The difference between simple interest and compound interest, both calculated annually on the same sum of money for 2 years at 4% per annum, is Rs. 1. What is the principal sum?

Introduction / Context:
This classic aptitude question tests the relationship between simple interest and compound interest over a short period. For two years, there is a well known formula for the difference between compound interest and simple interest at the same rate and on the same principal. Using this shortcut saves time in exams.

Given Data / Assumptions:

• Time period n = 2 years.
• Rate of interest r = 4% per annum.
• Difference between compound interest and simple interest = Rs. 1.
• Interest is compounded annually for the compound interest calculation.
• Principal sum P is unknown and needs to be found.

Concept / Approach:
For 2 years, the difference between compound interest and simple interest on the same principal P at rate r% per annum is given by:
Difference = P * (r^2) / 100^2 Here r is in percentage form. This formula comes from expanding the compound interest expression for 2 years and subtracting the simple interest amount.

Step-by-Step Solution:
Step 1: Write the formula: Difference = P * r^2 / 10000. Step 2: Substitute known values: 1 = P * 4^2 / 10000. Step 3: Simplify the square: 4^2 = 16, so 1 = P * 16 / 10000. Step 4: Rearrange for P: P = 1 * 10000 / 16. Step 5: Compute P = 10000 / 16 = 625. Therefore, the principal sum is Rs. 625.

Verification / Alternative Check:
Compute simple interest: SI = P * r * n / 100 = 625 * 4 * 2 / 100 = Rs. 50.
Compute amount with compound interest: A = 625 * (1 + 0.04)^2 = 625 * 1.0816 = Rs. 676.
Compound interest CI = A - P = 676 - 625 = Rs. 51.
Difference CI - SI = 51 - 50 = Rs. 1, which matches the given data.

Why Other Options Are Wrong:
Options A, B, and C (Rs. 620, Rs. 630, Rs. 640) do not satisfy the difference of exactly Rs. 1 when you compute the interests.
Option E (Rs. 600) also produces a different difference between CI and SI. Only Rs. 625 leads to a difference of one rupee.

Common Pitfalls:
Students sometimes try to compute full SI and CI amounts repeatedly without using the shortcut formula, which is time consuming. Others may mistakenly use r / 100 instead of r^2 / 100^2 in the formula for difference. Always remember that the difference for 2 years depends on the square of the rate.

Final Answer:
The principal sum of money is Rs. 625.