The difference between simple interest and compound interest, compounded annually, on a certain sum of money for 2 years at 4 percent per annum is Rs. 1. For this investment, what is the principal sum in rupees?

Introduction / Context:
This is a small numerical question that again uses the standard formula for the difference between compound interest (CI) and simple interest (SI) for 2 years at a given rate. The difference is very small (only Rs. 1), which suggests that the principal is also relatively small. The rate and the difference are provided, and we must determine the principal. Such questions are designed to reinforce the use of the CI–SI difference shortcut in a quick calculation.

Given Data / Assumptions:

• Annual interest rate r = 4 percent.
• Time period t = 2 years.
• Difference between CI and SI, D₂ = Rs. 1.
• Interest is compounded annually when calculating CI.
• Principal P is unknown and needs to be found.

Concept / Approach:
For 2 years at rate r percent per annum, the difference between compound interest and simple interest is:
D₂ = P * (r/100)^2.
This formula follows from expanding (1 + r/100)^2 and comparing it with 1 + 2r/100. Given D₂ and r, we substitute and solve directly for P. Because the rate is small and the difference is only Rs. 1, the principal will be easy to compute.

Step-by-Step Solution:
Step 1: Use the standard formula D₂ = P * (r/100)^2. Step 2: Substitute D₂ = 1 and r = 4: 1 = P * (4/100)^2. Step 3: Compute (4/100)^2 = 16/10000 = 0.0016. Step 4: So 1 = P * 0.0016. Step 5: Rearranging gives P = 1 / 0.0016. Step 6: Divide: 1 / 0.0016 = 625. Step 7: Therefore, the principal sum is Rs. 625.

Verification / Alternative check:
We can verify by calculating SI and CI for P = 625 at r = 4 percent over 2 years. Simple interest SI = (625 * 4 * 2) / 100 = (625 * 8) / 100 = 50. The compound amount A is A = 625 * (1.04)^2. Compute (1.04)^2 = 1.0816, so A = 625 * 1.0816 = 676.0. Compound interest CI = A − P = 676 − 625 = 51. The difference CI − SI = 51 − 50 = 1, which matches the given difference exactly.

Why Other Options Are Wrong:
If P were Rs. 600, then D₂ would be 600 * 0.0016 = 0.96, slightly less than 1. At P = 630, D₂ would be 630 * 0.0016 = 1.008, slightly more than 1. Similarly, 620 and 640 give differences that are close but not equal to 1. Only P = 625 yields exactly the required difference of Rs. 1 between CI and SI at 4 percent for 2 years.

Common Pitfalls:
Some students attempt to compute SI and CI from assumed principal values in the options rather than using the direct formula, which is slower. Others incorrectly compute (4/100)^2 as 4/100 or 16/100 instead of 16/10000, leading to large errors in P. Being careful with the square of the fraction and the decimal place is crucial in such small-difference problems.

Final Answer:
The principal sum on which the difference between simple interest and compound interest over 2 years at 4 percent per annum is Rs. 1 is Rs. 625.