How much will a sum of ₹53,000 become after one year if the interest is…

Concept: When compound interest is compounded half-yearly, the given annual rate is halved to get the rate per half-year, and the number of compounding periods for the given time is doubled (2 periods per year). The amount then follows A = P × (1 + r/100)n, where P is the principal, r is the half-yearly rate, and n is the number of half-year periods.

1. The given values are principal P = ₹53,000, annual rate R = 20% per annum, compounding half-yearly, and time t = 1 year.

2. Since compounding is half-yearly, the half-yearly rate is r = R/2 = 20/2 = 10%, and the number of half-year periods in 1 year is n = 2 × 1 = 2.

3. Applying the formula, A = 53,000 × (1 + 10/100)2 = 53,000 × (1.10)2.

4. Since (1.10)2 = 1.21, A = 53,000 × 1.21 = ₹64,130.

Cross-check: Two successive equal percentage increases of 10% each combine into a single equivalent increase of 10 + 10 + (10 × 10)/100 = 21%. Increasing ₹53,000 by 21% directly gives 53,000 + (53,000 × 21/100) = 53,000 + 11,130 = ₹64,130, which matches the amount obtained through the half-yearly compounding steps.

So, the sum of ₹53,000 becomes ₹64,130 after one year under half-yearly compounding at 20% per annum.