Time to earn a given interest

Statement

In how long does $3000\ \text{€}$ produce $900\ \text{€}$ of interest at $10\%$ compounded annually?

Solution

The accumulated value must reach $3000 + 900 = 3900\ \text{€}$. From $3000\,(1{.}10)^{n} = 3900$ we get $1{.}10^{n} = 1{.}3$. $n = \frac{\ln 1{.}3}{\ln 1{.}10} = \frac{0{.}262364}{0{.}095310} \approx \boxed{2{.}75\ \text{years}}.$