Question

Find the rate at which a sum of money will become four times the original amount in 2 years, if the interest is compounded half-yearly.

Answer 1

Let Principal (P) = Rs. 100
then Amount (A) = Rs. 400
Period (n) = 2 years or 4 half years.
Let R be the rate $\%$ half-yearly, then
$\frac{A}{P}={(1+\frac{R}{100})}^n\Rightarrow \frac{400}{100}={(1+\frac{R}{100})}^4$
$\Rightarrow {(1+\frac{R}{100})}^4=\frac{4}{1}$
$\Rightarrow {\left[{(1+\frac{R}{100})}^2\right]}^2=(2{)}^2$
$\Rightarrow {(1+\frac{R}{100})}^2=2\Rightarrow (1+\frac{R}{100})=\sqrt{2}$
$\Rightarrow 1+\frac{R}{100}=1.4142$
$\Rightarrow \frac{R}{100}=1.4142-1.0000$
$\Rightarrow \frac{R}{100}=0.4142\Rightarrow R=0.4142\times 100$
$\Rightarrow R=41.42$
$\therefore$ Rate $\%=41.42\%$ half yearly and $82.84\%$ p.a.