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. What is the principal sum in rupees?

Introduction / Context:
This question once again uses the formula for the difference between compound interest and simple interest over 2 years at the same rate and principal. The rate here is relatively small, 4 percent per annum, and the difference CI − SI is given as only Rs 1. We must find the principal that produces exactly this difference under the given conditions.

Given Data / Assumptions:
- Time period T = 2 years.
- Annual rate of interest R = 4 percent per annum.
- Interest is compounded annually for the CI calculation.
- Difference between CI and SI for 2 years is D = Rs 1.
- Principal P is to be determined.

Concept / Approach:
For 2 years at annual rate R on principal P, the difference between compound and simple interest is:
D = CI − SI = P * (R/100)^2This arises because in the second year, compound interest earns additional interest on the first year's interest, whereas simple interest does not. Knowing D and R, we can directly solve for P.

Step-by-Step Solution:
Given R = 4 %, so R/100 = 0.04.Compute (R/100)^2 = (0.04)^2 = 0.0016.Difference D is 1, so 1 = P * 0.0016.Solve for P: P = 1 / 0.0016.Compute P: 1 / 0.0016 = 625.Therefore, the principal sum is Rs 625.

Verification / Alternative check:
Let P = 625 and R = 4 %. Simple interest for 2 years is SI = (625 * 4 * 2) / 100 = 625 * 0.08 = 50. For compound interest, amount A = 625 * (1.04)^2. (1.04)^2 = 1.0816, so A = 625 * 1.0816 = 676.0000. Compound interest CI = A − P = 676 − 625 = 51. The difference CI − SI = 51 − 50 = 1, which matches the given difference. This confirms that P = 625 is correct.

Why Other Options Are Wrong:
If P were Rs 620, the difference D would be 620 * 0.0016 = 0.992, not exactly 1. For Rs 630, D would be about 1.008. For Rs 640, D would be about 1.024. For Rs 600, the difference would be 0.96. None of these values produce a difference of exactly 1 rupee. Only Rs 625 gives the correct difference between compound and simple interest at 4 percent per annum for 2 years.

Common Pitfalls:
Students sometimes mistakenly use 4 instead of 0.04 when squaring the rate, which leads to very large and incorrect values. Others try to compute CI and SI separately without recognizing the shortcut relation for 2 years. Correct handling of the percentage as a decimal and careful arithmetic with small numbers is essential here, since even small rounding errors can change the result when the difference is only 1 rupee.

Final Answer:
The principal sum is Rs 625.