What is the difference between the compound interest on Rs 5,000 for 1.5 years at 4% per annum when interest is compounded yearly and when it is compounded half yearly?

Introduction:
This question compares compound interest under two different compounding frequencies for the same principal, rate, and time. It tests understanding of how more frequent compounding (half yearly) slightly increases the total interest compared to yearly compounding for the same nominal annual rate.

Given Data / Assumptions:

• Principal P = Rs 5,000
• Annual nominal rate r = 4% per annum
• Total time t = 1.5 years
• Case 1: Interest compounded yearly
• Case 2: Interest compounded half yearly
• We need the difference CI_half_yearly minus CI_yearly

Concept / Approach:
For yearly compounding with a fractional year, we compound for full years and then apply simple interest for the remaining fractional year on the amount after the full years. For half yearly compounding, we convert the annual rate to a per half year rate and count the number of half year periods. Compound interest is always:
CI = A - P

Step-by-Step Solution:
Step 1: Yearly compounding for 1.5 years.After 1 year: A1 = 5,000 * (1 + 0.04) = 5,200For the next 0.5 year, use simple interest on 5,200 at 4% for half a year.Interest_half = 5,200 * 4 * 0.5 / 100 = 5,200 * 0.02 = Rs 104Final amount with yearly compounding: A_yearly = 5,200 + 104 = Rs 5,304CI_yearly = 5,304 - 5,000 = Rs 304Step 2: Half yearly compounding for 1.5 years.Rate per half year = 4% / 2 = 2% = 0.02Number of half years in 1.5 years = 3A_half = 5,000 * (1.02)^3(1.02)^3 = 1.061208A_half ≈ 5,000 * 1.061208 = Rs 5,306.04CI_half = 5,306.04 - 5,000 = Rs 306.04Step 3: Difference between the two compound interests.Difference = CI_half - CI_yearly = 306.04 - 304 = Rs 2.04

Verification / Alternative check:
The difference must be small because the rate is low and the time is only 1.5 years. A difference of a little more than Rs 2 is reasonable. Any much larger difference would indicate a miscalculation of either compounding method or time periods.

Why Other Options Are Wrong:

• Rs 3.04, Rs 4.04, Rs 5.04: These overstate the benefit of half yearly compounding for such a short period.
• Rs 1.04: This underestimates the impact of the extra compounding.

Common Pitfalls:
Common errors include using 1.5 directly as the exponent for yearly compounding, or forgetting that half yearly compounding means halving the rate and doubling the number of periods. Some also mistakenly compute simple interest in both cases rather than using compounding correctly.

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