A man borrows ₹1,024 at 50% per annum, compounded half-yearly, for 2 years.…

Concept

For compound interest, the amount after t years is A = P(1 + r/m)mt, where P is the principal, r is the annual rate in decimal form, and m is the number of compounding periods per year.

With half-yearly compounding, m = 2, so the annual rate is divided by 2 and the number of years is multiplied by 2.

Application

1. Convert the annual rate: r = 50/100 = 0.50. Therefore, the rate per half-year is 0.50/2 = 0.25.

2. Count the compounding periods: n = 2 × 2 = 4 half-years.

3. Substitute in the formula: A = 1,024(1 + 0.25)4 = 1,024(5/4)4.

4. Evaluate: A = 1,024 × 625/256 = 4 × 625 = ₹2,500.

Cross-check

The growth factor for one year is (5/4)2 = 25/16. Over two years it is (25/16)2 = 625/256, and ₹1,024 × 625/256 = ₹2,500, confirming the calculation.

Result

The amount to be repaid at the end of 2 years is ₹2,500.