The compound interest earned in 2 years at 12% per annum is Rs. 10176. What is the sum of money that was originally invested?

Introduction / Context:
Here, the total compound interest over a two year period is provided, together with the annual rate. The learner must find the original principal. This is similar in structure to an earlier problem but with a different rate, which means the growth factor will change. It tests flexibility in applying the general compound interest framework to recover the principal from the known interest.

Given Data / Assumptions:

• Compound interest CI for 2 years = Rs. 10176.
• Rate of interest r = 12% per annum.
• Time period n = 2 years.
• Interest is compounded annually.
• Relationship: CI = P * ((1 + r) ^ n - 1).

Concept / Approach:
We convert the rate into decimal form and compute the growth factor for two years. The generic relation CI = P * ((1 + r) ^ n - 1) allows us to express the given interest as a multiple of the unknown principal. We then solve for P by dividing the known interest by the net growth factor ((1 + r) ^ n - 1). This technique is particularly efficient when only the interest and not the final amount is supplied.

Step-by-Step Solution:
Step 1: Convert the rate to decimal form: r = 12% = 0.12. Step 2: Compute the two year growth factor: (1 + r) ^ n = (1.12) ^ 2. Step 3: Calculate (1.12) ^ 2 = 1.2544. Step 4: Then (1 + r) ^ n - 1 = 1.2544 - 1 = 0.2544. Step 5: The formula for compound interest is CI = P * 0.2544. Step 6: We are given CI = 10176, so 10176 = P * 0.2544. Step 7: Solve for P: P = 10176 / 0.2544 = 40000. Step 8: Therefore, the original principal is Rs. 40000.

Verification / Alternative check:
To verify, we compute the amount and the interest using the principal we found. With P = 40000, the amount is A = 40000 * (1.12) ^ 2 = 40000 * 1.2544 = 50176. The compound interest is CI = A - P = 50176 - 40000 = 10176, which matches the given interest exactly. This perfect match confirms that our principal value and calculations are correct and consistent with the data given in the problem.

Why Other Options Are Wrong:
If P were 50000, the compound interest would be 50000 * 0.2544 = 12720, which is larger than the required 10176. With P = 60000, CI becomes 60000 * 0.2544 = 15264, even further from the target. If P were 80000, the interest would double from the 40000 case to 20352, clearly too high. The option 45000 would give 45000 * 0.2544 = 11448, which is also not equal to 10176. Only a principal of 40000 leads to the specified interest of 10176.

Common Pitfalls:
A common error is to misinterpret the given interest as the final amount and attempt to subtract the principal from it, which leads to confusion. Another pitfall is to forget to square 1.12 correctly, resulting in an inaccurate growth factor. Some students may try to use the simple interest relation and divide the interest by 2 * 12%, which ignores compounding and gives the wrong principal. Maintaining a clear distinction between amount and interest and using the proper growth factor are essential to solving this kind of problem accurately.

Final Answer:
The sum invested, which earns Rs. 10176 as compound interest in 2 years at 12% per annum, is Rs. 40000, corresponding to option C.