A sum of money is lent at compound interest for 2 years at 20% per annum. If the interest were payable half-yearly instead of annually, it would fetch Rs. 482 more. What is the principal sum?

Introduction / Context:
This question compares compound interest under two different compounding frequencies: annually and half-yearly. The interest rate is 20% per annum, and the time is 2 years. You are told that half-yearly compounding yields Rs. 482 more interest than annual compounding. From this information, you must determine the principal sum. Such questions reinforce your understanding of how compounding frequency affects the total amount and how to work backward from a known difference in interest.

Given Data / Assumptions:

• Principal (P) = unknown (to be found).

• Time = 2 years.

• Rate of interest per annum = 20%.

• Case 1: Compounded annually.

• Case 2: Compounded half-yearly (twice a year).

• Additional interest when compounded half-yearly instead of annually = Rs. 482.

Concept / Approach:
We use the compound interest amount formula A = P * (1 + R/100)^n. For annual compounding, rate is 20% per year and n = 2 years. For half-yearly compounding, the rate per half-year is 10% (half of 20%), and the number of periods in 2 years is 4. We compute the amounts in both cases in terms of P, subtract them to find the difference in interest, and equate that difference to Rs. 482. Solving the resulting equation gives the principal P.

Step-by-Step Solution:
Step 1: For annual compounding, amount A1 = P * (1 + 20/100)^2. Step 2: 1 + 20/100 = 1.20, so A1 = P * (1.20)^2 = P * 1.44. Step 3: For half-yearly compounding, rate per half-year = 20/2 = 10%. Step 4: Number of half-year periods in 2 years = 2 * 2 = 4. Step 5: Amount A2 = P * (1 + 10/100)^4 = P * (1.10)^4. Step 6: (1.10)^2 = 1.21 and (1.10)^4 = (1.21)^2 = 1.4641, so A2 = P * 1.4641. Step 7: Difference in amounts (and hence in interest) = A2 − A1 = P * (1.4641 − 1.44). Step 8: 1.4641 − 1.44 = 0.0241, so difference = 0.0241 * P. Step 9: Given that this difference is Rs. 482, we have 0.0241 * P = 482. Step 10: Therefore, P = 482 / 0.0241 = Rs. 20,000.

Verification / Alternative check:
We can briefly check by computing the actual amounts for P = 20,000. A1 = 20,000 * 1.44 = Rs. 28,800. A2 = 20,000 * 1.4641 = Rs. 29,282. The difference A2 − A1 = 29,282 − 28,800 = Rs. 482, exactly matching the given difference. This confirms that the principal sum is indeed Rs. 20,000.

Why Other Options Are Wrong:
• Rs. 10,000: For this principal, the difference in interest would be half of 482, that is, Rs. 241, which does not match the given condition.
• Rs. 40,000: This would double the difference to Rs. 964, again not matching the stated difference of Rs. 482.
• Rs. 50,000: This would make the difference even larger, 0.0241 * 50,000 = Rs. 1205, far from the given figure.

Common Pitfalls:
Common mistakes include using the wrong number of compounding periods for the half-yearly case, or forgetting to convert the annual rate to the half-yearly rate. Some students also incorrectly subtract percentages instead of computing the difference in amounts. Staying disciplined with the formula and properly adjusting rate and number of periods for each compounding frequency prevents these errors.

Final Answer:
The principal sum of money is Rs. 20,000.