There are 44 students in a hostel. Later 15 new students join the hostel and the total mess expenses increase by Rs 33 per day, while the average expenditure per student per day decreases by Rs 3. What was the original total daily expenditure of the mess before the new students joined?

Introduction:
This question deals with average expenditure per person when the number of people changes. The mess expenses increase slightly when more students join, but the average cost per student decreases. We must translate this into equations and find the original total daily expenditure.

Given Data / Assumptions:
- Original number of students in the hostel = 44. - New students joining the hostel = 15, so new total students = 59. - Total daily mess expenses increase by Rs 33 when the new students join. - Average expenditure per student per day decreases by Rs 3. - We must find the original total daily expenditure of the mess.

Concept / Approach:
Let the original total daily mess expenditure be E. Then the original average per student is E / 44. After 15 students join, the total expenditure becomes E + 33, and the number of students becomes 59, so the new average is (E + 33) / 59. This new average is Rs 3 less than the original average. Setting up and solving this equation will give us E.

Step-by-Step Solution:
Step 1: Let E be the original total daily mess expenditure. Step 2: Original average per student = E / 44. Step 3: After 15 students join, total students = 44 + 15 = 59. Step 4: New total daily expenditure = E + 33. Step 5: New average per student = (E + 33) / 59. Step 6: Given that the new average is Rs 3 less than the old average: (E + 33) / 59 = E / 44 - 3. Step 7: Rewrite E / 44 - 3 as (E - 132) / 44. Step 8: Multiply both sides of the equation by 59 * 44 to clear denominators. Step 9: Left side becomes 44(E + 33), right side becomes 59(E - 132). Step 10: So 44E + 44 * 33 = 59E - 59 * 132. Step 11: Compute 44 * 33 = 1452 and 59 * 132 = 7788. Step 12: The equation is 44E + 1452 = 59E - 7788. Step 13: Bring like terms together: 1452 + 7788 = 59E - 44E which gives 9240 = 15E. Step 14: Solve for E: E = 9240 / 15 = 616.

Verification / Alternative Check:
Original total expenditure E = 616, so original average = 616 / 44 = 14. After 15 students join, total students = 59, new total expenditure = 616 + 33 = 649. New average = 649 / 59 = 11. The difference between the original and new averages is 14 - 11 = 3, which matches the given condition. Hence the value E = 616 is correct.

Why Other Options Are Wrong:
Options Rs 404, Rs 514, Rs 340 and Rs 480 do not satisfy the equation (E + 33) / 59 = E / 44 - 3. Substituting any of these values for E leads to inconsistent averages.

Common Pitfalls:
Learners sometimes misinterpret the statement and add 3 instead of subtracting, or they mistakenly apply the change to the total instead of the average. It is very important to write down the expressions for the average both before and after the change and then equate them correctly.

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