The compound interest earned in 2 years at 12% per annum, compounded annually, is Rs. 10,176. What is the original sum (principal) invested, in rupees?

Introduction / Context:
This problem asks you to find the principal when the compound interest earned over a known period at a given rate is provided. It is a reverse application of the standard compound interest formula and is common in bank deposit and loan questions.

Given Data / Assumptions:

Rate of interest r = 12% per annum.
Time t = 2 years.
Compound interest for 2 years = Rs. 10,176.
Compounding is annual.
Principal is P, amount is A.

Concept / Approach:
For annual compounding: A = P * (1 + r/100)^t Compound interest is: CI = A - P Given CI and the rate and time, we first write A in terms of P, then equate A - P to 10,176 and solve for P. For r = 12% and t = 2 years, the factor (1 + r/100)^2 becomes (1.12)^2 = 1.2544.

Step-by-Step Solution:
Step 1: Express A in terms of P. A = P * (1.12)^2 = P * 1.2544. Step 2: Express CI in terms of P. CI = A - P = P * 1.2544 - P = P * (1.2544 - 1) = P * 0.2544. Step 3: Use the given CI to solve for P. P * 0.2544 = 10,176. P = 10,176 / 0.2544 = 40,000.

Verification / Alternative check:
Check by recomputing the amount and interest. With P = Rs. 40,000: A = 40,000 * 1.2544 = Rs. 50,176. Compound interest = A - P = 50,176 - 40,000 = Rs. 10,176, matching the given value. Hence, the principal is correct.

Why Other Options Are Wrong:
If P were 50,000, CI for 2 years at 12% would be 50,000 * 0.2544 = 12,720. For 60,000 it would be 15,264, and for 80,000 it would be 20,352. None of these equals 10,176, so those options can be eliminated.

Common Pitfalls:
A common mistake is to treat the 2-year interest as simple interest and calculate CI = P * r * t / 100, which is incorrect. Others may square 12 instead of 1.12 or mis-handle the subtraction step when finding CI from A. Always convert the percentage correctly and square the growth factor, not the rate itself.

Final Answer:
The principal amount invested was Rs. 40,000.