Henry Ernest Dudeney/Puzzles and Curious Problems/68 - The Moving Staircase/Solution
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Puzzles and Curious Problems by Henry Ernest Dudeney: $68$
- The Moving Staircase
- "I counted $50$ steps that I made in going down the moving staircase," said Walker.
- "I counted $75$ steps," said Trotman; "but I was walking down three times as quickly as you."
- If the staircase were stopped, how many steps would be visible?
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:
- $n = 100$
$\blacksquare$
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $68$. -- The Moving Staircase
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $82$. The Escalator