The compound interest earned in 2 years at a rate of 12% per annum is Rs. 10,176. What was the original principal sum (in rupees) that was invested, assuming annual compounding?

Introduction:
This question reverses the usual compound interest calculation. Instead of finding the interest from a known principal, we are given the compound interest and asked to find the original principal that generated this interest at a known rate over 2 years.

Given Data / Assumptions:

• Compound interest (CI) for 2 years = Rs. 10,176
• Rate of interest (r) = 12% per annum
• Time (t) = 2 years
• Interest is compounded annually

Concept / Approach:
For annual compounding, the amount A after t years is:
A = P * (1 + r/100)^tThe compound interest is:
CI = A - PWe know CI and r and t, but not P. We first express CI in terms of P and then solve for P using algebra.

Step-by-Step Solution:
Step 1: Write the formula for the amount over 2 years: A = P * (1 + 12/100)^2 = P * (1.12)^2.Step 2: Compute (1.12)^2 = 1.2544.Step 3: So A = 1.2544 * P.Step 4: The compound interest CI is A - P, so CI = 1.2544 * P - P = (1.2544 - 1) * P = 0.2544 * P.Step 5: We are given CI = 10,176, so 0.2544 * P = 10,176.Step 6: Solve for P by dividing both sides: P = 10,176 / 0.2544.Step 7: Perform the division: P = 40,000.So the original principal sum invested is Rs. 40,000.

Verification / Alternative check:
To verify, compute the amount from P = 40,000 at 12% for 2 years. First year amount: 40,000 * 1.12 = 44,800. Second year amount: 44,800 * 1.12 = 50,176. The total compound interest is 50,176 - 40,000 = 10,176, which matches the given CI. Hence the calculation is consistent.

Why Other Options Are Wrong:
Rs. 50,000, Rs. 60,000, and Rs. 80,000 all give much larger compound interest at 12% over 2 years. Rs. 30,000 would give significantly less than Rs. 10,176 as compound interest. Only Rs. 40,000 produces exactly the stated interest figure.

Common Pitfalls:
Some learners plug numbers directly into the simple interest formula instead of using the compound interest relation. Others forget that CI depends on both the principal and accumulated interest, making linear methods incorrect. Always express CI in terms of P using the correct compound interest factor, then solve algebraically for P.

Final Answer:
The principal that generated Rs. 10,176 as compound interest at 12% per annum for 2 years is Rs. 40,000.