What is the difference between the compound interest on Rs. 5,000 for 1½ years at 4% per annum when interest is compounded yearly and when it is compounded half-yearly?

Introduction:
This question compares two different compounding frequencies on the same principal, rate, and time period. The goal is to understand how changing from yearly compounding to half yearly compounding slightly increases the compound interest earned for the same nominal annual rate.

Given Data / Assumptions:
Principal P = Rs. 5,000. Rate r = 4% per annum. Total time = 1½ years = 1.5 years. Case 1: Interest compounded yearly. Case 2: Interest compounded half-yearly.

Concept / Approach:
For yearly compounding with a broken period, we apply compound interest for the complete years and then simple interest on the amount for the remaining fraction of the year. For half yearly compounding, we convert the annual rate into a per half year rate and compute the amount over the corresponding number of half years. The required answer is the difference between the two compound interest values.

Step-by-Step Solution:
Case 1 (Yearly compounding): Amount after 1 year = 5000 * 1.04 = Rs. 5,200. Extra 0.5 year gives simple interest on 5,200. Simple interest = 5200 * 4 * 0.5 / 100 = 5200 * 0.02 = Rs. 104. Total interest (yearly) = 200 + 104 = Rs. 304. Case 2 (Half yearly compounding): Rate per half year = 4 / 2 = 2%. Number of half years in 1.5 years = 3. Amount = 5000 * (1.02)^3 = 5000 * 1.061208 ≈ Rs. 5,306.04. Interest (half yearly) ≈ 5306.04 − 5000 = Rs. 306.04. Difference = 306.04 − 304 = Rs. 2.04.

Verification / Alternative check:
We can verify by recomputing (1.02)^3 as 1.0404 * 1.02 = 1.061208. The multiplication 5000 * 1.061208 indeed gives approximately 5,306.04, confirming that the difference between the two interest amounts is Rs. 2.04.

Why Other Options Are Wrong:
Rs. 3.04, Rs. 4.04, and Rs. 5.04 assume much larger gaps between the amounts, which do not occur at such a small rate and period. Rs. 1.04 is smaller than the correct value and results from incorrect handling of the half yearly compounding or the broken period in yearly compounding.

Common Pitfalls:
Common errors include treating 1½ years as 1 year only or ignoring the fractional year, misusing the quarterly or monthly formula, or forgetting that half yearly compounding changes both the rate and the number of periods. Careful handling of the broken period for yearly compounding is also essential.

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