The difference between the simple interest on a certain sum at the rate of 10% per annum for 2 years and the compound interest on the same sum for the same period, when interest is compounded every 6 months, is Rs 124.05. What is the principal sum?

Introduction:
This question compares simple interest and compound interest over the same time span and at the same nominal annual rate, but with compounding done half yearly for the compound interest case. You are required to use both formulas and the given difference to determine the principal amount.

Given Data / Assumptions:

• Rate of interest (r) = 10% per annum.
• Time (t) = 2 years.
• Simple interest is computed annually.
• Compound interest is computed half yearly, so every 6 months.
• Difference (CI − SI) = Rs 124.05.
• We need to find the principal P.

Concept / Approach:
Simple interest for 2 years is given by SI = P * r * t / 100. For compound interest with half yearly compounding, the rate per half year is r/2 and the number of periods in t years is 2t. The compound amount is A = P * (1 + r/(2*100))^(2t), and CI = A − P. We set CI − SI equal to the given difference and solve for P.

Step-by-Step Solution:
Simple interest for 2 years: SI = P * 10 * 2 / 100 = 0.20PHalf yearly rate = 10/2 = 5% = 0.05Number of half year periods in 2 years = 4Compound amount: A = P * (1 + 0.05)^4 = P * (1.05)^4(1.05)^2 = 1.1025, so (1.05)^4 ≈ 1.21550625Thus CI = A − P = P * 1.21550625 − P = 0.21550625PDifference CI − SI = (0.21550625P − 0.20P) = 0.01550625PGiven CI − SI = 124.05, so 0.01550625P = 124.05Therefore P ≈ 124.05 / 0.01550625 ≈ Rs 8000

Verification / Alternative Check:
Using P = 8000, SI = 0.20 * 8000 = Rs 1600. Compound amount A = 8000 * (1.05)^4 ≈ 8000 * 1.21550625 ≈ Rs 9724.05. CI = 9724.05 − 8000 ≈ Rs 1724.05. Difference CI − SI ≈ 1724.05 − 1600 = Rs 124.05, which matches the given difference exactly.

Why Other Options Are Wrong:
Rs 6000 and Rs 12000: Substituting these in the formulas gives differences that are either too small or too large compared to Rs 124.05.Rs 10000: Also fails to produce the exact difference when SI and CI are computed.None of these: Incorrect, because Rs 8000 perfectly satisfies the condition.

Common Pitfalls:
It is common to forget to adjust the rate and number of periods when changing from annual to half yearly compounding. Some students also mistakenly compare the total amount rather than CI and SI themselves. Careful attention to the formulas and to the meaning of the difference helps avoid these errors.

Final Answer:
The required principal sum is Rs 8000.