The difference between compound interest (compounded every 6 months) and simple interest on a certain sum after 2 years is ₹248.10. If the annual rate of interest is 10%, what is the principal sum invested?

Introduction:
This question tests understanding of the difference between compound interest (CI) and simple interest (SI) when compounding is done semi-annually. For the same principal, rate, and time, CI is slightly higher than SI due to interest-on-interest. The given difference lets us solve for the principal by expressing both CI and SI in terms of P and isolating P using the computed factor.

Given Data / Assumptions:

• Difference (CI - SI) after 2 years = ₹248.10
• Annual rate = 10%
• Compounding every 6 months (semi-annually)
• Time = 2 years = 4 half-year periods
• Semi-annual rate = 10%/2 = 5% per half-year

Concept / Approach:
Compute SI for 2 years at 10%: SI = P * 10 * 2 / 100 = 0.2P. Compute CI factor for semi-annual compounding: amount factor = (1 + 5/100)^4 = 1.05^4. Then CI = P * (1.05^4 - 1). Difference is P * [(1.05^4 - 1) - 0.2]. Set this equal to ₹248.10 and solve for P.

Step-by-Step Solution:
SI for 2 years at 10%: SI = (P * 10 * 2) / 100 = 0.2P CI factor (semi-annual): (1.05)^4 (1.05)^2 = 1.1025, so (1.05)^4 = 1.1025^2 = 1.21550625 CI = P * (1.21550625 - 1) = P * 0.21550625 Difference CI - SI = P * (0.21550625 - 0.2) = P * 0.01550625 Given P * 0.01550625 = 248.10 P = 248.10 / 0.01550625 = 16000

Verification / Alternative check:
For P = 16000: SI = 0.2P = 3200. CI = 16000 * 0.21550625 = 3448.10. Difference = 3448.10 - 3200 = ₹248.10, matching exactly.

Why Other Options Are Wrong:
₹12,000 and ₹14,000 yield smaller differences than ₹248.10. ₹18,000 and ₹20,000 yield larger differences because the difference scales directly with P for fixed rate and time.

Common Pitfalls:
Using annual compounding instead of semi-annual, forgetting to use 4 half-year periods, or incorrectly computing the compound factor 1.05^4.

Final Answer:
The principal sum is ₹16,000.