The average annual revenue of a company over 9 consecutive years is Rs. 76 lakhs. The average revenue for the first 5 years is Rs. 71 lakhs and for the last 5 years is Rs. 83 lakhs. What is the revenue (in lakhs) for the 5th year?

Introduction / Context:
This problem is about average revenues over several years and uses overlapping groups of years to determine a single unknown year. It is a classic application of averages where the overlap between the first group and the last group helps us compute a specific term, here the 5th year's revenue.

Given Data / Assumptions:
- Total number of years considered = 9 consecutive years. - Average revenue over all 9 years = Rs. 76 lakhs. - Average revenue over the first 5 years = Rs. 71 lakhs. - Average revenue over the last 5 years = Rs. 83 lakhs. - Year numbers are 1 to 9, and the 5th year belongs to both the first 5 and last 5 groups.

Concept / Approach:
Average revenue is equal to total revenue divided by the number of years. We first compute the total revenue over all 9 years from the overall average. Then we compute the total revenue of the first 5 years and of the last 5 years separately. The sum of these two five year totals counts the 5th year revenue twice, and all other years once. By comparing the combined sum to the 9 year total, we can solve for the revenue of the 5th year using a simple subtraction.

Step-by-Step Solution:
Step 1: Total revenue for all 9 years = 9 * 76 = Rs. 684 lakhs. Step 2: Total revenue for the first 5 years = 5 * 71 = Rs. 355 lakhs. Step 3: Total revenue for the last 5 years = 5 * 83 = Rs. 415 lakhs. Step 4: Combined total of the first 5 years and last 5 years = 355 + 415 = Rs. 770 lakhs. Step 5: Let the revenue in the 5th year be R. Step 6: The combined total counts every year from 1 to 9 once, except the 5th year which is counted twice. Step 7: Therefore, combined total = total for 9 years + revenue of 5th year. Step 8: So, 770 = 684 + R. Step 9: Solve for R: R = 770 - 684 = Rs. 86 lakhs.

Verification / Alternative check:
To verify, imagine the individual revenues are unknown except the 5th year which we now set as Rs. 86 lakhs. If we sum all 9 years to get Rs. 684 lakhs, then the first 5 must total 355 and last 5 must total 415 by the given averages. Adding those two totals gives 770, which is 684 plus the 5th year's 86 again. This matches the counting logic we used, confirming that Rs. 86 lakhs is consistent with the data.

Why Other Options Are Wrong:
Option A (Rs. 88 lakhs): Would give combined total 684 + 88 = 772, which does not match 770. Option B (Rs. 84 lakhs): Leads to combined total 684 + 84 = 768, again mismatching 770. Option D (Rs. 82 lakhs): Gives combined total 684 + 82 = 766, incorrect. Option E (Rs. 80 lakhs): Also fails the equality with the combined total from the two averages.

Common Pitfalls:
Learners sometimes mistakenly average the two averages 71 and 83 or attempt to directly manipulate the averages without converting them to totals. Another mistake is to forget that the 5th year appears in both the first 5 years and the last 5 years, leading to an incorrect expression for the combined total. Always convert averages to totals and carefully account for how many times each year is counted.

Final Answer:
The revenue for the 5th year is Rs. 86 lakhs.