In compound interest, what is the difference in rupees between the compound interest on Rs. 1000 for 1 year at 10% per annum when the interest is compounded once yearly and when it is compounded half-yearly?

Introduction / Context:
This question tests the basic concept of compound interest and how the compounding frequency (yearly versus half-yearly) affects the final interest amount. Even though the rate of interest per annum is the same, changing the number of compounding periods in a year slightly changes the effective interest earned, which is important in banking and finance calculations.

Given Data / Assumptions:

• Principal P = Rs. 1000
• Nominal rate of interest r = 10% per annum
• Time T = 1 year
• Case 1: Compounded once yearly
• Case 2: Compounded half-yearly, that is twice in a year

Concept / Approach:
The compound amount A is calculated using the formula:
A = P * (1 + r / (100 * n))^(n * T)where n is the number of compounding periods per year. The compound interest CI is then:
CI = A - PWe compute CI for yearly compounding and half-yearly compounding, and then find the difference between the two interest amounts.

Step-by-Step Solution:
Step 1: For yearly compounding, n = 1.A1 = 1000 * (1 + 10 / 100)^1 = 1000 * 1.10 = 1100CI1 = A1 - P = 1100 - 1000 = Rs. 100Step 2: For half-yearly compounding, n = 2 and rate per half-year = 10 / 2 = 5%.A2 = 1000 * (1 + 5 / 100)^(2 * 1) = 1000 * (1.05)^2 = 1000 * 1.1025 = 1102.5CI2 = A2 - P = 1102.5 - 1000 = Rs. 102.5Step 3: Difference between the compound interests:Difference = CI2 - CI1 = 102.5 - 100 = Rs. 2.5

Verification / Alternative check:
Notice that for a single year, the difference between yearly and half-yearly compounding will be quite small because there are only two compounding periods. As a quick check, we can also compare the amounts directly: 1102.5 - 1100 = 2.5, which confirms that the difference in compound interest is Rs. 2.5.

Why Other Options Are Wrong:
1.5: This would arise from an incorrect calculation or using an approximate rate instead of the exact compounding formula.0.5: This is too small and indicates a misunderstanding of compounding frequency.3.5: This overestimates the effect of compounding for only one year.2.0: This is a rounded guess rather than the exact mathematical result.

Common Pitfalls:
Students often forget to adjust the rate when the compounding period changes, for example using 10% instead of 5% for each half-year. Another mistake is to subtract the principals incorrectly or compare the wrong quantities (amounts instead of interests, or vice versa). Careful substitution into the compound interest formula avoids these issues.

Final Answer:
The difference between the compound interest when interest is compounded yearly and half-yearly on Rs. 1000 at 10% per annum for 1 year is Rs. 2.5.