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

Introduction / Context:
This question compares compound interest with different compounding frequencies: yearly versus half-yearly, for the same nominal annual rate and time period. It tests your understanding of how changing the compounding frequency affects the final amount and, hence, the interest earned.

Given Data / Assumptions:
- Principal P = Rs. 5,000.
- Nominal annual rate r = 4% per annum.
- Time = 1½ years (that is, 1.5 years).
- Case 1: Interest compounded yearly.
- Case 2: Interest compounded half-yearly (every 6 months).
- We must find the difference between the two compound interest amounts.

Concept / Approach:
For yearly compounding over 1.5 years, we consider 1 year of compounding plus 0.5 year of simple interest on the amount after 1 year, because full-year compounding cannot be applied to the fractional period. For half-yearly compounding, we have 3 half-years (1.5 years = 3 × 0.5 years) at half the annual rate (4% / 2 = 2% per half-year). After we compute the amounts in both cases, we subtract their respective principals (which are the same) to get each interest, and then take the difference of these interests.

Step-by-Step Solution:
Step 1: Yearly compounding: For the first full year, amount A1 = 5000 * (1 + 4/100) = 5000 * 1.04 = 5200.Step 2: For the remaining half-year, apply simple interest on 5200 at 4% per annum for 0.5 year.Step 3: Interest for half-year = 5200 * 4/100 * 0.5 = 5200 * 0.02 = 104.Step 4: So, amount with yearly compounding after 1.5 years = 5200 + 104 = 5304, giving C.I._yearly = 5304 - 5000 = Rs. 304.Step 5: Half-yearly compounding: rate per half-year = 4% / 2 = 2%.Step 6: For 1.5 years, we have 3 half-year periods, so amount A_half = 5000 * (1.02)^3.Step 7: Compute (1.02)^3 ≈ 1.061208, so A_half ≈ 5000 * 1.061208 = 5306.04.Step 8: C.I._half-yearly = 5306.04 - 5000 = Rs. 306.04.Step 9: Difference in interests = 306.04 - 304 = Rs. 2.04.

Verification / Alternative check:
Notice that half-yearly compounding should produce slightly more interest than yearly compounding for the same nominal rate; hence the difference should be small and positive. Our result Rs. 2.04 fits that expectation and matches the option given.

Why Other Options Are Wrong:
Rs. 3.04, Rs. 4.04 and Rs. 5.04 come from miscomputing the number of compounding periods or using 4% instead of 2% per half-year. Rs. 1.04 is too small and suggests an arithmetic slip while subtracting. Only Rs. 2.04 correctly reflects the difference in compound interest between yearly and half-yearly compounding for this case.

Common Pitfalls:
Students often forget that 1.5 years with yearly compounding involves one full year plus half-year simple interest, not full compounding for 1.5 years. Another mistake is using 4% per half-year instead of halving the rate. Careful interpretation of the compounding frequency and period length is crucial to solving such problems accurately.

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