Find the compound interest on Rs. 10,000 in 2 years at 4% per annum, when the interest is compounded half-yearly (that is, at 2% every half-year).

Introduction / Context:
This problem asks you to calculate compound interest when the interest is compounded half-yearly. The annual rate is 4%, which translates into 2% for each half-year period. With a principal of Rs. 10,000 and a total time of 2 years, you must account for four compounding periods. Such questions are common in examinations and reinforce your ability to apply the compound interest formula with adjusted rates and multiple compounding periods.

Given Data / Assumptions:

• Principal (P) = Rs. 10,000.

• Rate of interest per annum = 4%.

• Interest is compounded half-yearly.

• Time (T) = 2 years.

• Therefore, rate per half-year = 4% / 2 = 2%.

• Number of half-year periods in 2 years = 2 * 2 = 4.

Concept / Approach:
When interest is compounded half-yearly, we adjust the annual rate and the number of compounding periods accordingly. The compound amount formula becomes A = P * (1 + r/100)^n, where r is the rate per half-year and n is the number of half-year periods. After computing the final amount A, we subtract the principal to get the compound interest. This method ensures you properly account for the effect of compounding more frequently than once per year.

Step-by-Step Solution:
Step 1: Determine the rate per compounding period. Step 2: Annual rate = 4%, so half-yearly rate r = 4 / 2 = 2% per half-year. Step 3: Time = 2 years, so number of half-year periods n = 2 * 2 = 4. Step 4: Use the compound amount formula: A = P * (1 + r/100)^n. Step 5: Substitute the values: A = 10,000 * (1 + 2/100)^4 = 10,000 * (1.02)^4. Step 6: Compute (1.02)^2 = 1.0404. Step 7: Then (1.02)^4 = (1.0404)^2 ≈ 1.08243216. Step 8: Amount A = 10,000 * 1.08243216 ≈ Rs. 10,824.32. Step 9: Compound interest CI = A − P = 10,824.32 − 10,000 = Rs. 824.32.

Verification / Alternative check:
We can also compute stepwise. After the first half-year, amount = 10,000 * 1.02 = 10,200. After one year (two half-years), amount = 10,200 * 1.02 ≈ 10,404. After 1.5 years, amount ≈ 10,404 * 1.02 ≈ 10,612.08. After 2 years, amount ≈ 10,612.08 * 1.02 ≈ 10,824.32. The total interest is again 10,824.32 − 10,000 = Rs. 824.32, confirming our earlier calculation.

Why Other Options Are Wrong:
• Rs. 524.32: This significantly underestimates the effect of four compounding periods at 2% each and does not match the computed amount.
• Rs. 624.32: This is closer but still below the true interest, reflecting fewer effective compounding periods than actually occur.
• Rs. 724.32: This is also less than the correct value and arises from approximate or incomplete calculations.

Common Pitfalls:
Some students mistakenly use 4% as the rate per half-year or forget to double the number of periods when compounding half-yearly. Others may incorrectly apply simple interest formulas or perform rounding too early in intermediate steps. To avoid errors, always adjust the rate and number of periods carefully and perform calculations with sufficient precision before rounding at the final step.

Final Answer:
The compound interest on Rs. 10,000 at 4% per annum for 2 years, compounded half-yearly, is Rs. 824.32.