x = 1125 / 30 = 37.5.Step 10: New total expenditure = 125 * (x - 5) = 125 * 32.5 = 4062.5 rupees.

Verification / Alternative check:
Initial total = 95 * 37.5 = 3562.5. Adding Rs. 500 gives 3562.5 + 500 = 4062.5, which matches the computed new total, so the result is consistent from both directions.

Why Other Options Are Wrong:
- Rs. 4420, Rs. 4680, Rs. 4960 and Rs. 5480 do not satisfy the equation that links the old and new totals and the change in average. If you plug these values back and compute the implied averages, they will not give a decrease of exactly Rs. 5 with an increase of 30 students.

Common Pitfalls:
- A common mistake is to assume the new total equals 125x instead of 125(x - 5).
- Another error is to add Rs. 5 to the average instead of subtracting, misreading the direction of the change.
- Some also forget to increase the number of students in the new scenario and use 95 in both totals.

Final Answer:
The total expenditure of the hostel after the new students join is Rs. 4062.50.