A sum of Rs. 25,000 earns compound interest for 1 year at a nominal rate of 20% per annum. What is the difference, in rupees, between the compound interest earned when interest is compounded semi-annually and when it is compounded annually?

Introduction / Context:
This question examines how the compounding frequency affects the amount of compound interest earned for the same nominal annual rate. With more frequent compounding (semi-annual instead of annual), the effective interest rate increases slightly, leading to a higher compound interest amount. We must quantify this difference over one year for a principal of Rs. 25,000 at 20% per annum.

Given Data / Assumptions:

Principal P = Rs. 25,000
Nominal annual interest rate = 20% per annum
Time t = 1 year
Case 1: Compounded annually (once per year)
Case 2: Compounded semi-annually (twice per year)
We must find CI(semi-annual) − CI(annual)

Concept / Approach:
For annual compounding, the amount after 1 year is:
A1 = P * (1 + 0.20). For semi-annual compounding, the periodic rate is 20% / 2 = 10% per half-year, and there are 2 periods in 1 year, so:
A2 = P * (1 + 0.10)^2. Compound interest in each case is amount minus principal. The difference in compound interest is then (A2 − P) − (A1 − P) = A2 − A1.

Step-by-Step Solution:
Case 1 (Annual compounding): A1 = 25,000 * 1.20 = Rs. 30,000. CI1 = A1 − P = 30,000 − 25,000 = Rs. 5,000. Case 2 (Semi-annual compounding): Periodic rate i = 10% per half-year. Number of periods n = 2. A2 = 25,000 * (1.10)^2 = 25,000 * 1.21 = Rs. 30,250. CI2 = A2 − P = 30,250 − 25,000 = Rs. 5,250. Difference in compound interest = CI2 − CI1 = 5,250 − 5,000 = Rs. 250.

Verification / Alternative check:
Instead of computing both interests outright, notice that the only difference is between the amounts A2 and A1. Directly computing 25,000 * 1.21 − 25,000 * 1.20 gives 25,000 * (1.21 − 1.20) = 25,000 * 0.01 = 250. This reconfirms that semi-annual compounding generates exactly Rs. 250 more interest than annual compounding at the same nominal rate over one year.

Why Other Options Are Wrong:
Rs. 125 and Rs. 375 represent half and one and a half times the correct difference, respectively, and do not align with the precise calculation. Rs. 500 would correspond to a much larger gap in effective interest than actually occurs between annual and semi-annual compounding at 20%. Only Rs. 250 matches the exact difference.

Common Pitfalls:
A common mistake is to forget to divide the nominal annual rate by 2 when switching to semi-annual compounding, instead using 20% per half-year, which is incorrect. Another error is to compare the interest amounts without subtracting the principal or to think that the nominal rate alone determines the interest, ignoring compounding frequency. Careful identification of the periodic rate and number of periods avoids such errors.

Final Answer:
The difference in compound interest between semi-annual and annual compounding is Rs. 250.