Henry Ernest Dudeney/Puzzles and Curious Problems/68 - The Moving Staircase/Solution

by : $68$

 * The Moving Staircase

Solution

 * $100$ steps.

Proof
Let $n$ be then number of visible steps when the escalator is stopped.

Let $t$ be the unit of time taken for one step to vanish at the bottom.

If you stand still on the escalator, it takes time $n t$ to reach the bottom.

If you take $x$ steps down the escalator, it takes time $\paren {n - x} t$ to reach the bottom.

Trotman takes $75$ steps in $\paren {n - 75} t$.

That is, he takes $3$ steps in $\dfrac {n - 75} {25} t$.

Walker takes $50$ steps in $\paren {n - 50} t$.

That is, he takes $1$ step in $\dfrac {n - 50} {50} t$.

But $3$ steps taken by Trotman take as much time as $1$ step taken by Walker.

Hence:
 * $\dfrac {n - 75} {25} = \dfrac {n - 50} {50}$

which, after algebra, gives:
 * $n = 100$