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. For this investment, what is the principal amount in rupees?

Introduction / Context:
This question uses a well-known shortcut formula for the difference between compound interest (CI) and simple interest (SI) for 2 years at a given rate. When interest is compounded annually, the difference over 2 years depends only on the principal and the square of the rate. The problem gives us this difference and the rate, and asks us to find the principal. It is a classic formula-based question that saves time in competitive exams.

Given Data / Assumptions:

• Annual rate of interest r = 9 percent.
• Time t = 2 years.
• Difference between CI and SI for 2 years, D₂ = Rs. 405.
• Interest is compounded annually when calculating CI.
• Principal P is unknown and must be determined.

Concept / Approach:
For 2 years at rate r percent per annum, the difference between compound interest and simple interest is:
D₂ = P * (r/100)^2.
This result comes from expanding the compound interest formula and comparing it to simple interest for 2 years on the same principal. Given D₂ and r, we can directly solve for P. This avoids computing CI and SI separately and is much faster.

Step-by-Step Solution:
Step 1: Use the shortcut formula for 2 years: D₂ = P * (r/100)^2. Step 2: Substitute D₂ = 405 and r = 9: 405 = P * (9/100)^2. Step 3: Compute (9/100)^2 = 81/10000 = 0.0081. Step 4: So 405 = P * 0.0081. Step 5: Rearranging, P = 405 / 0.0081. Step 6: Divide: 405 / 0.0081 = 50000. Step 7: Hence, the principal amount is Rs. 50000.

Verification / Alternative check:
We can verify by calculating SI and CI explicitly. For P = 50000, r = 9 percent, t = 2 years, simple interest SI = (50000 * 9 * 2) / 100 = 9000. For compound interest, the amount A = 50000 * (1.09)^2. Compute (1.09)^2 = 1.1881, so A = 50000 * 1.1881 = 59405. CI = A − P = 59405 − 50000 = 9405. The difference CI − SI = 9405 − 9000 = 405, matching the given value. This confirms the principal is indeed Rs. 50000.

Why Other Options Are Wrong:
If P were Rs. 25000, then D₂ would be 25000 * 0.0081 = 202.5, which is too small. For P = 100000, D₂ would be 810, which is too large. Similarly, 150000 and 200000 produce even larger differences. Only P = 50000 yields the exact difference of Rs. 405, so all other options are inconsistent with the given information.

Common Pitfalls:
Some students attempt to compute CI and SI separately from scratch without knowing the shortcut formula, which is more time consuming and prone to calculation errors. Others forget to square r/100 or mistakenly use r instead of r/100 in the formula, giving wildly incorrect values for P. Remembering that for 2 years, the difference depends on the squared rate factor is crucial for speed and accuracy.

Final Answer:
The principal amount on which the difference between compound and simple interest over 2 years at 9 percent per annum is Rs. 405 is Rs. 50000.