A sum of money is lent at compound interest for 2 years at the rate of 20% per annum. If the interest were payable half yearly instead of annually, the amount of interest earned over 2 years would be Rs 482 more. What is the principal (original sum of money)?

Introduction:
This question examines your understanding of how compounding frequency affects the amount of compound interest. The nominal rate remains 20% per annum, but the interest may be compounded annually or semi annually, creating different accumulated amounts and hence a difference in interest earned.

Given Data / Assumptions:

• Annual rate of interest r = 20% per annum.
• Time period = 2 years.
• Case 1: Interest compounded annually.
• Case 2: Interest compounded half yearly (every 6 months).
• Difference in interest between Case 2 and Case 1 over 2 years = Rs 482.
• We must find the principal P.

Concept / Approach:
For compound interest, the amount depends on both the nominal rate and the compounding frequency. With half yearly compounding, the rate per period is half the annual rate and the number of periods doubles. We compute the interest expressions in both cases, form an equation for the given difference and solve for the principal amount P.

Step-by-Step Solution:
Case 1 (annual compounding): A1 = P * (1 + 20/100)^2 = P * (1.2)^2 = P * 1.44Interest1 = A1 - P = 0.44PCase 2 (half yearly compounding): rate per half year = 20/2 = 10%Number of half year periods in 2 years = 4A2 = P * (1 + 10/100)^4 = P * (1.1)^4(1.1)^2 = 1.21 and (1.1)^4 = 1.21^2 = 1.4641Interest2 = A2 - P = 0.4641PDifference in interest: Interest2 - Interest1 = 0.4641P - 0.44P = 0.0241PGiven 0.0241P = 482, so P = 482 / 0.0241 = Rs 20000

Verification / Alternative Check:
Using P = 20000, Interest1 = 0.44 * 20000 = Rs 8800. Interest2 = 0.4641 * 20000 = Rs 9282. Difference = 9282 - 8800 = Rs 482, which matches the given data, confirming that the principal is correct.

Why Other Options Are Wrong:
Rs 10000: Gives a difference of about Rs 241, only half of what is required.Rs 40000 and Rs 50000: Produce differences that are too large compared to Rs 482.Rs 15000: Also fails to match the required difference when substituted in the formula.

Common Pitfalls:
Students sometimes forget to double the number of compounding periods for half yearly interest or incorrectly use 20% directly as the half yearly rate. Another frequent mistake is equating the amounts instead of the difference in interest as specified in the question.

Final Answer:
The principal (original sum of money) is Rs 20000.