When a sum of money is put on loan for one and a half years at a rate of 20%,…

Concept: For compound interest at a periodic rate r, the amount after n periods is A = P(1 + r)n, so the interest earned is A - P. When a scheme compounds annually but the total time includes a fractional year, the leftover fraction is settled by applying the same periodic rate as simple interest on the amount already accumulated. Comparing two compounding schemes over the same nominal rate and time means finding this amount separately under each scheme and subtracting the two interests.

Applying it to this sum:

1. Annual compounding for 1.5 years: after the first year at 20%, amount = 1.2P. The remaining half-year is settled as simple interest at 10% (half of 20%) on 1.2P, giving amount = 1.2P x 1.1 = 1.32P, so interest = 0.32P.

2. Half-yearly compounding for 1.5 years = 3 half-year periods at 10% each: amount = P(1.1)3 = 1.331P, so interest = 0.331P.

3. Difference between the two interests = 0.331P - 0.32P = 0.011P.

4. This difference equals the ₹264 given in the question: 0.011P = 264, so P = 264 / 0.011 = ₹24,000.

Cross-check: At P = ₹24,000, the annual-compounding interest is 0.32 x 24,000 = ₹7,680 and the half-yearly-compounding interest is 0.331 x 24,000 = ₹7,944. Their difference is ₹7,944 - ₹7,680 = ₹264, matching the value given in the question.

Sum = ₹24,000.