The difference between the simple interest and the compound interest on a certain sum of money for 2 years at 4% per annum, compounded annually, is Re. 1. What is the principal amount (in rupees)?

Introduction:
This problem asks us to find the principal when we know the difference between compound interest and simple interest for 2 years at a given rate. Questions like these test the understanding of how simple interest and compound interest differ, especially over more than one year.

Given Data / Assumptions:

• Difference between compound interest and simple interest for 2 years = Re. 1
• Time period, t = 2 years
• Annual rate of interest, r = 4% per annum
• Interest under compound interest is compounded annually
• Principal amount is P rupees, which we need to find

Concept / Approach:
For simple interest, the interest for t years is: SI = P * r * t / 100 For compound interest (compounded annually) for 2 years, the total amount is: A = P * (1 + r / 100)^2 So compound interest is: CI = A - P = P * ((1 + r / 100)^2 - 1) The difference between CI and SI for 2 years at rate r is: Difference = CI - SI

Step-by-Step Solution:
Step 1: Write expressions for SI and CI for 2 years. SI = P * 4 * 2 / 100 = 0.08 * P CI = P * ((1 + 0.04)^2 - 1) Step 2: Compute (1 + 0.04)^2. (1.04)^2 = 1.0816 So CI = P * (1.0816 - 1) = P * 0.0816 Step 3: Compute the difference between CI and SI. Difference = CI - SI = P * 0.0816 - P * 0.08 Difference = P * (0.0816 - 0.08) = P * 0.0016 Step 4: Use the given difference to find P. P * 0.0016 = 1 P = 1 / 0.0016 P = 625 rupees

Verification / Alternative check:
We can verify by directly calculating SI and CI for P = 625 rupees. SI = 625 * 4 * 2 / 100 = 625 * 0.08 = 50 rupees CI = 625 * (1.04)^2 - 625 = 625 * 1.0816 - 625 = 676 rupees - 625 rupees = 51 rupees Difference = 51 - 50 = 1 rupee This matches the given difference, so P = 625 is confirmed as correct.

Why Other Options Are Wrong:
635, 645, 655, 600 rupees: For each of these values, the difference between the calculated compound interest and simple interest for 2 years at 4% would not be exactly 1 rupee. They do not satisfy the equation P * 0.0016 = 1. 625 rupees: This is the only principal value that produces the correct difference of exactly 1 rupee.

Common Pitfalls:
Some learners incorrectly assume that the difference between CI and SI for 2 years depends only on the rate and time, not on the principal. Others may miscalculate (1.04)^2 or forget to convert 4% to 0.04. Mixing up the formula for CI and SI, or failing to subtract the principal when calculating CI, are also common errors. It is important to work systematically through the expressions and carefully compute the difference.

Final Answer:
The principal amount that makes the difference between compound interest and simple interest equal to Re. 1 for 2 years at 4% per annum is 625 rupees.