What is the compound interest for 1 year on a sum of Rs 20000 at a rate of 40% per annum, if the interest is compounded half yearly (every 6 months)?

Introduction / Context:
This question involves a high annual interest rate of 40% with half yearly compounding. Because interest is compounded twice in a year, we must convert the annual rate to a half yearly rate and compute the amount after 1 year. The compound interest is then the difference between this amount and the original principal.

Given Data / Assumptions:

• Principal P = Rs 20000.
• Annual rate R = 40% per annum.
• Compounding frequency = half yearly.
• Total time = 1 year.
• We need the compound interest for the year.

Concept / Approach:
For half yearly compounding, there are two periods in a year and the rate per period is R / 2. The amount after 1 year is:
A = P * (1 + (R / 100) / 2)^2 The compound interest is:
CI = A - P With R = 40, the half yearly rate is 20% per half year.

Step-by-Step Solution:
Half yearly rate = 40 / 2 = 20%. Number of half years in 1 year = 2. Amount factor = (1.20)^2. Compute (1.20)^2 = 1.44. Amount A = 20000 * 1.44 = Rs 28800. Compound interest CI = A - P = 28800 - 20000 = Rs 8800.

Verification / Alternative Check:
We can also verify period by period. After first half year: amount = 20000 * 1.20 = 24000. After second half year: amount = 24000 * 1.20 = 28800. Interest over the year = 28800 - 20000 = Rs 8800, confirming our calculation.

Why Other Options Are Wrong:
8000 would be the simple interest at 40% for 1 year on 20000, ignoring compounding. 8650 and 8750 are intermediate values and not consistent with exact half yearly compounding at 20% per period.

Common Pitfalls:
A typical error is to apply 40% once for the whole year as simple interest. Some candidates incorrectly use 40% per half year instead of 20%, badly overestimating the amount.

Final Answer:
The compound interest for 1 year at 40% per annum compounded half yearly is Rs. 8800.