The average revenues of a company over 7 consecutive years are Rs. 79 lakhs. The average revenue of the first 4 years is Rs. 74 lakhs and the average revenue of the last 4 years is Rs. 86 lakhs. What is the revenue (in lakhs of rupees) in the 4th year?

Introduction / Context:
This problem deals with averages over overlapping time periods. We know the overall average revenue for 7 years, as well as the average for the first 4 years and the last 4 years. Because the 4th year lies in both groups, we can use overlapping totals to find the revenue for that particular year.

Given Data / Assumptions:

• Average revenue over 7 years = Rs. 79 lakhs.
• Average revenue of first 4 years = Rs. 74 lakhs.
• Average revenue of last 4 years = Rs. 86 lakhs.
• Years are consecutive and numbered 1 to 7.
• The 4th year is common to both the first 4 and the last 4 years.

Concept / Approach:
Let the revenues in years 1 to 7 be R1, R2, R3, R4, R5, R6, R7. Then:

• (R1 + R2 + R3 + R4 + R5 + R6 + R7) / 7 = 79.
• (R1 + R2 + R3 + R4) / 4 = 74.
• (R4 + R5 + R6 + R7) / 4 = 86.

We convert these into sums, then use the fact that the 4th year appears in both groups to form an equation that yields R4.

Step-by-Step Solution:
Step 1: Find total revenue over all 7 years. Total over 7 years = 79 * 7 = 553 lakhs. Step 2: Find total revenue for the first 4 years. R1 + R2 + R3 + R4 = 74 * 4 = 296 lakhs. Step 3: Find total revenue for the last 4 years. R4 + R5 + R6 + R7 = 86 * 4 = 344 lakhs. Step 4: Express total revenue in terms of these sums. We know total over 7 years = (R1 + R2 + R3) + R4 + (R5 + R6 + R7). From Step 2: R1 + R2 + R3 = 296 - R4. From Step 3: R5 + R6 + R7 = 344 - R4. So total = (296 - R4) + R4 + (344 - R4) = 640 - R4. Step 5: Set this equal to the known total 553 and solve for R4. 640 - R4 = 553. R4 = 640 - 553 = 87 lakhs.

Verification / Alternative check:
If R4 = 87, then:

• First 4 years total = 296, so R1 + R2 + R3 = 296 - 87 = 209.
• Last 4 years total = 344, so R5 + R6 + R7 = 344 - 87 = 257.

Total for all 7 years = 209 + 87 + 257 = 553, and the average is 553 / 7 = 79 lakhs, which matches the given data. The first 4 year average is 296 / 4 = 74, and the last 4 year average is 344 / 4 = 86, so all conditions are satisfied.

Why Other Options Are Wrong:
If R4 were 89 or 85 or 83, the recomputed totals would not equal 553 and at least one of the averages would not match the given values. Only R4 = 87 ensures consistency with all three average conditions simultaneously.

Common Pitfalls:
Students sometimes forget that the 4th year is counted in both the first and last group, or they may try to average 74 and 86 directly. Another mistake is to ignore the overall average and only work with partial sums. The systematic approach is to express everything in terms of sums and carefully account for overlaps when constructing the total.

Final Answer:
The revenue in the 4th year is Rs. 87 lakhs.