At what rate of interest per annum will a sum of ₹8,000 amount to ₹15,625 in 1 year and 6 months, if the interest is compounded half-yearly?​

Explanation

Given: Principal amount (P) = ₹8,000 Amount after 1 year and 6 months (A) = ₹15,625 Time period (T) = 1 year 6 months = 1.5 years The interest is compounded half-yearly. Formula Used: The compound interest formula when interest is compounded more than once a year is: nt A = P (1 + ) ​ r n × 100 ​ Where: A = Amount P = Principal r = Annual interest rate (in decimal) n = Number of times the interest is compounded per year t = Time period in years Solution: Since the interest is compounded half-yearly, n = 2 and t = 1.5 years Substituting the given values into the compound interest formula: 3 625 = 8, 000 (1 + ) r 15, ​ 2 × 100 ​ 15, 625 r 3 = (1 + ) 8, 000 200 ​ ​ 125 r 3 = (1 + ) 64 200 ​ r 125 (1 + )= 3 200 64 ​ ​ ​ r 5 (1 + )= 200 4 ​ ​ r 5 = −1 200 4 ​ ​ r 1 = 200 4 ​ ​ r = 50%​ Thus, rate of interest is 50%