If the difference between the compound interest and the simple interest on a certain sum at the rate of 5% per annum for 2 years is Rs. 20, then what is the principal (sum invested)? Assume annual compounding for compound interest.

Introduction / Context:
This question again uses the relationship between simple interest and compound interest on the same principal, rate and time. The difference between CI and SI over 2 years has a compact formula and allows us to solve quickly for the principal when the rate and difference are known.

Given Data / Assumptions:

Rate of interest r = 5% per annum.
Time t = 2 years.
Difference between compound interest and simple interest = Rs. 20.
Interest is compounded annually for CI.
Principal P must be determined.

Concept / Approach:
For 2 years, the difference between CI and SI is: CI − SI = P * (r^2 / 100^2). This arises because CI for 2 years equals SI for 2 years plus interest on the first year's interest. Substituting r and the known difference allows direct solution for P.

Step-by-Step Solution:
Step 1: Write the difference formula. CI − SI = P * r^2 / 100^2. Given CI − SI = 20 and r = 5. Step 2: Substitute values. 20 = P * 5^2 / 100^2. 5^2 = 25 and 100^2 = 10,000. So 20 = P * 25 / 10,000. Step 3: Solve for P. P = 20 * 10,000 / 25. P = 200,000 / 25 = 8,000.

Verification / Alternative check:
Check explicitly. SI for 2 years: SI = P * r * t / 100 = 8,000 * 5 * 2 / 100 = 800. CI amount: A = 8,000 * (1.05)^2 = 8,000 * 1.1025 = 8,820. CI = 8,820 − 8,000 = 820. Difference CI − SI = 820 − 800 = 20, matching the given difference.

Why Other Options Are Wrong:
For a principal of 2,000, 4,000 or 6,000, the difference CI − SI at 5% for 2 years would be 5, 10 and 15 respectively, not 20. Therefore those options are inconsistent with the problem data.

Common Pitfalls:
Some learners misapply the simple interest formula or forget that the difference formula uses r^2 / 100^2. Others attempt long methods by setting up full SI and CI equations instead of using the shortcut. Using the standard relationship saves time and reduces calculation errors.

Final Answer:
The principal (sum invested) is Rs. 8000.