Mr. Shah decided to walk down the escalator of a tube station. He found that if he walks down 26 steps, he requires 30 seconds to reach the bottom. However, if he steps down 34 stairs he would only require 18 seconds to get to the bottom. If the time is measured from the moment the top step begins to descend to the time he steps off the last step at the bottom, find out the height of the stairway in steps.

### Concept & Strategy The total number of steps on an escalator is constant. The total steps is equal to the steps walked by the person plus the steps the escalator moves on its own during the time it took the person to reach the bottom. $$ \text{Total Steps} = \text{Steps Walked} + (\text{Escalator Speed} \times \text{Time}) $$ ### Step-by-Step Solution * **Set up the variables:** Let $N$ be the total number of steps, and $e$ be the speed of the escalator in steps per second. * **Formulate Equation 1 (Case 1):** He walks 26 steps in 30 seconds. * $N = 26 + 30e$ * **Formulate Equation 2 (Case 2):** He walks 34 steps in 18 seconds. * $N = 34 + 18e$ * **Equate and Solve:** Since $N$ is the same in both cases: * $26 + 30e = 34 + 18e$ * $30e - 18e = 34 - 26$ * $12e = 8$ * $e = \frac{8}{12} = \frac{2}{3}$ steps per second. * **Calculate Total Steps ($N$):** Substitute $e$ back into the first equation. * $N = 26 + 30 \times \left(\frac{2}{3}\right)$ * $N = 26 + 20 = 46$. ### Exam Strategy & Shortcut Observe the differences directly. Difference in steps walked = $34 - 26 = 8$ steps. Difference in time taken = $30 - 18 = 12$ seconds. This means the escalator covers those 8 steps in 12 seconds, giving an escalator speed of $\frac{8}{12}$ or $\frac{2}{3}$ steps per second. In the first scenario (30 seconds), the escalator contributes $30 \times \left(\frac{2}{3}\right) = 20$ steps. Total steps = $26 (\text{walked}) + 20 (\text{escalator}) = 46$. ### Common Pitfall Assuming that speed equals $\frac{\text{Distance}}{\text{Time}}$ for the person alone (e.g., trying to calculate $26/30$). You must account for the fact that the escalator acts as a moving frame of reference, adding its own speed to the person's descent. ### Final Answer **Therefore, the correct answer is 46.**