A certain sum of money is invested for 2 years at 20% per annum compound interest. If the interest earned is Rs 723 more when interest is compounded half-yearly instead of annually, what is the principal sum?

Introduction / Context:
This question compares compound interest under two different compounding frequencies: annually and half-yearly, for the same nominal rate and time period. The rate is 20% per annum, and the time is 2 years. We are told that the interest when compounding half-yearly exceeds the interest when compounding annually by Rs 723, and we must use this information to find the original principal. This is a classic comparison problem that tests understanding of effective rates under different compounding intervals.

Given Data / Assumptions:

• Principal = P rupees (unknown).
• Nominal rate r = 20% per annum.
• Time t = 2 years.
• Case 1: Compounded annually at 20%.
• Case 2: Compounded half-yearly at 20% per annum, i.e., 10% per half-year.
• Interest in half-yearly case is Rs 723 more than interest in annual case.

Concept / Approach:
When compounding annually, the amount after 2 years is A1 = P * (1 + 0.20)^2. When compounding half-yearly, the rate per half-year is 20% / 2 = 10% and there are 2 * 2 = 4 compounding periods, so A2 = P * (1 + 0.10)^4. The corresponding interests are I1 = A1 - P and I2 = A2 - P. The question tells us that I2 - I1 = 723. We can factor P out of both expressions and set up an equation in P using the numerical values of (1.10)^4 and (1.20)^2.

Step-by-Step Solution:
Amount with annual compounding: A1 = P * (1.20)^2 = P * 1.44. So interest I1 = A1 - P = P * 1.44 - P = 0.44P. Amount with half-yearly compounding: A2 = P * (1.10)^4. Compute (1.10)^2 = 1.21 and (1.10)^4 = (1.21)^2 = 1.4641. So interest I2 = A2 - P = P * 1.4641 - P = 0.4641P. Difference in interest: I2 - I1 = 0.4641P - 0.44P = 0.0241P. Given I2 - I1 = 723, so 0.0241P = 723. Thus, P = 723 / 0.0241 = 30,000. Therefore, the principal is Rs 30,000.

Verification / Alternative check:
Take P = 30,000. Annual compounding for 2 years: A1 = 30,000 * 1.44 = 43,200, so I1 = 43,200 - 30,000 = 13,200. Half-yearly compounding: A2 = 30,000 * 1.4641 = 43,923 (using 1.4641), so I2 = 43,923 - 30,000 = 13,923. Difference I2 - I1 = 13,923 - 13,200 = 723, which matches the condition perfectly. This confirms that our calculated principal is correct.

Why Other Options Are Wrong:
If P = 20,000, the difference would be 20,000 * 0.0241 = 482, not 723. For P = 15,000, the difference drops further to around 361.5. For P = 25,000 or P = 40,000, the difference scales to about 602.5 or 964 respectively. None of these equal exactly Rs 723, so those options cannot be correct.

Common Pitfalls:
A common error is to mistakenly use simple interest for one of the cases or to forget that half-yearly compounding changes both the rate and the number of periods. Another pitfall is incorrect rounding of the factor 1.4641, which can slightly disturb the final difference. In exams, where options are well separated, using 1.4641 is accurate enough. Always set up the algebraic equation carefully as I2 - I1 = 723 and avoid mixing up the order of subtraction.

Final Answer:
The principal sum invested is Rs 30,000.