Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time ${\mathrm{t}}_1$. On other days, if she remains stationary on the moving 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

Preeti travelled a height h on a stationary escalator in t₁ seconds.
So, velocity of Preeti is,

$v_m=\frac{h}{t_1}.$

Escalator takes her to the same height in t₂ s, so velocity of escalator is,

$v_e=\frac{h}{t_2}$.

When Preeti walks on a moving escalator, time taken will be,

${\mathrm{t}}_3=\frac{\mathrm{h}}{{\mathrm{v}}_{\mathrm{e}}+{\mathrm{v}}_{\mathrm{m}}}$.

Substituting the value of V_m and V_e, we get,

${\mathrm{t}}_3=\frac{\mathrm{h}}{\frac{\mathrm{h}}{{\mathrm{t}}_1}+\frac{\mathrm{h}}{{\mathrm{t}}_2}}=\frac{{\mathrm{t}}_1{\mathrm{t}}_2}{{\mathrm{t}}_1+{\mathrm{t}}_2}.$