Preeti reached the metro station and found that the escalator was...

Let $d$ be the displacement of the escalator.

Velocity of girl w.r.t. elevator $= \dfrac{d}{{t}_{1}} = {v}_{ge}$

Velocity of elevator w.r.t ground ${v}_{eG} = \dfrac{d}{{t}_{2}}$

Then velocity of girl w.r.t ground

${\overset{\to}{v}}_{gG} = {\overset{\to}{v}}_{ge} + {\overset{\to}{v}}_{eG}$

i.e., ${V}_{gG} = {V}_{ge} + {V}_{eG}$

$\dfrac{d}{t} = \dfrac{d}{{t}_{1}} + \dfrac{d}{{t}_{2}}$

$\dfrac{1}{t} = \dfrac{1}{{t}_{1}} + \dfrac{1}{{t}_{2}}$

$t = \dfrac{{t}_{1}{t}_{2}}{\left({t}_{1} + {t}_{2}\right)}$