The difference between simple interest and compound interest, compounded annually, on a certain sum of money for 2 years at 12 percent per annum is Rs. 72. For this investment, what is the value of the principal sum in rupees?

Introduction / Context:
This question is another application of the standard formula for the difference between compound interest (CI) and simple interest (SI) for 2 years at a given rate. The difference over 2 years is proportional to the principal and to the square of the rate. The question supplies the difference and the rate, and asks for the principal. This kind of formula-based problem is very common in interest-related aptitude tests.

Given Data / Assumptions:

• Annual interest rate r = 12 percent.
• Time period t = 2 years.
• Difference between CI and SI for 2 years, D₂ = Rs. 72.
• Interest is compounded annually when computing CI.
• Principal P is unknown and needs to be determined.

Concept / Approach:
For 2 years at r percent per annum, the difference between compound interest and simple interest is given by:
D₂ = P * (r/100)^2.
This comes from comparing the expression for compound amount with the linear simple interest formula over 2 years. Once D₂ and r are known, we substitute them into the formula and solve for P directly, which is much faster than computing CI and SI separately.

Step-by-Step Solution:
Step 1: Use the standard formula D₂ = P * (r/100)^2. Step 2: Substitute D₂ = 72 and r = 12: 72 = P * (12/100)^2. Step 3: Compute (12/100)^2 = 144/10000 = 0.0144. Step 4: So 72 = P * 0.0144. Step 5: Rearranging, P = 72 / 0.0144. Step 6: Evaluate the division: 72 / 0.0144 = 5000. Step 7: Therefore, the principal sum is Rs. 5000.

Verification / Alternative check:
We can verify this result by computing CI and SI explicitly. For P = 5000, r = 12 percent, t = 2 years, simple interest SI = (5000 * 12 * 2) / 100 = 1200. The compound amount A under annual compounding is A = 5000 * (1.12)^2. Compute (1.12)^2 = 1.2544, so A = 5000 * 1.2544 = 6272. Compound interest CI = A − P = 6272 − 5000 = 1272. The difference CI − SI = 1272 − 1200 = 72, which matches the given value exactly.

Why Other Options Are Wrong:
If P were Rs. 10000, the difference would be 10000 * 0.0144 = 144, which is double the required 72. At P = 20000, the difference becomes 288, and at P = 15000 it becomes 216. For P = 2500, the difference is 36. None of these match the given difference of Rs. 72. Only a principal of Rs. 5000 produces exactly the specified difference between compound and simple interest.

Common Pitfalls:
Some candidates wrongly compute (r/100)^2 as r^2/100 or forget to square the fraction, leading to large errors. Others attempt a more complicated approach by computing CI and SI separately with incomplete arithmetic precision. Remembering and correctly applying the simple formula D₂ = P * (r/100)^2 makes such questions fast and reliable. Careful handling of decimals is also important to avoid calculation errors.

Final Answer:
The principal sum for which the difference between simple interest and compound interest over 2 years at 12 percent per annum is Rs. 72 is Rs. 5000.