Question

A sum of money doubles itself in 3 years at compound interest, when the interest is compounded annually. In how many years will the sum amount to 16 times of itself?

Answer 1

The principal amount and the interest rate is same in both cases.
Therefore,
$2P=P{(1+\frac{r}{100})}^n$, here $P$ is the principal, $r$ is the rate of interest and $n$ is the time period.
$\Rightarrow 2={(1+\frac{r}{100})}^3$
$\Rightarrow 1+\frac{r}{100}=2^{\frac{1}{3}}\dots \dots \left(1\right)$

Assume the money becomes 16 times itself in $n$ years.
$\Rightarrow 16P=P{(1+\frac{r}{100})}^n$
$\Rightarrow 1+\frac{r}{100}=2^{\frac{4}{n}}\dots \dots \left(2\right)$
From (1) and (2)
$2^{\frac{1}{3}}=2^{\frac{4}{n}}$
$\Rightarrow \frac{1}{3}=\frac{4}{n}$
$\Rightarrow n=12$ years