The difference between the simple interest and the compound interest on a certain sum of money for 2 years at 4 percent per annum, compounded annually, is Re. 1. What is the sum?

Introduction / Context:
This question also uses the special relation between compound interest and simple interest for 2 years on the same principal at the same rate. We know the difference between these two types of interest is Re. 1 and the rate is 4 percent per annum. We are asked to find the principal sum. Recognizing and applying the correct formula can make this problem very quick to solve.

Given Data / Assumptions:

• Difference between simple interest and compound interest for 2 years is Re. 1.
• Rate of interest r = 4 percent per annum.
• Time n = 2 years.
• Interest is compounded annually for the compound interest part.
• We need to find the principal P.

Concept / Approach:
For 2 years on the same principal P at rate r, the difference between compound interest and simple interest is: Difference = P * (r/100)^2 In this question, the difference is Re. 1. We substitute r = 4 and difference = 1 into the formula and solve for P. The key is to correctly square the rate fraction and then divide the difference by it.

Step-by-Step Solution:
Given Difference = 1 Formula: Difference = P * (r/100)^2 So 1 = P * (4/100)^2 (4/100)^2 = (0.04)^2 = 0.0016 Therefore, 1 = P * 0.0016 P = 1 / 0.0016 P = 625 Thus the principal sum is Rs. 625

Verification / Alternative check:
We can verify by explicitly computing simple and compound interest. For P = 625, r = 4 percent, n = 2 years: Simple interest SI = 625 * 4 * 2 / 100 = 625 * 0.08 = 50 Compound amount A = 625 * (1.04)^2 (1.04)^2 = 1.0816 A = 625 * 1.0816 = 676 Compound interest CI = 676 - 625 = 51 Difference = CI - SI = 51 - 50 = 1 This matches the given difference of Re. 1, confirming that the principal is Rs. 625.

Why Other Options Are Wrong:
If we tried P = 630, 640, or 650, then multiplying by 0.0016 would give differences different from 1. For example, with P = 630, the difference would be 630 * 0.0016 = 1.008, which is not exactly 1. Similarly, P = 600 would give a difference of 0.96, again not matching the given value. Only P = 625 satisfies the relation exactly and also works out when we compute SI and CI directly.

Common Pitfalls:
Some students mistakenly treat the difference as simple interest instead of the difference between CI and SI. Others may use an incorrect formula or forget to square the rate fraction, using P * r/100 instead of P * (r/100)^2. Miscalculating 4 squared or misplacing decimal points when squaring 0.04 can also lead to wrong answers. Memorizing and correctly applying this special relation helps solve such problems efficiently.

Final Answer:
The required principal sum is Rs. 625.