The average revenues of a company over 9 consecutive years is Rs 80 lakhs. The average revenue for the first 5 years is Rs 75 lakhs, and the average revenue for the last 5 years is Rs 87 lakhs. What is the revenue (in lakhs of rupees) in the 5th year?

Introduction / Context:
This question uses averages of overlapping time periods to find the revenue in a specific year. By knowing the overall average for all 9 years, as well as averages for the first 5 and last 5 years, we can set up equations that allow us to isolate the revenue for the 5th year, which lies in the overlap between these periods.

Given Data / Assumptions:

• Total number of years = 9.
• Average revenue for all 9 years = Rs 80 lakhs.
• Average revenue for first 5 years = Rs 75 lakhs.
• Average revenue for last 5 years = Rs 87 lakhs.
• The 5th year is common to both the first 5 and the last 5 years.
• We need to find the revenue in the 5th year.

Concept / Approach:
Let the revenues for the 9 years be R1, R2, R3, R4, R5, R6, R7, R8 and R9. We can express sums of certain groups using the given averages. The sum of all 9 years comes from the overall average. The sum of the first 5 years and the last 5 years are found from their respective averages. Since the 5th year appears in both these sums, adding them together will count R5 twice. We then use this overlap to extract R5.

Step-by-Step Solution:
Total revenue for 9 years = 80 * 9 = Rs 720 lakhs. Sum of first 5 years (R1 to R5) = 75 * 5 = Rs 375 lakhs. Sum of last 5 years (R5 to R9) = 87 * 5 = Rs 435 lakhs. Add these two sums: (R1 + R2 + R3 + R4 + R5) + (R5 + R6 + R7 + R8 + R9). This equals Rs 375 lakhs + Rs 435 lakhs = Rs 810 lakhs. But R1 + R2 + R3 + R4 + R5 + R6 + R7 + R8 + R9 = Rs 720 lakhs (total of 9 years), and R5 is added one extra time. So, Rs 810 lakhs = Rs 720 lakhs + R5. Therefore, R5 = 810 - 720 = Rs 90 lakhs.
Verification / Alternative check:
If R5 = Rs 90 lakhs, then sum of first 5 years is Rs 375 lakhs, so R1 + R2 + R3 + R4 = 375 - 90 = Rs 285 lakhs. Sum of last 5 years is Rs 435 lakhs, so R6 + R7 + R8 + R9 = 435 - 90 = Rs 345 lakhs. Total for 9 years = 285 + 90 + 345 = Rs 720 lakhs, which gives average 720 / 9 = Rs 80 lakhs, matching the data.
Why Other Options Are Wrong:
Option B (Rs 92 lakhs): Makes the combined sums inconsistent with the overall average. Option C (Rs 88 lakhs): Too low to reconcile both given averages. Option D (Rs 86 lakhs): Also cannot satisfy both the first 5 year and last 5 year averages simultaneously. Option A (Rs 90 lakhs): Satisfies all given conditions and is therefore correct.
Common Pitfalls:
Students may forget that the 5th year is counted twice when adding the sums of first 5 and last 5 years. Another common error is to mistakenly assume that the 5th year revenue is simply the average of 75 and 87, which is not correct. Some may also miscalculate the total revenue for 9 years, leading to a wrong final answer.
Final Answer:
The revenue in the 5th year is Rs 90 lakhs.