What is the difference, in rupees, between the compound interest on Rs 1000 for 1 year at 10% per annum when interest is compounded yearly and when interest is compounded half yearly?

Introduction / Context:
This problem compares two different compounding frequencies applied to the same principal, rate, and total time. Even though the nominal annual rate is the same in both cases, changing the compounding period from yearly to half yearly slightly changes the amount of compound interest earned. Understanding this difference is useful for evaluating investment products and loans, where the compounding frequency can influence the true return or cost.

Given Data / Assumptions:

• Principal P = Rs 1000
• Nominal annual rate r = 10% per annum
• Total time T = 1 year
• Case 1: Interest compounded yearly (once per year)
• Case 2: Interest compounded half yearly (twice per year)

Concept / Approach:
For annual compounding, the amount after 1 year is A1 = P * (1 + r). The compound interest is A1 - P. For half yearly compounding, the yearly rate is divided by 2 to get the half yearly rate, and the number of periods is multiplied by 2. The amount after T years is A2 = P * (1 + r / 2)^(2T). The difference in compound interest is then (A2 - P) minus (A1 - P), which simplifies to A2 - A1. Since P and r are small, we can compute these values exactly without approximation.

Step-by-Step Solution:
Case 1 (yearly compounding): A1 = P * (1 + r) = 1000 * (1 + 0.10) = 1000 * 1.10 = 1100 CI1 = A1 - P = 1100 - 1000 = Rs 100 Case 2 (half yearly compounding): Half yearly rate = r / 2 = 10% / 2 = 5% = 0.05 Number of half years in 1 year = 2 A2 = P * (1 + 0.05)^2 = 1000 * 1.05^2 = 1000 * 1.1025 = 1102.50 CI2 = A2 - P = 1102.50 - 1000 = Rs 102.50 Difference in compound interest = CI2 - CI1 = 102.50 - 100 = Rs 2.50

Verification / Alternative check:
Another way to check is to compute the difference directly from the amounts: A2 - A1 = 1102.50 - 1100 = Rs 2.50. This is the additional interest earned due to more frequent compounding. Both methods are consistent and confirm that the extra interest is Rs 2.50.

Why Other Options Are Wrong:
A difference of Rs 1.50 or Rs 0.50 would imply much closer amounts for the two compounding schemes and does not match the correctly calculated gap. Rs 3.50 is larger than the true difference and is inconsistent with the computed amounts. Only Rs 2.50 exactly matches the difference between the two correct compound interest values.

Common Pitfalls:
Students sometimes forget to square the term for half yearly compounding and incorrectly use only one period. Another common error is to compare simple interest versus compound interest instead of comparing two compound interest cases. Some also mistakenly subtract the wrong amounts, mixing up principal and amount. Carefully writing the formulas and performing calculations step by step helps to avoid these mistakes.

Final Answer:
The difference between the two compound interests is Rs 2.50 in favor of half yearly compounding.