At a metro station, a girl walks up a stationary escalator in time ${\mathrm{t}}_1$. If she remains stationary on the escalator, then the escalator takes  her up in time ${\mathrm{t}}_2.$The time taken by her to walk up on the moving escalator will be

We have to find net velocity of the girl  with respect to the Earth that will be equal to velocity of the girl relative to the escalator plus velocity of escalator. Let displacement be L which is equal to the length of the escalator, then

Velocity of girl ${\mathrm{v}}_{\mathrm{g}}= \frac{\mathrm{L}}{{\mathrm{t}}_1}$

Velocity of escalator ${\mathrm{v}}_{\mathrm{e} }= \frac{\mathrm{L}}{{\mathrm{t}}_2}$

Net velocity of the girl = ${\mathrm{v}}_{\mathrm{g}}+{\mathrm{v}}_{\mathrm{e}} = \frac{\mathrm{L}}{{\mathrm{t}}_1}+\frac{\mathrm{L}}{{\mathrm{t}}_2}$

If t is total time taken in covering distamce L, then

$\frac{\mathrm{L}}{\mathrm{t}}=\frac{\mathrm{L}}{{\mathrm{t}}_1}+\frac{\mathrm{L}}{{\mathrm{t}}_2} \Rightarrow \mathrm{t} = \frac{{\mathrm{t}}_1{\mathrm{t}}_2}{{\mathrm{t}}_1+{\mathrm{t}}_2}$