The difference between simple interest and compound interest, compounded annually, on a certain sum of money for 2 years at 9 percent per annum is Rs 405. What is the principal amount?

Introduction / Context:
This question uses the well known relationship between simple interest and compound interest for 2 years at the same rate and principal. The difference between compound and simple interest for 2 years at an annual rate R has a simple formula. The problem gives this difference and the rate and asks us to find the principal. This is a common type in banking and aptitude exams.

Given Data / Assumptions:
- The investment period is 2 years.
- The rate of interest R is 9 percent per annum.
- Compound interest is calculated with annual compounding.
- The difference between compound interest (CI) and simple interest (SI) for the 2 year period is Rs 405.
- We must determine the principal sum P.

Concept / Approach:
For a principal P and annual rate R, the difference D between compound interest and simple interest over 2 years is:
D = CI − SI = P * (R/100)^2This formula arises because for 2 years, CI includes an extra interest on the first year's interest, while SI does not. Knowing D and R, we can solve directly for P without computing CI and SI separately.

Step-by-Step Solution:
Given R = 9 %, so R/100 = 0.09.Compute (R/100)^2 = (0.09)^2 = 0.0081.Difference D is 405, so D = P * 0.0081.Thus, 405 = P * 0.0081.Solve for P: P = 405 / 0.0081.Compute P: 405 / 0.0081 = 50000.Therefore, the principal amount is Rs 50000.

Verification / Alternative check:
We can verify by computing CI and SI separately. Let P = 50000 and R = 9 %. Simple interest for 2 years is SI = (50000 * 9 * 2) / 100 = 50000 * 0.18 = 9000. For compound interest, amount A = 50000 * (1.09)^2. (1.09)^2 = 1.1881, so A = 50000 * 1.1881 = 59405. CI = A − P = 59405 − 50000 = 9405. The difference CI − SI = 9405 − 9000 = 405, which matches the given value. This confirms that the principal is correct.

Why Other Options Are Wrong:
If P were Rs 100000, the difference D would be 100000 * 0.0081 = 810, which is greater than 405. For Rs 200000, D would be 1620, and for Rs 150000, it would be 1215. Rs 25000 would give a difference of 202.5, which is less than 405. Only Rs 50000 produces the exact difference of 405 between compound and simple interest for 2 years at 9 percent.

Common Pitfalls:
Some students mistakenly use the full simple interest formula or attempt to compute CI and SI separately without using the shortcut D = P * (R/100)^2, which is more complicated. Others confuse R as a decimal and incorrectly square 9 instead of 0.09. Carefully applying the correct formula and keeping track of percentages and decimals helps avoid these errors.

Final Answer:
The principal sum is Rs 50000.