The difference between simple interest and compound interest, compounded annually, on a certain sum of money for 2 years at 16 percent per annum is Rs 320. What is the principal sum in rupees?

Introduction / Context:
This is another problem based on the standard formula for the difference between compound interest and simple interest over 2 years at the same rate and principal. The rate is 16 percent per annum, and we are told that the difference between CI and SI for 2 years is Rs 320. The task is to find the principal. Such questions reinforce understanding of how the extra interest in compound interest arises from interest on interest.

Given Data / Assumptions:
- Time period T = 2 years.
- Annual rate of interest R = 16 percent per annum.
- Interest is compounded annually for CI.
- Difference between CI and SI over 2 years is D = Rs 320.
- We must find the principal P.

Concept / Approach:
For 2 years, the difference between compound interest and simple interest on a principal P at an annual rate R is given by:
D = CI − SI = P * (R/100)^2This formula comes from the fact that, in the second year, compound interest earns interest on the first year's interest as well, while simple interest does not. Once we know R and D, solving for P is straightforward.

Step-by-Step Solution:
Given R = 16 %, so R/100 = 0.16.Compute (R/100)^2 = (0.16)^2 = 0.0256.Difference D is 320, so we have 320 = P * 0.0256.Solve for P: P = 320 / 0.0256.Compute P: 320 / 0.0256 = 12500.Therefore, the principal sum is Rs 12500.

Verification / Alternative check:
Let P = 12500 and R = 16 %. Simple interest for 2 years is SI = (12500 * 16 * 2) / 100 = 12500 * 0.32 = 4000. For compound interest, amount A = 12500 * (1.16)^2. (1.16)^2 = 1.3456, so A = 12500 * 1.3456 = 16820. Compound interest CI = A − P = 16820 − 12500 = 4320. The difference CI − SI = 4320 − 4000 = 320, matching the given difference and confirming that P = 12500 is correct.

Why Other Options Are Wrong:
For P = 25000, the difference D would be 25000 * 0.0256 = 640, not 320. For P = 50000, D would be 1280. For P = 37500, the difference would be 960. A principal of Rs 10000 would give D = 256. Only a principal of Rs 12500 gives the exact difference of Rs 320 under the stated conditions.

Common Pitfalls:
Typical errors include miscalculating (0.16)^2, using 16 instead of 0.16 in the formula, or mistakenly applying the simple interest formula for the difference. Another mistake is to forget that the formula D = P * (R/100)^2 specifically applies to a 2 year period with annual compounding. Paying attention to these details and carefully handling decimals avoids such mistakes.

Final Answer:
The principal sum is Rs 12500.