The average revenue of a company over 7 consecutive years is Rs 75 lakhs per year. If the average revenue for the first 4 years is Rs 70 lakhs and the average revenue for the last 4 years is Rs 82 lakhs, what is the revenue (in Rs lakhs) for the 4th year?

Introduction / Context:
This problem tests understanding of averages over consecutive years and how overlapping groups can be used to determine an unknown yearly revenue. Instead of handling each year separately, we use the relationship between the overall average, the average of the first subset of years, and the average of the last subset of years to find the revenue in the 4th year.

Given Data / Assumptions:
• There are 7 consecutive years of revenue.• Average revenue of all 7 years = Rs 75 lakhs per year.• Average revenue of the first 4 years = Rs 70 lakhs per year.• Average revenue of the last 4 years = Rs 82 lakhs per year.• Revenues are measured in lakhs of rupees.

Concept / Approach:
The total revenue over a period equals average revenue multiplied by the number of years. When we know averages for overlapping groups (first 4 years and last 4 years) and the overall group (7 years), we can add the sums of the two overlapping groups. This double counts the common year, which in this case is the 4th year. Using this fact, we can isolate and compute the revenue for that overlapping year.

Step-by-Step Solution:
Step 1: Let the revenues for the 7 consecutive years be R1, R2, R3, R4, R5, R6, R7.Step 2: Total revenue for all 7 years = 7 * 75 = 525 lakhs.Step 3: Total revenue for the first 4 years = (R1 + R2 + R3 + R4) = 4 * 70 = 280 lakhs.Step 4: Total revenue for the last 4 years = (R4 + R5 + R6 + R7) = 4 * 82 = 328 lakhs.Step 5: Add the two partial sums: (R1 + R2 + R3 + R4) + (R4 + R5 + R6 + R7) = 280 + 328 = 608 lakhs.Step 6: The left side equals (R1 + R2 + R3 + R4 + R5 + R6 + R7) + R4, which is the total of all 7 years plus R4 once more.Step 7: Total of all 7 years is 525 lakhs, so 525 + R4 = 608.Step 8: Therefore, R4 = 608 - 525 = 83 lakhs.

Verification / Alternative check:
We can verify by constructing a simple hypothetical distribution that respects the sums. Once we fix R4 = 83 lakhs, we can choose values for other years so that the first 4 years sum to 280 lakhs and the last 4 years sum to 328 lakhs, and the overall sum is 525 lakhs. The consistency of these totals confirms that R4 = 83 lakhs is correct.

Why Other Options Are Wrong:
Rs 85 lakhs: This would make the combined sum of first and last 4 years too large compared with the seven year total.Rs 81 lakhs: This is obtained if arithmetic steps are reversed or if the double counting logic is misapplied.Rs 79 lakhs: This is too low and does not satisfy both partial averages simultaneously.

Common Pitfalls:
Students often forget that the 4th year is included in both the first 4 years and the last 4 years, leading to double counting mistakes. Another common mistake is to average the two partial averages directly without considering the number of years in each subset. Care must be taken to convert averages to totals and then use algebra to isolate the unknown year.

Final Answer:
The revenue for the 4th year is Rs 83 lakhs.