There are 44 students in a hostel. Later, 15 new students join the hostel, and as a result the total mess expense increases by Rs. 33 per day. However, the average expenditure per head decreases by Rs. 3. What was the original total daily expenditure of the mess?

Introduction / Context:
This is a typical average and total cost problem in which the number of people sharing a common expense changes. When more students join the hostel, the total mess expense increases, but because more people are sharing the cost, the average expenditure per head decreases. You must use these two effects together to work out the original total daily expenditure of the mess.

Given Data / Assumptions:

• Original number of students = 44.
• New number of students = 44 + 15 = 59.
• Original total mess expense per day = E rupees.
• After 15 students join, the total mess expense per day becomes E + 33 rupees.
• The average expenditure per head decreases by Rs. 3 after the new students join.
• We need to find E, the original total daily mess expenditure.

Concept / Approach:
The original average per student is E / 44. After 15 new students join, the new average per student becomes (E + 33) / 59. We are told that this new average is Rs. 3 less than the original average. Therefore, we can set up an equation expressing this relationship and solve for E. This is a straightforward application of the definition of average and basic algebra.

Step-by-Step Solution:
Step 1: Let the original total expense per day be E rupees. Step 2: Original average per head = E / 44. Step 3: After 15 new students join, total students = 59 and new total expense = E + 33. Step 4: New average per head = (E + 33) / 59. Step 5: The average decreases by Rs. 3, so (E + 33) / 59 = (E / 44) - 3. Step 6: Multiply through by 59 * 44 to clear denominators: 44(E + 33) = 59E - 3 * 59 * 44. Step 7: Expand the left side: 44E + 44 * 33 = 44E + 1452. Step 8: Compute 3 * 59 * 44 = 3 * 2596 = 7788, so the equation is 44E + 1452 = 59E - 7788. Step 9: Rearrange: 1452 + 7788 = 59E - 44E ⇒ 9240 = 15E. Step 10: Solve for E: E = 9240 / 15 = 616. Step 11: Therefore, the original total mess expense per day was Rs. 616.

Verification / Alternative check:
Check the averages. Original average = 616 / 44 = 14. After 15 students join, new total expense = 616 + 33 = 649 and students = 59. New average = 649 / 59 = 11. This is indeed Rs. 3 less than 14, matching the description. Also, the increase in total expense is Rs. 33 as given. This confirms that E = 616 is consistent with all parts of the problem.

Why Other Options Are Wrong:
If the original expense were Rs. 404, 514 or 340, then using the same reasoning would not produce an average decrease of exactly Rs. 3 when 15 students join and the total increases by Rs. 33. These values fail to satisfy the equation derived from the average relationship. The option Rs. 580 also does not satisfy the required conditions. Only Rs. 616 gives an original average of 14 and a new average of 11, exactly as required.

Common Pitfalls:
Some students confuse the change in total expense with the change in average and try to directly divide 33 by 3, thinking that 11 is the number of new students, which is incorrect. Others forget to use 59 in the denominator for the new average or misinterpret the direction of the change in average. Careful translation of the problem into algebraic equations is crucial to arrive at the correct answer.

Final Answer:
The original total daily expenditure of the mess was Rs. 616.