What is the difference (in rupees) between the compound interests on Rs. 4,000 for 1 year at 12% per annum when interest is compounded yearly and when it is compounded half yearly?

Introduction / Context:
This question compares compound interest earned under two different compounding frequencies, yearly and half yearly, for the same time period, rate, and principal. It illustrates how more frequent compounding slightly increases the interest earned, even when the nominal annual rate is the same.

Given Data / Assumptions:

• Principal P = Rs. 4,000.
• Annual nominal interest rate r = 12% per annum.
• Time period n = 1 year.
• Case 1: Interest compounded yearly.
• Case 2: Interest compounded half yearly (two times per year).
• We must find the difference between the compound interests in the two cases.

Concept / Approach:
For yearly compounding, interest is added once at the end of the year using the full 12% rate. For half yearly compounding, the periodic rate is half of the annual nominal rate, that is 6% per half year, and compounding occurs twice. We compute the amount in each case, subtract the principal to find the interest in each case, and then take the difference between these two interest values.

Step-by-Step Solution:
Step 1: Case 1, yearly compounding. Amount A1 = P * (1 + r) = 4,000 * (1 + 0.12).Step 2: Compute A1 = 4,000 * 1.12 = 4,480. Interest I1 = A1 - P = 4,480 - 4,000 = Rs. 480.Step 3: Case 2, half yearly compounding. Periodic rate i = 0.12 / 2 = 0.06 (6%).Step 4: There are 2 periods in 1 year, so amount A2 = P * (1 + i)^2 = 4,000 * (1.06)^2.Step 5: Compute (1.06)^2 = 1.1236 and A2 = 4,000 * 1.1236 = 4,494.4.Step 6: Interest I2 = A2 - P = 4,494.4 - 4,000 = Rs. 494.4.Step 7: Difference in interests = I2 - I1 = 494.4 - 480 = Rs. 14.4.

Verification / Alternative check:
We can note that the only difference comes from the extra interest earned in the second half of the year because interest from the first half is added to the principal and also earns interest in the second half. The additional amount gained is roughly 12% of the first half year interest, and for these numbers that additional gain works out to about Rs. 14.4. This aligns with the detailed calculation, helping to confirm the answer.

Why Other Options Are Wrong:

• 12.4: This is close but does not match the accurate difference between 494.4 and 480.
• 10.4: This underestimates the effect of the extra compounding over the second half of the year.
• 16.4: This overestimates the extra interest gained from half yearly compounding.

Common Pitfalls:
Some students treat both cases as simple interest and see no difference at all. Others correctly use the compound interest formula for yearly compounding but incorrectly handle the half yearly case, for example by multiplying by 1.12 twice or by 1.06 only once. It is important to divide the annual rate by the number of compounding periods and raise the factor to the appropriate power. Only then can the difference in interest be computed accurately.

Final Answer:
The difference between the compound interests is Rs. 14.4 more when interest is compounded half yearly than when it is compounded yearly.