A person invests a certain sum of money for two years at an annual rate of 8% and finds that the amount received under compound interest is Rs 48 more than the amount that would have been received under simple interest; what is the original principal sum invested in this interest comparison problem?

Introduction / Context:
This question compares compound interest and simple interest on the same principal, rate, and time. Over two years, compound interest yields a slightly higher amount than simple interest, and the difference in the amounts is given as Rs 48. The task is to work backwards from this difference to find the original principal invested. Such problems test understanding of both interest types and the special relation between them for a two year period.

Given Data / Assumptions:
- Principal invested = P (unknown). - Annual rate of interest R = 8 percent per annum. - Time period T = 2 years. - Amount with compound interest exceeds the amount with simple interest by Rs 48. - Interest is calculated annually, with compounding once per year.

Concept / Approach:
For simple interest over two years, total interest is SI = (P * R * T) / 100. For compound interest compounded annually, amount after two years is A = P * (1 + R / 100)^2. The difference between compound interest and simple interest for two years at the same rate is a known relation: CI − SI = P * (R^2) / 100^2. Using this relation, we can directly equate P * (R^2) / 100^2 to the given difference and solve for P.

Step-by-Step Solution:
Step 1: Note the relation for two years: CI − SI = P * (R^2) / 100^2. Step 2: Here, CI − SI = Rs 48 and R = 8 percent. Step 3: Substitute values: 48 = P * (8^2) / 100^2. Step 4: Compute 8^2 = 64, so 48 = P * 64 / 10,000. Step 5: Rearrange to solve for P: P = 48 * 10,000 / 64. Step 6: Compute 48 * 10,000 = 4,80,000. Step 7: Divide: 4,80,000 / 64 = 7,500. Step 8: Therefore, the original principal is P = Rs 7,500.

Verification / Alternative check:
Using P = Rs 7,500, compute simple and compound interest. Simple interest for two years: SI = (7,500 * 8 * 2) / 100 = (7,500 * 16) / 100 = 1,20,000 / 100 = Rs 1,200. Amount with simple interest = 7,500 + 1,200 = 8,700. For compound interest, amount A = 7,500 * (1 + 8 / 100)^2 = 7,500 * (1.08)^2. Now (1.08)^2 = 1.1664, so A = 7,500 * 1.1664 = Rs 8,748. The difference A − 8,700 = 8,748 − 8,700 = Rs 48, which matches the given data.

Why Other Options Are Wrong:
If P were Rs 8,000, the difference between compound and simple interest at 8 percent for two years would be 8,000 * 64 / 10,000 = Rs 51.20, not Rs 48. For P = 6,500, the difference becomes 6,500 * 64 / 10,000 = Rs 41.60. For P = 5,000, the difference is 5,000 * 64 / 10,000 = Rs 32. None of these equal the required Rs 48. Only P = Rs 7,500 satisfies the condition exactly.

Common Pitfalls:
Learners often compute full compound and simple interest amounts separately and may make arithmetic errors, instead of using the compact relation for two years. Another error is to apply the relation for a wrong time period or wrong rate. Forgetting to square the rate in the special formula CI − SI = P * R^2 / 100^2 also leads to incorrect results. Using the standard relation carefully avoids unnecessary complications.

Final Answer:
The original principal sum invested is Rs 7,500.