What is the difference between the compound interest on Rs 5000 for 1 1/2 years at 4% per annum when interest is compounded yearly and when it is compounded half-yearly?

Introduction / Context:
This question compares two ways of compounding interest for the same principal, rate, and total time period of 1 1/2 years (that is, 1.5 years). In the first case, interest is compounded yearly at 4% per annum. In the second case, interest is compounded half-yearly at an effective rate of 2% per half-year. We are asked to find the difference between the compound interests obtained in these two scenarios. This problem highlights how compounding frequency affects the total interest earned over a given period.

Given Data / Assumptions:

• Principal P = Rs 5,000.
• Nominal annual rate r = 4% per annum.
• Total time T = 1.5 years (1 1/2 years).
• Case 1: CI compounded yearly at 4%.
• Case 2: CI compounded half-yearly at 4% nominal (2% per half-year).
• We must compute the difference between the two compound interests.

Concept / Approach:
For yearly compounding over 1.5 years, one standard approach is to treat the first full year under compound interest and the remaining half-year under simple interest calculated on the amount at the end of the first year. For half-yearly compounding, we count the number of half-year periods: 1.5 years = 3 half-years, with an interest rate of 2% per half-year. We compute the final amounts in both cases, subtract the principal to obtain each CI, and finally take the difference. Careful handling of the half-year simple interest in the first case is important.

Step-by-Step Solution:
Case 1 (Yearly compounding): After 1 year at 4%, amount A1 = 5,000 * 1.04 = 5,200. For the remaining 0.5 year, interest is taken as simple interest on 5,200 at 4% per annum for half a year. That half-year interest = 5,200 * 4/100 * 1/2 = 5,200 * 0.02 = 104. So final amount in Case 1 = 5,200 + 104 = 5,304, and CI1 = 5,304 - 5,000 = 304. Case 2 (Half-yearly compounding): Rate per half-year = 4% / 2 = 2% = 0.02. Number of half-year periods in 1.5 years = 3. Amount A2 = 5,000 * (1.02)^3. Compute (1.02)^2 = 1.0404; then (1.02)^3 = 1.0404 * 1.02 = 1.061208. So A2 = 5,000 * 1.061208 = 5,306.04; CI2 = 5,306.04 - 5,000 = 306.04. Difference in interest = CI2 - CI1 = 306.04 - 304 = 2.04.

Verification / Alternative check:
We can round the second amount to two decimal places at each step to see that the effect of more frequent compounding is modest over this short period and low rate. A small difference of Rs 2.04 is reasonable. If the difference were very large, such as Rs 5 or Rs 10, it would suggest a miscalculation. Our stepwise computations align with the exact formula and typical exam answers, confirming that Rs 2.04 is correct.

Why Other Options Are Wrong:
Rs 3.06 and Rs 3.65 are larger than the difference created by these mild changes in compounding frequency and would correspond to either higher rates or longer durations. Rs 5.40 is even more unrealistic for such a small principal and short period. Rs 1.50 is too small and does not match the calculated difference of Rs 2.04. Thus, only Rs 2.04 corresponds to the exact arithmetic difference between the two compound interest amounts.

Common Pitfalls:
One common mistake is to treat 1.5 years as exactly 1.5 compounding periods in the yearly compounding case, which is not valid since compounding is done once per year. Another error is to forget that under half-yearly compounding the rate per period is 2%, not 4%. Students also sometimes subtract the two final amounts incorrectly or round numbers too aggressively before subtraction. Performing the calculations in a structured order and retaining sufficient precision prevents such errors.

Final Answer:
The difference between the two compound interests is Rs 2.04.