The average revenues of 7 consecutive years of a company is Rs 75 lakhs. The average revenue of the first 4 years is Rs 70 lakhs and the average revenue of the last 4 years is Rs 82 lakhs. What is the revenue for the 4th year?

Introduction / Context:
Similar to a previous revenue question, this problem uses overlapping averages over consecutive years to determine the revenue in a specific year. The 4th year revenue lies at the overlap of the first 4 year block and the last 4 year block, which allows us to find it using algebra.

Given Data / Assumptions:

• There are 7 consecutive years of revenue figures, years 1 to 7.
• The average revenue over all 7 years is Rs 75 lakhs.
• The average revenue of the first 4 years is Rs 70 lakhs.
• The average revenue of the last 4 years (years 4 to 7) is Rs 82 lakhs.
• We must find the revenue in the 4th year.

Concept / Approach:
We convert each average into a total revenue. Let the yearly revenues be Y1, Y2, Y3, Y4, Y5, Y6 and Y7. The first four year total is Y1 + Y2 + Y3 + Y4, and the last four year total is Y4 + Y5 + Y6 + Y7. We also know the total for all seven years. By expressing the totals in terms of Y4 and the sums of other years, we can isolate Y4 by solving a simple equation.

Step-by-Step Solution:
Step 1: Total revenue for all 7 years = 75 * 7 = 525 lakhs. Step 2: Total revenue for the first 4 years = 70 * 4 = 280 lakhs. Step 3: Total revenue for the last 4 years = 82 * 4 = 328 lakhs. Step 4: Let Y4 be the revenue of the 4th year. Then Y1 + Y2 + Y3 = 280 - Y4 and Y5 + Y6 + Y7 = 328 - Y4. Step 5: The total for all 7 years is (Y1 + Y2 + Y3) + Y4 + (Y5 + Y6 + Y7) = (280 - Y4) + Y4 + (328 - Y4) = 608 - Y4. Step 6: Set this equal to 525: 608 - Y4 = 525, so Y4 = 608 - 525 = 83.

Verification / Alternative check:
If the 4th year revenue is Rs 83 lakhs, then the first 4 year total is 280, so Y1 + Y2 + Y3 = 280 - 83 = 197. The last 4 year total is 328, so Y5 + Y6 + Y7 = 328 - 83 = 245. The sum of all 7 years is 197 + 83 + 245 = 525, and 525 / 7 = 75, matching the given seven year average. Thus the value Rs 83 lakhs is consistent.

Why Other Options Are Wrong:
If Y4 were 85, 81 or 79 lakhs, the equation 608 - Y4 = 525 would not hold, and either the first 4 year total or the last 4 year total would no longer produce the correct averages. Substituting any of those values leads to inconsistencies when checked against the provided averages.

Common Pitfalls:
Learners sometimes confuse which years form the last 4 year block or forget that the 4th year is common to both blocks. Another mistake is trying to directly average the three given averages without using totals. Always convert each average to a sum, write the equations and solve for the overlapping year carefully.

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