Question

If a certain sum becomes double in $3$ years at a certain rate of interest compounded annually then in how many years it will become $16$ times.

Answer 1

Let us assume that the initial sum was $P$ and the rate of interest is $R\%$ per annum.

It is given that in $3$ years the sum doubles,

$\begin{array}{cc}A& =P{\left(1+\frac{R}{100}\right)}^n\\ 2P& =P{\left(1+\frac{R}{100}\right)}^3\\ 2& ={\left(1+\frac{R}{100}\right)}^3\dots \left(i\right)\end{array}$

Let us assume that the sum becomes $16$ times in $t$ years. So,

$\begin{array}{cc}A& =P{\left(1+\frac{R}{100}\right)}^n\\ 16P& =P{\left(1+\frac{R}{100}\right)}^t\\ 16& ={\left(1+\frac{R}{100}\right)}^t\dots \left(ii\right)\end{array}$

Now, if we exponentiate left side of equation $\left(i\right)$ by $4$ the it becomes equal to left side of the equation $\left(ii\right)$. So, from this observation we can conclude that,

$t=3\times 4$

$=12$

So, in a period of $12$ years the amount becomes $16$ times of the original sum.