Question

if a sum of money Doubles itself in 6 years when compounded annually in what time will it become 8 times of itself at the same rate of interest​

Answer 1

$\fontsize{18}{10}{\textup{\textbf{The required time is 18 years}}.}$

Step-by-step explanation:  Given that a sum of money doubles itself in 6 years when compounded annually.

We are to find the time in which the same amount of money becomes 8 times of itself at the same rate of interest.

Let P, r and n represents the principal amount, rate of interest and the number of years.

Then, according to the given information, we have

$2P=P\left(1+\dfrac{r}{100}\right)^n\\\\\\\Rightarrow 2P=P\left(1+\dfrac{r}{100}\right)^6\\\\\\\Rightarrow \left(1+\dfrac{r}{100}\right)=2^\frac{1}{6}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

If the principal amount becomes 8 time of itself in n years with the same rate of interest, then

$8P=P\left(1+\dfrac{r}{100}\right)^n\\\\\\\Rightarrow 8=(2^\frac{1}{6})^n\\\\\Rightarrow 2^3=2^\frac{n}{6}\\\\\Rightarrow \dfrac{n}{6}=3\\\\\Rightarrow n=18.$

Thus, the required time is 18 years.