There were 35 students in a hostel. If the number of students is increased by 7, the total expenditure on food increases by Rs. 42 per day, while the average expenditure per student decreases by Rs. 1 per day. What was the initial total expenditure on food per day for the hostel?

Introduction / Context:
This question tests the concept of average expenditure and how total cost and per head cost change when the number of people sharing the total amount increases. Such problems are very common in aptitude tests related to averages and linear equations, especially in hostel or group expense scenarios.

Given Data / Assumptions:
- Original number of students in the hostel = 35. - Number of students after increase = 35 + 7 = 42. - Increase in total food expenditure per day = Rs. 42. - Decrease in average expenditure per student per day = Rs. 1. - Let the original total expenditure per day be E rupees.

Concept / Approach:
The key idea is that average expenditure per student is equal to total expenditure divided by the number of students. If the number of students changes and the total expenditure also changes, we can write two average expressions and relate them. Then we form a linear equation in one variable and solve it. The relation is:
average = total expenditure / number of students. We use the information that the new average is exactly Rs. 1 less than the old average to set up an equation.

Step-by-Step Solution:
Step 1: Original average expenditure per student per day = E / 35. Step 2: After 7 more students join, total number of students = 42. Step 3: New total expenditure per day = E + 42 (since it increases by Rs. 42). Step 4: New average expenditure per student per day = (E + 42) / 42. Step 5: According to the question, the new average is Rs. 1 less than the old average: (E + 42) / 42 = (E / 35) - 1. Step 6: Multiply throughout to clear denominators. First compute left side and right side with common denominator 210: 5 * (E + 42) = 6 * E - 210. Step 7: Expand and simplify: 5E + 210 = 6E - 210. Step 8: Bring like terms together: 210 + 210 = 6E - 5E. Step 9: 420 = E, so the original total expenditure per day = Rs. 420.

Verification / Alternative check:
If E = Rs. 420, original average per student = 420 / 35 = Rs. 12. New total expenditure = 420 + 42 = Rs. 462. New number of students = 42, so new average = 462 / 42 = Rs. 11. The new average is exactly Rs. 1 less than the original (12 - 1 = 11), which matches the condition. So E = Rs. 420 is correct.

Why Other Options Are Wrong:
Option A (Rs. 432): Gives different averages that do not differ by exactly Rs. 1. Option B (Rs. 442): Similarly fails the condition for the drop of Rs. 1 per student. Option D (Rs. 400): Leads to original average 400 / 35 and the new average does not become exactly Rs. 1 less. Option E (Rs. 412): Also does not satisfy the equation relating old and new averages.

Common Pitfalls:
Students often confuse the change in total expenditure with the change in average and sometimes subtract Rs. 1 directly from the total instead of from the average. Another frequent mistake is to assume the number of students remains the same and to form an incorrect equation. It is also easy to forget that the increased total expenditure is E + 42, not E - 42. Careful reading and translating each line into a mathematical expression is essential.

Final Answer:
The initial total expenditure on food per day for the hostel was Rs. 420.