Find the compound interest on Rs 10000 in 2 years at 4% per annum, when the interest is compounded half yearly (that is, every six months).

Introduction:
This problem tests your understanding of compound interest when interest is compounded more frequently than once a year. Here, interest is compounded half yearly, which means the rate per period and the number of periods both change from the standard annual case.

Given Data / Assumptions:

• Principal (P) = Rs 10000.
• Rate of interest (r) = 4% per annum.
• Time period = 2 years.
• Interest is compounded every 6 months (half yearly).
• We are asked to find the compound interest (CI), not the total amount.

Concept / Approach:
When interest is compounded half yearly, the annual rate is split into two, and the number of periods doubles. Thus, the rate per half year is r/2 and the number of compounding periods in t years is 2t. We use the amount formula A = P * (1 + r/(2*100))^(2t) and then compute CI as A - P.

Step-by-Step Solution:
Rate per half year = 4% / 2 = 2% = 0.02Number of half yearly periods in 2 years = 2 * 2 = 4Amount after 4 periods: A = 10000 * (1 + 0.02)^4(1.02)^2 = 1.0404 and (1.02)^4 = (1.0404)^2 ≈ 1.08243216So A ≈ 10000 * 1.08243216 = Rs 10824.32 (rounded)Compound interest CI = A - P = 10824.32 - 10000 = Rs 824.32

Verification / Alternative Check:
You may compute step by step: after first half year, amount = 10000 * 1.02 = 10200. After second half year, multiply again by 1.02 and continue for four multiplications. The final amount will again be approximately Rs 10824.32, giving CI ≈ Rs 824.32.

Why Other Options Are Wrong:
Rs 524.32 and Rs 624.32: These underestimate the interest by ignoring some of the compounding effect.Rs 724.32: Close but still smaller than the correct calculated compound interest.Rs 432.32: This is far too low and might come from using simple interest or treating the time as only 1 year.

Common Pitfalls:
Students often forget to double the number of periods or divide the interest rate correctly when switching to half yearly compounding. Some also mistakenly apply the simple interest formula, which does not capture the growth on interest itself. Careful handling of the rate per period and the number of periods avoids these errors.

Final Answer:
The compound interest on Rs 10000 in 2 years at 4% per annum, compounded half yearly, is Rs 824.32.