What is the difference, in rupees, between the compound interest on Rs. 1000 for 1 year at 10% per annum when interest is compounded yearly and when interest is compounded half-yearly?

Introduction / Context:
This problem compares two different compounding frequencies for the same nominal annual interest rate. The principal, rate and total time are fixed, but the compounding interval changes from yearly to half-yearly. This changes the effective amount of interest earned, and we must find the difference between the two resulting compound interests.

Given Data / Assumptions:

Principal P = Rs. 1000
Nominal annual interest rate = 10% per annum
Total time t = 1 year
Case 1: Interest compounded yearly (once per year)
Case 2: Interest compounded half-yearly (twice per year)
We must find CI(half-yearly) − CI(yearly) in rupees

Concept / Approach:
For annual compounding, we use the formula:
A = P * (1 + r/100)^t. For half-yearly compounding, the periodic rate is r/2 and the number of periods n = 2 * t. The amount becomes:
A = P * (1 + (r/2)/100)^(2t). We compute the amount and interest in both cases and then subtract to get the difference.

Step-by-Step Solution:
Case 1 (Yearly compounding): A1 = 1000 * (1 + 10/100)^1 = 1000 * 1.10 = 1100. CI1 = A1 − P = 1100 − 1000 = Rs. 100. Case 2 (Half-yearly compounding): Periodic rate = 10% / 2 = 5% per half-year. Number of periods in 1 year n = 2. A2 = 1000 * (1.05)^2 = 1000 * 1.1025 = 1102.50. CI2 = A2 − P = 1102.50 − 1000 = Rs. 102.50. Difference in compound interest = CI2 − CI1 = 102.50 − 100 = Rs. 2.50.

Verification / Alternative check:
We can also think of this as the extra interest generated by earning interest for an additional half-year on the first half-year's interest. That extra interest is roughly 5% of Rs. 50 (the first half-year's interest), which is Rs. 2.50. This intuitive reasoning matches our exact calculation.

Why Other Options Are Wrong:
Rs. 0.5 and Rs. 1.5 are too small; they would imply very weak effect from compounding more frequently. Rs. 3.5 is slightly too large. Only Rs. 2.5 exactly matches the calculated difference in compound interest between the two compounding methods.

Common Pitfalls:
Many learners forget to divide the annual rate by 2 when converting from yearly to half-yearly compounding, or they incorrectly double the rate. Others mistakenly assume that the interest will be the same regardless of compounding frequency. In reality, more frequent compounding at the same nominal rate always yields slightly more interest, as seen here.

Final Answer:
The difference in compound interest is Rs. 2.5.