The average annual revenue of a company over 11 consecutive years is Rs 69 lakhs. The average for the first 6 years is Rs 64 lakhs, and the average for the last 6 years is Rs 76 lakhs. What is the revenue (in Rs lakhs) for the 6th year?

Introduction / Context:
This question deals with averages over overlapping time periods. The company's average revenue over 11 years is known, as well as the averages for the first 6 and the last 6 years. Notice that the 6th year is included in both the first 6 and the last 6. We are asked to compute the revenue for this overlapping year. Solving this requires writing sums in terms of yearly revenues and using the fact that the 6th year is counted twice when adding the two 6 year totals.

Given Data / Assumptions:

• Average revenue for 11 years = Rs 69 lakhs per year.
• Average revenue for the first 6 years = Rs 64 lakhs per year.
• Average revenue for the last 6 years = Rs 76 lakhs per year.
• Years are consecutive and the 6th year belongs to both first 6 and last 6 years.
• We must find revenue in the 6th year, call it X lakhs.

Concept / Approach:
First, convert each average into a total revenue. Let the revenues for the 11 years be R1, R2, ..., R11. The total for the first 6 years is R1 + R2 + R3 + R4 + R5 + R6, and the total for the last 6 years is R6 + R7 + R8 + R9 + R10 + R11. Adding these two totals counts R6 twice, while all other years are counted once. On the other hand, the total revenue for all 11 years is given by 11 * 69. Using these relationships, we can solve for R6.

Step-by-Step Solution:
Step 1: Let total revenue for all 11 years be T. T = 11 * 69 = 759 lakhs. Step 2: Compute total for the first 6 years. Sum of first 6 years = 6 * 64 = 384 lakhs. Denote this as S1_6 = R1 + R2 + R3 + R4 + R5 + R6. Step 3: Compute total for the last 6 years. Sum of last 6 years = 6 * 76 = 456 lakhs. Denote this as S6_11 = R6 + R7 + R8 + R9 + R10 + R11. Step 4: Add S1_6 and S6_11. S1_6 + S6_11 = (R1 + R2 + R3 + R4 + R5 + R6) + (R6 + R7 + R8 + R9 + R10 + R11). This equals (R1 + R2 + R3 + R4 + R5 + R7 + R8 + R9 + R10 + R11) + 2R6. But R1 + R2 + ... + R11 = T, so S1_6 + S6_11 = T + R6. Step 5: Use the numerical values. Left side S1_6 + S6_11 = 384 + 456 = 840. Right side is T + R6 = 759 + X. So 840 = 759 + X. Step 6: Solve for X. X = 840 − 759 = 81.
Verification / Alternative check:
If R6 = 81, we can confirm that all averages match. Total for first 6 years is 384; total for last 6 years is 456. Adding gives 840, which equals T + 81, so T = 759. Overall average = 759 / 11 = 69 lakhs per year, as given.
Why Other Options Are Wrong:
Values like 77, 79, 83 or 75 would not satisfy the equation 384 + 456 = 759 + X. Each of these alternatives would lead to a total T different from 759, and hence to a different overall average than 69 lakhs.
Common Pitfalls:
A common mistake is to think that the 6th year's revenue is simply the difference between the two averages 76 and 64, which is not correct. Another error is to forget that the 6th year appears in both 6 year averages, leading to miscounting in the equations.
Final Answer:
The revenue for the 6th year is Rs 81 lakhs.