A sum of money lent out at C.I. at a certain rate per annum becomes three times of itself in 10 years. Find in how many years will the money become twenty-seven times of itself at the same rate of interest p.a.

Let rate of interest be r% and sum of money be ₹ P.

By formula,

A = $P\Big(1 + \dfrac{r}{100}\Big)^n$

Given,

₹ P becomes three times of itself in 10 years.

$\therefore 3P = P\Big(1 + \dfrac{r}{100}\Big)^{10} \\[1em] \Rightarrow \dfrac{3P}{P} = \Big(1 + \dfrac{r}{100}\Big)^{10} \\[1em] \Rightarrow 3 = \Big(1 + \dfrac{r}{100}\Big)^{10} …..(1)$

Let in n years money will become 27 times.

$\Rightarrow P\Big(1 + \dfrac{r}{100}\Big)^n = 27P \\[1em] \Rightarrow \Big(1 + \dfrac{r}{100}\Big)^n = \dfrac{27P}{P} \\[1em] \Rightarrow \Big(1 + \dfrac{r}{100}\Big)^n = 27 \\[1em] \Rightarrow \Big(1 + \dfrac{r}{100}\Big)^n = 3^3 \\[1em]$

From equation (1)

$\Rightarrow \Big(1 + \dfrac{r}{100}\Big)^n = \Big[\Big(1 + \dfrac{r}{100}\Big)^{10}\Big]^3 \\[1em] \Rightarrow \Big(1 + \dfrac{r}{100}\Big)^n =\Big(1 + \dfrac{r}{100}\Big)^{30} \\[1em] \Rightarrow n = 30 \text{ years}.$

Hence, in 30 years money will becomes 27 times of itself.