The average revenues of 11 consecutive years of a company is Rs 69 lakhs. The average revenue of the first 6 years is Rs 64 lakhs and the average revenue of the last 6 years is Rs 76 lakhs. What is the revenue for the 6th year?

Introduction / Context:
This problem deals with average revenues over overlapping periods of years. You are told the average for 11 consecutive years, as well as the averages for the first 6 and the last 6 years. Because the 6th year is counted in both of these six year averages, we can use this overlap to find the revenue for that particular year.

Given Data / Assumptions:

• There are 11 consecutive years of revenue figures.
• The overall average revenue for the 11 years is Rs 69 lakhs.
• The average revenue for the first 6 years is Rs 64 lakhs.
• The average revenue for the last 6 years is Rs 76 lakhs.
• Year 6 is common to both the first 6 and the last 6 years.
• We need to find the revenue in the 6th year.

Concept / Approach:
We convert each average into a total revenue for that period. Let the revenues in years 1 to 11 be Y1, Y2, ..., Y11. Then we express the total of years 1 to 6 and the total of years 6 to 11 in terms of these symbols. Because both range sums include Y6, we can add or subtract equations to isolate Y6. We also use the total for all 11 years from the overall average.

Step-by-Step Solution:
Step 1: Total revenue for all 11 years = 69 * 11 = 759 lakhs. Step 2: Total revenue for the first 6 years = 64 * 6 = 384 lakhs. Step 3: Total revenue for the last 6 years (years 6 to 11) = 76 * 6 = 456 lakhs. Step 4: Let Y6 be the revenue in year 6. Then sum of years 1 to 5 is 384 - Y6 and sum of years 7 to 11 is 456 - Y6. Step 5: Total revenue for all 11 years is (sum of years 1 to 5) + Y6 + (sum of years 7 to 11) = (384 - Y6) + Y6 + (456 - Y6) = 840 - Y6. Step 6: Set this equal to the total 759: 840 - Y6 = 759, so Y6 = 840 - 759 = 81.

Verification / Alternative check:
If the 6th year revenue is Rs 81 lakhs, then the first 6 years total 384, so the first 5 years sum to 384 - 81 = 303 lakhs. The last 6 years total 456, so years 7 to 11 sum to 456 - 81 = 375 lakhs. Adding 303 + 81 + 375 gives 759, and 759 / 11 is 69, which matches the overall average, confirming the result.

Why Other Options Are Wrong:
Values such as 83, 79 or 77 lakhs would lead to totals for the first and last sets of years that do not match 384 and 456 simultaneously when combined into 11 years. Substituting any of these alternatives for Y6 would break the equation 840 - Y6 = 759, so they cannot be correct.

Common Pitfalls:
A frequent mistake is to think that the 6th year revenue is simply the difference between two averages or to misidentify which years are included in each range. Another pitfall is not converting averages to totals and trying to reason only with averages. Always write out total revenues and use clear algebra when years overlap.

Final Answer:
The revenue in the 6th year is Rs 81 lakhs.