A sum of ₹ 12,000 amounts to ₹ 15,972 at a certain rate percent per annum in 1…

Compound interest formula: for a rate compounded k times per year, the amount after t years is Principal multiplied by (1 + rate ÷ (100 × k)) raised to the power (k × t). When the given time span does not divide evenly into whole compounding cycles — for example, annual compounding applied over one and a half years — the standard convention is to compound for each completed year and then apply simple interest, at the same annual rate, for the leftover fraction of the year.

1. Half-yearly compounding over one and a half years covers 3 half-year periods, so Amount = Principal × (1 + rate/200)3.

2. Substituting the given values: 15,972 = 12,000 × (1 + rate/200)3, so (1 + rate/200)3 = 15,972 ÷ 12,000 = 1.331.

3. Since 1.331 = 1.13, comparing the bases gives 1 + rate/200 = 1.1, so rate/200 = 0.1, and the annual rate = 20%.

4. Applying annual compounding for the first full year at 20% per annum: Amount = 12,000 × (1 + 20/100) = 12,000 × 1.2 = ₹14,400.

5. For the remaining half year, simple interest is applied at the same 20% per annum rate on the amount already accumulated: Amount = 14,400 × (1 + 10/100) = 14,400 × 1.1 = ₹15,840.

As an independent check, the combined multiplying factor for one and a half years of annual compounding at 20% per annum is 1.2 × 1.1 = 1.32, and 12,000 × 1.32 = 15,840, which matches the step-by-step result above. This amount is naturally lower than the ₹15,972 obtained with half-yearly compounding at the same nominal rate, since more frequent compounding always produces a larger amount for the same nominal annual rate.