The difference between the compound interest and the simple interest on the same principal at the rate of 12% per annum for 3 years is Rs. 112.32, when interest is compounded annually. What is the principal amount?

Introduction / Context:
This question examines the relationship between simple interest and compound interest on the same principal, at the same rate, over the same time period. Because compound interest includes interest on interest, it is always at least as large as simple interest, and usually larger when the time period has more than one compounding interval. Here we are told by how much compound interest exceeds simple interest over 3 years at 12 percent per annum, and we must find the principal that leads to this difference.

Given Data / Assumptions:
Rate of interest r is 12 percent per annum.
Time period t is 3 years.
Interest is compounded annually for the compound interest calculation.
The difference between compound interest and simple interest on the same principal is Rs. 112.32.
Principal P is unknown and must be determined.

Concept / Approach:
For simple interest, I_si = P * r * t / 100. For compound interest with annual compounding, amount A = P * (1 + r / 100)^t and compound interest I_ci = A - P. The difference D between compound and simple interest is D = I_ci - I_si. We compute A symbolically and then express D in terms of P, r, and t. With r = 12 and t = 3 fixed, the difference can be written as a constant multiplier times P. Setting this equal to 112.32 allows us to solve for P.

Step-by-Step Solution:
For simple interest at 12 percent per annum for 3 years, I_si = P * 12 * 3 / 100 = P * 36 / 100 = 0.36P.For compound interest with annual compounding, amount A = P * (1 + 12 / 100)^3.Compute the factor (1 + 12 / 100) = 1.12. Then A = P * (1.12)^3.Calculate (1.12)^3 = 1.404928.Compound interest is I_ci = A - P = P * 1.404928 - P = P * (1.404928 - 1) = 0.404928P.Difference D between compound and simple interest is D = I_ci - I_si = 0.404928P - 0.36P.Simplify D to D = 0.044928P.The question states D = 112.32, so 0.044928P = 112.32.Solve for P: P = 112.32 / 0.044928 = 2500 rupees.Therefore the principal amount is Rs. 2500.

Verification / Alternative check:
Check with P = 2500. Simple interest I_si equals 2500 * 12 * 3 / 100 = 2500 * 36 / 100 = 900 rupees. For compound interest, amount A = 2500 * (1.12)^3 = 2500 * 1.404928 = 3512.32 rupees. Compound interest I_ci is A - P = 3512.32 - 2500 = 1012.32 rupees. The difference I_ci - I_si is 1012.32 - 900 = 112.32 rupees, which matches the given value. This confirms the correctness of the principal determined.

Why Other Options Are Wrong:
If principal were Rs. 25000 or Rs. 50000, the difference between compound and simple interest would be ten or twenty times larger than 112.32, respectively.
For Rs. 5000 or Rs. 3000, the computed difference would not equal 112.32 when the formulas are applied correctly.
Only P = 2500 produces the exact difference of Rs. 112.32, so the other options are incorrect.

Common Pitfalls:
Errors often arise from miscomputing the power (1.12)^3 or from forgetting to subtract principal when finding compound interest. Some learners accidentally compute the difference between amounts instead of between interests, or they use simple interest formulas in both cases. Attention to the definition of each term and to the correct formula for compound interest is critical. Working with decimals carefully and using the difference expression D = 0.044928P helps keep the calculation structured.

Final Answer:
The principal amount for which compound interest exceeds simple interest by Rs. 112.32 at 12 percent per annum over 3 years is Rs. 2500.