The difference between the simple interest and the compound interest on Rs 1,200 for 1 year at 10% per annum, when compound interest is reckoned half-yearly, is equal to how many rupees?

Introduction / Context:
This question compares simple interest (SI) and compound interest (CI) on the same principal for 1 year at a rate of 10% per annum, where the compound interest is calculated half-yearly. Because the time is only 1 year, SI is straightforward, and CI with half-yearly compounding involves two periods of 6 months each. We are asked specifically for the difference CI - SI, which is a standard exam pattern for illustrating the effect of more frequent compounding over a short period.

Given Data / Assumptions:

• Principal P = Rs 1,200.
• Nominal annual rate r = 10% per annum.
• Total time t = 1 year.
• Simple interest uses annual rate 10% for 1 year.
• Compound interest is calculated half-yearly at 5% per half-year.
• We must find the numerical difference between CI and SI.

Concept / Approach:
Simple interest for 1 year is SI = P * r * t / 100. For half-yearly compounding, the rate per period is r/2 and the number of periods is 2 per year. So the amount under CI is A = P * (1 + r/(2 * 100))^2. The compound interest CI is A - P. After we compute SI and CI, their difference CI - SI will give the answer. Because the rate and time are small, we can calculate all numbers exactly without approximation.

Step-by-Step Solution:
Simple interest SI = P * r * t / 100 = 1,200 * 10 * 1 / 100. So SI = 1,200 * 0.10 = Rs 120. For compound interest with half-yearly compounding, rate per half-year = 10% / 2 = 5% = 0.05. Number of half-year periods in 1 year = 2. Amount A under CI = 1,200 * (1.05)^2. Compute (1.05)^2 = 1.1025. So A = 1,200 * 1.1025 = 1,323. Therefore, CI = A - P = 1,323 - 1,200 = Rs 123. Difference CI - SI = 123 - 120 = Rs 3.

Verification / Alternative check:
We can also compute CI stepwise. After the first 6 months, interest is 1,200 * 5/100 = 60, so the amount becomes 1,260. For the next 6 months, interest is 1,260 * 5/100 = 63, making the final amount 1,323. The total CI is 60 + 63 = 123, which is again Rs 3 more than the simple interest of 120. The agreement between both methods confirms the accuracy of the result.

Why Other Options Are Wrong:
Differences like Rs 2 or Rs 4 would result from miscalculating one of the half-year periods or from rounding errors. Rs 5 and Rs 6 are too large for a small principal and a moderate rate over only one year. The exact computation shows that the difference is Rs 3, which matches only one of the given options.

Common Pitfalls:
A common error is to mistakenly apply 10% for each half-year (making it effectively 20% per annum), which greatly overstates CI. Another mistake is to forget to square the factor 1.05 and instead use it only once. Students may also subtract SI from P instead of subtracting SI from CI. Carefully distinguishing the two interest types and tracking each step avoids these mistakes.

Final Answer:
The difference between the compound interest and the simple interest is Rs 3.