At what rate of interest per annum will a sum of ₹1,000 amount to ₹1,728 in 1 year and 6 months, if the interest is compounded half-yearly?​

Correct option is D

Given:

Principal P = ₹1,000

Amount A = ₹1,728

Time t = 1.5 years = 3 half-years

Compounding: Half-yearly

Concept Used:

In half-yearly compounding, the annual rate is halved and the number of periods is doubled.

Formula Used:

A = $$P \left(1 + \frac{r}{2 \times 100} \right)^{2t}$$​​

Solution: 

Now, Using formula;

​$$1728 = 1000 \left(1 + \frac{r}{200} \right)^3$$​​

​$$\left(1 + \frac{r}{200} \right)^3 = \frac{1728}{1000}$$​​

​$$1 + \frac{r}{200} = \sqrt[3]{\frac{1728}{1000}}$$​​

​$$\frac{r}{200} = \frac{12}{10}-1 \\ \ \\ r = \frac{2}{10}\times 200$$​​

r = 40%

Thus, rate of interest is 40% per annum