The average revenues of a company over 13 consecutive years is Rs 82 lakhs. The average of the first 7 years is Rs 77 lakhs and the average of the last 7 years is Rs 89 lakhs. What is the revenue (in lakhs) in the 7th year?

Introduction / Context:
This question again uses overlapping time intervals with known averages to determine a specific year's revenue. The 7th year lies in the overlap of the first 7 years and the last 7 years (years 7 to 13). By converting averages to sums, we can use the overlap to isolate the revenue for the 7th year.

Given Data / Assumptions:

• Number of consecutive years = 13.
• Average revenue for all 13 years = Rs 82 lakhs.
• Average revenue for first 7 years = Rs 77 lakhs.
• Average revenue for last 7 years = Rs 89 lakhs.
• Year 7 is common to both the first 7 and last 7 years.
• We need to find the revenue in the 7th year.

Concept / Approach:
Let the revenues for the 13 years be R1, R2, ..., R13. The overall average gives us the total of all 13 years. The average of the first 7 years provides the sum R1 + R2 + ... + R7. The average of the last 7 years gives the sum R7 + R8 + ... + R13. Adding these two sums counts R7 twice and also includes all other years once. Using this relationship, we can solve for R7.

Step-by-Step Solution:
Total revenue for 13 years = 82 * 13 = Rs 1066 lakhs. Sum of first 7 years (R1 to R7) = 77 * 7 = Rs 539 lakhs. Sum of last 7 years (R7 to R13) = 89 * 7 = Rs 623 lakhs. Add these two sums: (R1 + R2 + R3 + R4 + R5 + R6 + R7) + (R7 + R8 + R9 + R10 + R11 + R12 + R13) = 539 + 623 = Rs 1162 lakhs. The total revenue for 13 years is Rs 1066 lakhs, so On the left side, all 13 years are included once, and R7 is included one extra time. Therefore, Rs 1162 lakhs = Rs 1066 lakhs + R7. R7 = 1162 - 1066 = Rs 96 lakhs.
Verification / Alternative check:
If R7 = Rs 96 lakhs, then sum of first 7 years is 539, so sum of R1 to R6 = 539 - 96 = Rs 443 lakhs. Sum of last 7 years is 623, so sum of R8 to R13 = 623 - 96 = Rs 527 lakhs. Total for all 13 years = 443 + 96 + 527 = Rs 1066 lakhs. Average = 1066 / 13 = 82 lakhs, consistent with the given information.
Why Other Options Are Wrong:
Option A (Rs 98 lakhs): Would make combined sums exceed the total implied by the overall average. Option C (Rs 94 lakhs): Too low to satisfy both subset averages simultaneously. Option D (Rs 92 lakhs): Also inconsistent with the given averages. Option B (Rs 96 lakhs): Satisfies all three average conditions and is therefore correct.
Common Pitfalls:
Students sometimes forget that the 7th year is counted twice when the two partial sums are added. Another mistake is to average 77 and 89 and assume that gives the 7th year revenue, which is incorrect. Errors in computing total revenue for 13 years or in basic subtraction can also cause wrong answers.
Final Answer:
The revenue in the 7th year is Rs 96 lakhs.