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

Introduction / Context:
This problem compares two different compounding frequencies for the same principal, time, and annual rate. We must find the difference between the compound interest calculated with yearly compounding and that computed with half yearly compounding over 1 1/2 years at 4 percent per annum. Such questions highlight the effect of more frequent compounding and reinforce the correct handling of mixed time periods.

Given Data / Assumptions:

• Principal P = Rs. 5000.
• Annual rate r = 4 percent.
• Total time = 1 1/2 years (that is, 1.5 years).
• Case 1: interest compounded yearly.
• Case 2: interest compounded half yearly.
• We need the difference between the two compound interest amounts.

Concept / Approach:
For yearly compounding over 1 1/2 years, we treat the first full year with compound interest and the remaining half year with simple interest on the amount after 1 year. For half yearly compounding, we convert the annual rate to a half yearly rate and the time to number of half year periods, then apply the standard compound interest formula. The difference between the resulting interest amounts is then found by subtraction.

Step-by-Step Solution:
Case 1: Compounded yearly. After 1 year: A1 = 5000 * (1 + 4/100) = 5000 * 1.04 = 5200 For the next half year, use simple interest on 5200 for 0.5 year at 4 percent: Interest for 0.5 year = 5200 * 4 * 0.5 / 100 = 104 Amount after 1.5 years in yearly case = 5200 + 104 = 5304 Compound interest in yearly case = 5304 - 5000 = 304 Case 2: Compounded half yearly. Half yearly rate = 4 percent / 2 = 2 percent per half year Number of half years in 1.5 years = 3 Amount A2 = 5000 * (1 + 2/100)^3 = 5000 * (1.02)^3 (1.02)^3 ≈ 1.061208, so A2 ≈ 5000 * 1.061208 = 5306.04 Compound interest in half yearly case = 5306.04 - 5000 = 306.04 Difference between the two interests = 306.04 - 304 = 2.04

Verification / Alternative check:
We can double check by recomputing the second case with more precise intermediate calculations and confirm that the amount remains approximately 5306.04. As long as the yearly case gives 5304 and the half yearly case gives slightly more, the difference must be a small positive value close to 2.04. The fact that the half yearly compounded amount is higher is consistent with the general principle that more frequent compounding at the same nominal rate yields a larger amount.

Why Other Options Are Wrong:
Rs. 1.80 and Rs. 3.18 are different small values but do not correspond to the exact calculations. Rs. 4.15 and Rs. 5.00 are too large for such a modest rate and time frame. The precise difference obtained from the computations is approximately Rs. 2.04, which matches option B exactly.

Common Pitfalls:
Students sometimes misinterpret 1 1/2 years as 2 years or treat both cases identically, forgetting about the half year component in the yearly compounding case. Another frequent error is to use the annual rate directly in the half yearly formula without dividing by 2, or to use the wrong number of compounding periods. Carefully distinguishing between the two compounding frequencies and handling the mixed time period correctly is essential.

Final Answer:
The difference between the compound interests with yearly and half yearly compounding is Rs. 2.04.