Question

The interest is supposed compound unless the contrary is state.If $25$ years' purchase must be paid for an annuity to continue $n$ years, and $30$ years' purchase for an annuity to continue $2n$ years, find the rate per cent.

Answer 1

Given: $25years$ and $30years$ are the present values of the annuity for $n,2n$ years respectively.

Let $r$ be the rate of interest.

The, $25=\frac{1-r^{-n}}{r-1}...\left(i\right)$

and $30=\frac{1-r^{-2n}}{r-1}...\left(ii\right)$

Divide $\left(ii\right)$ by $\left(i\right)$, we get

$\frac{30}{25}=\frac{1-r^{-2n}}{1-r^{-n}}$

$\Rightarrow \frac{6}{5}=\frac{(1-r^{-n})(1+r^{-n})}{1-r^{-n}}$

$\Rightarrow \frac{6}{5}=1+r^{-n}$

$\Rightarrow r^{-n}=\frac{6}{5}-1$

$\Rightarrow \frac{1}{r^n}=\frac{1}{5}$

$\Rightarrow r=5^{\frac{1}{n}}\%$