At a metro station, a girl walks up a stationary escalator in time $t_1$. If she remains stationary on the escalator, then the escalator take her up in time $t_2$. The time taken by her to walk up on the moving escalator will be

Let l be the length of the escalator
$v_{ge}$ = velocity of girl relative to the escalator = v
$v_{egr}$= velocity of escalator relative to the ground = u and
$v_{ggr}$= velocity of the girl relative to the ground
       = $v_{ggr}$+ $v_{egr}$ = v + u
Now, $t_1$ = l/v; $t_2$ = l/u
 

$\therefore t_3=\frac{l}{v_{ggr}}=\frac{l}{v+u}=\frac{l}{{\displaystyle \frac{l}{t_1}}+{\displaystyle \frac{l}{t_2}}}\Rightarrow t_3=\frac{t_1{t}_2}{t_1+t_2}$