The difference between the compound interest (compounded every 6 months) and the simple interest on a certain sum of money after 2 years is Rs. 248.10. The annual rate of interest is 10%. What is the principal sum?

Introduction / Context:
This question combines both compound interest and simple interest. You are told that at the same nominal annual rate and over the same 2-year period, the compound interest (with half yearly compounding) exceeds the simple interest by Rs. 248.10. From this information, you must find the original principal.

Given Data / Assumptions:

• Principal (P) is unknown.
• Nominal annual rate of interest = 10%.
• Time = 2 years.
• Compound interest is calculated with half yearly compounding.
• Simple interest is calculated at the same nominal annual rate of 10%.
• Difference between compound interest and simple interest after 2 years = Rs. 248.10.

Concept / Approach:
For simple interest, SI = (P * R * T) / 100. For compound interest with half yearly compounding at a nominal annual rate R, the rate per half year is R/2, and the number of periods in 2 years is 4. The amount with compound interest is A = P * (1 + (R / 2) / 100)^(4). CI is then A - P. The difference CI - SI is given as Rs. 248.10, so you can write an equation in P and solve.

Step-by-Step Solution:
Step 1: Simple interest for 2 years at 10% per annum: SI = (P * 10 * 2) / 100 = 0.20 * P. Step 2: For half yearly compounding, rate per half year = 10 / 2 = 5%. Step 3: Number of half yearly periods in 2 years = 4. Step 4: Compound amount after 4 periods: A = P * (1 + 5 / 100)^4 = P * (1.05)^4. Step 5: Compute (1.05)^4 = 1.21550625 (approximately). Step 6: Compound interest CI = A - P = P * (1.21550625 - 1) = P * 0.21550625. Step 7: Difference between CI and SI: CI - SI = P * 0.21550625 - 0.20 * P = P * (0.01550625). Step 8: Given that CI - SI = 248.10, so P * 0.01550625 = 248.10. Step 9: Solve for P: P = 248.10 / 0.01550625 = 16000.

Verification / Alternative check:
Check the result by recomputing both interests for P = 16000. Simple interest: SI = (16000 * 10 * 2) / 100 = 3200. Compound amount: A = 16000 * (1.05)^4 ≈ 16000 * 1.21550625 ≈ 19448.10. CI ≈ 19448.10 - 16000 = 3448.10. The difference CI - SI ≈ 3448.10 - 3200 = 248.10, which matches the given value.

Why Other Options Are Wrong:
12000, 14000, and 18000 would lead to different differences between compound and simple interest over 2 years at 10%, and none of them would produce exactly Rs. 248.10. Only P = 16000 satisfies the condition.

Common Pitfalls:
Students sometimes incorrectly use annual compounding instead of half yearly compounding or confuse the amount with the interest. Another common error is subtracting simple interest from the amount instead of from the compound interest. Carefully distinguishing SI, CI, and the amount, and correctly setting up the difference, is key.

Final Answer:
The required principal sum is 16000.