A sum of Rs 8000 is invested at a compound interest rate of 5% per annum, compounded annually, for 3 years. What is the difference between the compound interest earned for 3 years and the compound interest earned for only 2 years on the same principal?

Introduction / Context:
This problem focuses on compound interest for different time periods on the same principal and at the same rate. Instead of asking for the total amount, it asks how much extra interest is earned in the third year compared with only the first two years. This is a classic way to test the understanding that compound interest grows on both principal and accumulated interest.

Given Data / Assumptions:

• Principal P = Rs 8000.
• Rate of interest r = 5% per annum.
• Interest is compounded annually.
• Time period 1 = 2 years.
• Time period 2 = 3 years.
• We need CI for 3 years minus CI for 2 years.

Concept / Approach:
The amount A after n years at rate r percent per annum compounded annually is:
A = P * (1 + r / 100)^n The compound interest CI for n years is:
CI = A - P The difference between CI for 3 years and CI for 2 years is equal to the interest earned in the third year on the amount after 2 years.

Step-by-Step Solution:
P = 8000 and r = 5%. Amount after 2 years: A2 = 8000 * (1.05)^2. Compute (1.05)^2 = 1.1025, so A2 = 8000 * 1.1025 = Rs 8820. Amount after 3 years: A3 = 8000 * (1.05)^3. Compute (1.05)^3 = 1.157625, so A3 = 8000 * 1.157625 = Rs 9261. CI for 2 years = A2 - P = 8820 - 8000 = Rs 820. CI for 3 years = A3 - P = 9261 - 8000 = Rs 1261. Difference in interest = 1261 - 820 = Rs 441.

Verification / Alternative Check:
The third year interest is CI for year three on A2. Third year interest = A2 * r / 100 = 8820 * 5 / 100 = 441. This equals the difference between CI for 3 years and 2 years, confirming our result.

Why Other Options Are Wrong:
Rs. 387 is too low and does not match the computed third year interest. Rs. 469 and Rs. 503 are higher than the correct value and result from incorrect calculations or using a wrong base.

Common Pitfalls:
Some learners mistakenly take simple interest for each year, which ignores compounding. Another error is to compute CI for 3 years incorrectly by not using the power of 3 in the formula. Using the wrong principal for the third year (for example, using 8000 instead of 8820) is also a frequent mistake.

Final Answer:
The correct difference between the compound interest for 3 years and 2 years is Rs. 441.