The average expenditure of a hotel is Rs. 60 per guest when there are 10 guests, and Rs. 40 per guest when there are 20 guests. Assuming that the hotel has some fixed daily expenses (independent of the number of guests) and the rest of the expenditure varies directly with the number of guests, what will be the average expenditure per guest when there are 40 guests?

Introduction / Context:
This question involves a mixture of fixed and variable costs and asks for the average expenditure per guest at different levels of occupancy. Such problems commonly appear in quantitative aptitude tests to check your ability to set up simple linear equations representing fixed and variable components and then use them to predict costs at other levels.

Given Data / Assumptions:

• When there are 10 guests, the average expenditure per guest is Rs. 60.
• When there are 20 guests, the average expenditure per guest is Rs. 40.
• There is a fixed daily expenditure F (which does not depend on the number of guests).
• There is a variable expenditure v per guest.
• We must find the average expenditure per guest when there are 40 guests.

Concept / Approach:
Let F be the fixed cost and v be the variable cost per guest. Then for N guests total expenditure is F + vN. The average expenditure per guest is (F + vN) / N. We are given two such situations (for N = 10 and N = 20). From these two equations we can solve for F and v. Once we know F and v, we can compute the average cost when N = 40.

Step-by-Step Solution:
Step 1: When N = 10, average cost is 60, so (F + 10v) / 10 = 60. Step 2: Multiply both sides by 10: F + 10v = 600. Step 3: When N = 20, average cost is 40, so (F + 20v) / 20 = 40. Step 4: Multiply both sides by 20: F + 20v = 800. Step 5: Subtract the first equation from the second: (F + 20v) - (F + 10v) = 800 - 600. Step 6: This gives 10v = 200, so v = 20. Step 7: Substitute v = 20 into F + 10v = 600: F + 10 * 20 = 600 ⇒ F + 200 = 600 ⇒ F = 400. Step 8: For N = 40 guests, total expenditure = F + 40v = 400 + 40 * 20 = 400 + 800 = 1200. Step 9: Average expenditure per guest when N = 40 is 1200 / 40 = 30.

Verification / Alternative check:
We can cross check by computing the total expenditure for the earlier cases using F = 400 and v = 20. For 10 guests, total = 400 + 10 * 20 = 400 + 200 = 600, average = 600 / 10 = 60, which matches the given data. For 20 guests, total = 400 + 20 * 20 = 400 + 400 = 800, average = 800 / 20 = 40, also matching the problem statement. Since these match, using the same F and v for 40 guests is valid, leading to an average of 30.

Why Other Options Are Wrong:
An average of Rs. 25 or Rs. 20 per guest would imply a much lower variable cost or fixed cost than indicated by the given data. The option Rs. 35 does not satisfy the linear relationship defined by the two known points. Saying that the value cannot be determined ignores the fact that two equations in the two unknowns F and v are sufficient to find a unique solution. Only Rs. 30 is consistent with all the given conditions.

Common Pitfalls:
A common mistake is to assume that the average cost varies linearly with the number of guests and simply interpolate averages rather than expressing the total cost in terms of fixed and variable components. Another error is to forget that the fixed cost does not change with the number of guests, leading to incorrect equations. Careful identification of F and v and setting up the equations correctly ensures a straightforward solution.

Final Answer:
The average expenditure per guest when there are 40 guests in the hotel is Rs. 30.