$I=I_0\left(1-{\text{e}}^{-\frac{t}{\tau }}\right)$

$\frac{9}{10}{I}_0=I_0\left(1-{\text{e}}^{-\frac{t_1}{\tau }}\right)$

$\Rightarrow {\text{e}}^{-\frac{t_1}{\tau }}=\frac{1}{10}$

$\Rightarrow {\text{e}}^{\frac{t_1}{\tau }}=10$

$\Rightarrow t_1=\tau ln10$    ............(1)

and $\frac{99}{100}{I}_0=I_0\left(1-{\text{e}}^{-\frac{t_2}{\tau }}\right)$

$\Rightarrow {\text{e}}^{-\frac{t_2}{\tau }}=\frac{1}{100}$

$\Rightarrow t_2=\tau ln100=2\tau ln10$   ..............(2)

From equations (1) and (2), we have
$\Rightarrow \frac{t_1}{t_2}=\frac{1}{2}$