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 the average revenue for the last 5 years is Rs. 83 lakhs. What is the revenue of the 5th year in lakhs?

Introduction / Context:
This problem involves overlapping averages over several years of company revenue. You are given the average revenue over 9 years, the average over the first 5 years, and the average over the last 5 years. The 5th year is common to both the first group and the last group, and you must find the revenue in that particular year. This is a textbook example of how to handle overlapping intervals in average calculations for business mathematics.

Given Data / Assumptions:
- The company has revenue figures for 9 consecutive years.
- Average revenue for all 9 years = Rs. 76 lakhs per year.
- Average revenue for the first 5 years = Rs. 71 lakhs per year.
- Average revenue for the last 5 years = Rs. 83 lakhs per year.
- Year numbering is assumed as year 1 to year 9, and the last 5 years are years 5 to 9.
- We must find the revenue for year 5 in lakhs.

Concept / Approach:
Let the revenues in years 1 to 9 be R1, R2, ..., R9. The first 5 year average gives the sum R1 + R2 + R3 + R4 + R5 and the last 5 year average gives R5 + R6 + R7 + R8 + R9. Notice that year 5 appears in both sums. The total over 9 years is R1 + ... + R9. If we add the first 5 year sum and the last 5 year sum, we count R5 twice, while all other years are counted once. Therefore the sum of the two 5 year totals equals the 9 year total plus R5. This leads to a simple equation for R5.

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: Let the revenue of the 5th year be R5 lakhs.Step 5: The sum of the first 5 years plus the sum of the last 5 years is (R1 + R2 + R3 + R4 + R5) + (R5 + R6 + R7 + R8 + R9).Step 6: This combined sum equals the total of all 9 years plus R5, because year 5 is counted twice: 355 + 415 = 684 + R5.Step 7: Add the left side: 355 + 415 = 770.Step 8: Set up the equation 770 = 684 + R5, so R5 = 770 - 684 = 86.Step 9: Therefore, revenue in the 5th year = Rs. 86 lakhs.

Verification / Alternative check:
Check the logic. If year 5 revenue is 86 lakhs, then the total for nine years is still 684 lakhs. The first 5 year sum must be 355, which includes R5, and the last 5 year sum must be 415, also including R5. Adding 355 and 415 produces 770, and subtracting the 9 year total of 684 leaves 86, agreeing with our computed value for R5. This confirms that the overlapping sum reasoning is correct.

Why Other Options Are Wrong:
Values like Rs. 88 lakhs or Rs. 82 lakhs do not satisfy the relation 355 + 415 = 684 + R5. Substituting any of those figures leads to a mismatch between the combined 5 year totals and the 9 year total. Only Rs. 86 lakhs keeps all averages and totals consistent with the information given in the problem.

Common Pitfalls:
Some students incorrectly think the last 5 years are years 4 to 8 instead of 5 to 9, while others forget that year 5 is double counted when adding the two 5 year sums. Another frequent error is trying to guess R5 directly from the three averages without using the total sums. Always convert averages to totals and use the double counting idea for overlapping intervals.

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