The difference between the compound interest and the simple interest on a certain principal at 12% per annum for 2 years is Rs 90. If interest is compounded annually, what will be the total amount (principal plus compound interest) at the end of 3 years at the same rate?

Introduction / Context:
This question combines simple interest and compound interest concepts. The difference between compound interest and simple interest for 2 years is given. From this, we must deduce the principal, then use compound interest at the same rate to find the amount at the end of 3 years. It tests understanding of the special formula for the difference between compound and simple interest over 2 years.

Given Data / Assumptions:

• Rate of interest r = 12% per annum.
• Time t = 2 years for the given difference.
• Difference between CI and SI for 2 years = Rs 90.
• Interest is compounded annually.
• We need the amount A after 3 years at the same rate on the same principal.

Concept / Approach:
For 2 years, the difference between compound interest and simple interest on principal P at rate r is:
Difference = P * r^2 / 100^2 This formula comes from expanding the compound interest expression for 2 years. Once P is known, the amount after 3 years at rate r is:
A = P * (1 + r / 100)^3

Step-by-Step Solution:
Given difference = 90 and r = 12%. Use formula: Difference = P * r^2 / 100^2. So 90 = P * 12^2 / 10000 = P * 144 / 10000. Therefore P = 90 * 10000 / 144. P = 900000 / 144 = Rs 6250. Now compute amount after 3 years at 12% compound interest. Amount factor: (1.12)^3. Compute (1.12)^2 = 1.2544 and (1.12)^3 = 1.2544 * 1.12 = 1.404928. Amount A = 6250 * 1.404928 ≈ Rs 8780.80.

Verification / Alternative Check:
Check the difference for 2 years using P = 6250. Difference = P * r^2 / 100^2 = 6250 * 144 / 10000 = 6250 * 0.0144 = 90. This matches the given difference, so P is correct. Recomputing A confirms that the amount at the end of 3 years is approximately Rs 8780.80.

Why Other Options Are Wrong:
8560 and 8673 are less than the correct amount and correspond to incorrect or incomplete compounding. 8746 is close but does not match the precise compound interest calculation.

Common Pitfalls:
Ignoring the special formula for the difference between CI and SI for 2 years and attempting a long method can lead to algebra errors. Using simple interest instead of compound interest when computing the amount for 3 years leads to an underestimation. Rounding intermediate values too early can slightly distort the final result, so rounding should be done at the end.

Final Answer:
The amount at the end of 3 years at 12% compound interest is Rs. 8780.80.