For a certain principal sum of money, the difference between the compound interest (compounded half-yearly) for 1 year and the simple interest for the same 1 year period, when the rate of interest is 8% per annum, is Rs. 64. What is the principal amount (in rupees)?

Introduction / Context:
In many bank and aptitude examinations, you are often asked to compare simple interest and compound interest for the same principal, rate, and time period. This question focuses on the extra interest earned due to compounding when interest is calculated half-yearly instead of using simple interest for 1 year at the same annual rate of 8% per annum. By understanding how to compute both simple and compound interest, and then taking their difference, we can easily back-calculate the original principal from the given difference of Rs. 64.

Given Data / Assumptions:

• The rate of interest is 8% per annum.
• Time period is 1 year.
• Compound interest is calculated half-yearly (two compounding periods in a year).
• The difference between compound interest and simple interest for this 1 year is Rs. 64.
• We assume standard banking calculation and ignore any taxes or additional charges.

Concept / Approach:
For simple interest, the formula is: SI = P * R * T / 100, where P is principal, R is rate per annum, and T is time in years. For compound interest with half-yearly compounding, we use the formula: Amount = P * (1 + r/100) ^ n, where r is the rate per compounding period and n is the number of periods. Compound interest is CI = Amount - P. The difference (CI - SI) is given as Rs. 64, so we set up an equation in terms of P and solve for the principal.

Step-by-Step Solution:
Step 1: Let the principal be P rupees. Step 2: Simple interest for 1 year at 8% per annum is SI = P * 8 * 1 / 100 = 0.08 * P. Step 3: For half-yearly compounding, rate per half-year = 8% / 2 = 4%. Step 4: Number of half-yearly periods in 1 year = 2. Step 5: Amount under compound interest after 1 year = P * (1 + 4/100) ^ 2 = P * (1.04) ^ 2 = P * 1.0816. Step 6: Compound interest for 1 year = CI = P * 1.0816 - P = P * 0.0816. Step 7: Difference between CI and SI = P * 0.0816 - P * 0.08 = P * (0.0016). Step 8: This difference is given as Rs. 64, so P * 0.0016 = 64. Step 9: Therefore P = 64 / 0.0016 = 40000. Step 10: Hence, the required principal amount is Rs. 40000.

Verification / Alternative check:
We can quickly verify the answer by plugging P = 40000 back into the formulas. Simple interest in 1 year at 8% is 40000 * 8 / 100 = 3200. Under half-yearly compounding, the amount is 40000 * (1.04) ^ 2 = 40000 * 1.0816 = 43264, so the compound interest is 43264 - 40000 = 3264. The difference between compound interest and simple interest is 3264 - 3200 = 64, which matches the given condition. This confirms that our principal is correct.

Why Other Options Are Wrong:
If the principal were Rs. 42000, the difference between CI and SI would be 42000 * 0.0016 = 67.2, not 64. For Rs. 44000, the difference would be 70.4, and for Rs. 44800, it would be 71.68. None of these match the required difference of Rs. 64, so these options are incorrect.

Common Pitfalls:
A common mistake is to use 8% directly as the half-yearly rate instead of dividing by 2. Another frequent error is to forget that there are two compounding periods in 1 year when compounding is half-yearly. Some students also subtract the rate percentages directly rather than computing the actual rupee difference between compound and simple interest. Carefully distinguishing between annual rate and period rate, and between amount and interest, prevents these errors.

Final Answer:
Thus, the required principal sum of money is Rs. 40000.