For a certain principal, the difference between the compound interest and the simple interest on that sum for 2 years at 5% per annum is Rs. 41. What is the value of the principal (sum of money)?

Introduction / Context:
This question compares simple interest (SI) and compound interest (CI) for the same principal, rate and time. Over two years, CI exceeds SI by a small amount that depends on the square of the rate. Using a known shortcut formula for the difference between CI and SI for 2 years, we can quickly find the principal without computing both interests separately.

Given Data / Assumptions:

Difference (CI − SI) over 2 years = Rs. 41
Rate of interest r = 5% per annum
Time t = 2 years
Interest is compounded annually for CI
Principal P is unknown

Concept / Approach:
For 2 years at r% per annum, the difference between compound interest and simple interest on principal P is given by:
CI − SI = P * (r^2 / 100^2). This comes from the fact that compound interest in 2 years equals simple interest in 2 years plus interest on the interest of the first year. Once we plug in the known values of r and CI − SI, we can solve directly for P.

Step-by-Step Solution:
Step 1: Write the formula. CI − SI = P * r^2 / 100^2. Step 2: Substitute r = 5 and CI − SI = 41. 41 = P * 5^2 / 10,000. 5^2 = 25, so 41 = P * 25 / 10,000. Step 3: Solve for P. P = 41 * 10,000 / 25. P = 410,000 / 25 = 16,400.

Verification / Alternative check:
We can verify by actually computing SI and CI. Simple interest for 2 years at 5%:
SI = P * r * t / 100 = 16,400 * 5 * 2 / 100 = 1,640. Amount under CI after 2 years:
A = 16,400 * (1.05)^2 = 16,400 * 1.1025 = 18,081. CI = A − P = 18,081 − 16,400 = 1,681. Difference CI − SI = 1,681 − 1,640 = 41, matching the given value exactly.

Why Other Options Are Wrong:
For P = 7,200, difference would be 7,200 * 25 / 10,000 = 18, not 41. For 9,600, difference is 24; for 8,400 it is 21. None of these match 41, so those principals cannot be correct.

Common Pitfalls:
Some learners try to separately calculate CI and SI with an unknown principal, leading to two equations and more algebra than necessary. Others confuse the formula for the 2-year difference and forget to square the rate. Remembering the compact relationship CI − SI = P * r^2 / 100^2 for 2 years makes such problems much faster and less error-prone.

Final Answer:
The principal (sum of money) is Rs. 16,400.