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?​

Correct option is B

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:

A = $$P \left(1 + \frac{r}{n\times 100}\right)^{nt}$$​​

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:

​$$15,625 = 8,000 \left(1 + \frac{r}{2\times 100}\right)^{3} \\ \ \\\frac{15,625}{8,000} = \left(1 + \frac{r}{200}\right)^{3}\\ \ \\\frac{125}{64} = \left(1 + \frac{r}{200}\right)^{3}\\ \ \\\left(1 + \frac{r}{200}\right) = \sqrt[3]{\frac{125}{64}}\\ \ \\\left(1 + \frac{r}{200}\right) = \frac54 \\ \ \\\frac{r}{200} = \frac{5}4 - 1 \\ \ \\\frac r{200} = \frac 1 4 \\ \ \\r = 50\%$$​​

Thus, rate of interest is 50%