To solve this problem, we need to understand the relationship between Preeti's walking time, the escalator's time, and the combined time when both are moving.$\\$Let's denote:$\\$• $s$ as the total distance of the escalator.$\\$• $v_p$ as Preeti's walking speed.$\\$• $v_e$ as the speed of the escalator.$\\$Given:$\\$• Preeti walks up the stationary escalator in time $t_1,$ so $s = v_p \cdot t_1. \\$• The escalator takes her up in time $t_2$ when she is stationary, so $s = v_e \cdot t_2. \\$We need to find the time $t$ taken by Preeti to walk up on the moving escalator.$\\$When both Preeti and the escalator are moving, their relative speed is $v_p + v_e. \\$The time taken $t$ is given by:$\\ t = \frac{s}{v_p + v_e} \\$Substitute $s = v_p \cdot t_1$ and $s = v_e \cdot t_2$ into the equation:$\\ t = \frac{v_p \cdot t_1}{v_p + v_e} \\$Since $v_p = \frac{s}{t_1}$ and $v_e = \frac{s}{t_2},$ we can write:$\\ t = \frac{\frac{s}{t_1} \cdot t_1}{\frac{s}{t_1} + \frac{s}{t_2}} \\$Simplify the expression:$\\ t = \frac{s}{\frac{s}{t_1} + \frac{s}{t_2}} \\ t = \frac{s}{\frac{s \cdot t_2 + s \cdot t_1}{t_1 \cdot t_2}} \\ t = \frac{s \cdot t_1 \cdot t_2}{s \cdot (t_2 + t_1)} \\ t = \frac{t_1 \cdot t_2}{t_2 + t_1} \\$Therefore, the time taken by Preeti to walk up on the moving escalator is $\frac{t_1 t_2}{t_2 + t_1}. \\$This corresponds to Option 2.