The difference between compound interest and simple interest on a certain sum for 2 years at 8% per annum, when interest is compounded annually, is Rs. 16. If the interest were instead compounded half-yearly, what would be the approximate difference between compound interest and simple interest for the same 2 years?

Introduction / Context:
This question contrasts simple interest (SI) with compound interest (CI) at the same annual rate, but under different compounding frequencies. You are told the difference between CI and SI when interest is compounded annually and asked to find the difference if the interest were compounded half-yearly instead. This requires first determining the principal using the annual-compounding difference, and then recomputing the CI under half-yearly compounding for the same time period and rate.

Given Data / Assumptions:

• Let the principal be P (unknown initially).

• Rate of interest (R) = 8% per annum.

• Time (T) = 2 years.

• When interest is compounded annually, CI − SI for 2 years = Rs. 16.

• We must find the new difference CI − SI if interest is compounded half-yearly over the same 2 years.

Concept / Approach:
First, use the formula for CI and SI with annual compounding to find P from the given difference of Rs. 16. For 2 years at 8% per annum, simple interest is P * R * T / 100, and compound interest is P * (1 + R/100)^2 − P. The difference CI − SI depends only on P and R. Once we compute P, we then calculate: (1) simple interest for 2 years at 8% per annum, and (2) compound interest for 2 years when compounded half-yearly, i.e., at 4% every half-year for 4 periods. The difference between these new CI and SI values is the required answer.

Step-by-Step Solution:
Step 1: Under annual compounding, SI for 2 years = P * 8/100 * 2 = 0.16P. Step 2: CI for 2 years (annual) = P * (1 + 8/100)^2 − P = P * (1.08)^2 − P. Step 3: (1.08)^2 = 1.1664, so CI = P * 1.1664 − P = 0.1664P. Step 4: Given CI − SI = 0.1664P − 0.16P = 0.0064P = Rs. 16. Step 5: Therefore, P = 16 / 0.0064 = Rs. 2500. Step 6: Compute SI for 2 years at 8% per annum on P = 2500. Step 7: SI = 2500 * 8/100 * 2 = 2500 * 0.16 = Rs. 400. Step 8: Now consider half-yearly compounding. Rate per half-year = 8/2 = 4%. Step 9: Number of half-year periods in 2 years = 4. Step 10: CI (half-yearly) amount = 2500 * (1 + 4/100)^4 = 2500 * (1.04)^4. Step 11: (1.04)^2 = 1.0816 and (1.04)^4 = (1.0816)^2 ≈ 1.16985856. Step 12: Amount with CI = 2500 * 1.16985856 ≈ Rs. 2924.65. Step 13: CI (half-yearly) = 2924.65 − 2500 ≈ Rs. 424.65. Step 14: New difference CI − SI = 424.65 − 400 ≈ Rs. 24.65, which is very close to Rs. 24.64.

Verification / Alternative check:
We can compute a more precise value directly: CI (half-yearly) = P * [(1.04)^4 − 1]. Since P = 2500 and (1.04)^4 ≈ 1.16985856, CI ≈ 2500 * 0.16985856 ≈ 424.65. The simple interest remains exactly Rs. 400, so CI − SI ≈ 24.65. The nearest given option is Rs. 24.64, which matches the standard rounded value used in such exam questions.

Why Other Options Are Wrong:
• Rs. 21.85: This underestimates the true difference and does not arise from correct calculation of (1.04)^4.
• Rs. 16.00: This is the original difference under annual compounding and does not apply to half-yearly compounding.
• Rs. 16.80: This is still significantly lower than the accurately computed difference of about Rs. 24.65.

Common Pitfalls:
Errors often occur when students forget to first determine the principal from the initial difference of Rs. 16. Others may incorrectly treat half-yearly compounding as simply doubling the annual rate or using the wrong number of periods. Carefully applying the compound interest formula with the adjusted rate and number of periods, and using the same principal for both SI and CI comparisons, is critical to obtaining the correct answer.

Final Answer:
If the interest is compounded half-yearly instead of annually, the approximate difference between compound interest and simple interest over 2 years is Rs. 24.64.