A sum of money is invested at simple interest. The difference between compound interest (compounded half-yearly) for 1 year and simple interest for the same 1 year at 8% per annum is ₹64. What is the principal amount (in rupees)?

Introduction:
This is the same concept as the earlier CI vs SI difference problem: half-yearly compounding makes CI slightly higher than SI for the same 1-year period. The difference becomes a small fraction of principal. Once you compute that fraction at 8% with two half-year periods, the principal is found by dividing ₹64 by that fraction. This checks compounding frequency handling and precise subtraction between CI and SI.

Given Data / Assumptions:

• CI is compounded half-yearly for 1 year
• SI is for 1 year
• Annual rate = 8%
• CI - SI = ₹64
• Principal = P

Concept / Approach:
Half-yearly rate = 8%/2 = 4%. Number of half-years in 1 year = 2. CI = P * [(1.04)^2 - 1] = 0.0816P. SI for 1 year at 8% = 0.08P. Difference = 0.0016P. Set 0.0016P = 64 and solve for P.

Step-by-Step Solution:
SI = 0.08P CI factor = (1.04)^2 = 1.0816 CI = 0.0816P CI - SI = 0.0816P - 0.08P = 0.0016P 0.0016P = 64 P = 64 / 0.0016 = 40000

Verification / Alternative check:
For P=40000: CI=3264 and SI=3200, difference=64. Matches exactly.

Why Other Options Are Wrong:
All other values produce a difference not equal to ₹64 since the difference scales directly with P.

Common Pitfalls:
Using 8% per half-year, using only one period, or using amount instead of interest when computing CI.

Final Answer:
The principal amount is ₹40,000.