A sum of money is invested at 20% per annum compound interest. Over a period of 2 years, it would fetch Rs. 723 more if interest were compounded half-yearly instead of annually. What is the principal sum invested (in rupees)?

Introduction:
This problem compares two different compounding frequencies for the same nominal rate and time period. The sum is invested at 20% per annum, and we are told how much extra amount is earned when the interest is compounded half yearly instead of annually over 2 years. Using this additional amount, we must determine the original principal.

Given Data / Assumptions:

• Nominal annual rate R = 20% per annum.
• Time period T = 2 years.
• Case 1: Compounding annually at 20%.
• Case 2: Compounding half yearly at 10% per half year (since 20% per year divided by 2).
• Difference in amounts after 2 years between the two cases = Rs. 723.
• Principal P is the same in both cases and is unknown.

Concept / Approach:
First, we write the amount in each case as a function of P. For annual compounding over 2 years we use:
A1 = P * (1 + 0.20)^2For half yearly compounding over 2 years, there are 4 half year periods at 10% per period:
A2 = P * (1 + 0.10)^4The problem states that the extra amount earned due to half yearly compounding is A2 - A1 = 723. We can use this to solve for P.

Step-by-Step Solution:
Step 1: Compute the annual compounding factor: (1 + 0.20)^2 = 1.2^2 = 1.44.Step 2: Compute the half yearly compounding factor: (1 + 0.10)^4.Step 3: First square 1.10: (1.10)^2 = 1.21.Step 4: Square again to get the fourth power: (1.10)^4 = (1.21)^2 = 1.4641.Step 5: Write A1 and A2 explicitly: A1 = 1.44 * P and A2 = 1.4641 * P.Step 6: The extra amount earned is A2 - A1 = 1.4641 * P - 1.44 * P = (1.4641 - 1.44) * P.Step 7: Compute the difference in factors: 1.4641 - 1.44 = 0.0241.Step 8: Therefore 0.0241 * P = 723.Step 9: Solve for P: P = 723 / 0.0241.Step 10: Perform the division: P ≈ 30,000.So the principal sum invested is Rs. 30,000.

Verification / Alternative check:
Check the amounts using P = 30,000. Annual compounding: A1 = 30,000 * 1.44 = 43,200. Half yearly compounding: A2 = 30,000 * 1.4641 = 43,923. The difference A2 - A1 = 43,923 - 43,200 = 723. This matches the given extra amount, confirming that P = 30,000 is correct.

Why Other Options Are Wrong:
If P were Rs. 20,000, the extra amount would be 0.0241 * 20,000 = 482, which is too small. For Rs. 7,500 the difference would be far smaller, and for Rs. 15,000 or Rs. 10,000 it would still not reach 723. Only Rs. 30,000 produces the additional 723 when switching from annual to half yearly compounding at 20% per annum over 2 years.

Common Pitfalls:
Many students mistakenly apply simple interest or forget to adjust the rate and number of periods when changing to half yearly compounding. Others subtract the rates directly instead of subtracting the growth factors. Always compute the two amounts using the correct formula and frequency, then subtract the amounts, not the rates. Careful calculation of (1.10)^4 and (1.20)^2 is essential.

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