Henry Ernest Dudeney/Modern Puzzles/34 - The Moving Stairway/Solution

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Modern Puzzles by Henry Ernest Dudeney: $34$

The Moving Stairway
On one of the moving stairways on the London Tube Railway I find that if I walk down $26$ steps I require $30$ seconds to get to the bottom,
but if I take $34$ steps I require only $18$ seconds to reach the bottom.
What is the height of the stairway in steps?


Solution

$46$ steps.


Proof

Let $S$ be the number of steps on the stairway.

Let $v$ be the speed of descent of the stairway in steps per second.

We are given that:

\(\ds S - 26\) \(=\) \(\ds 30 v\)
\(\ds S - 34\) \(=\) \(\ds 18 v\)
\(\ds \leadsto \ \ \) \(\ds 3 \paren {S - 26}\) \(=\) \(\ds 90 v\)
\(\ds 5 \paren {S - 34}\) \(=\) \(\ds 90 v\)
\(\ds \leadsto \ \ \) \(\ds 3 S - 78\) \(=\) \(\ds 5 S - 170\)
\(\ds \leadsto \ \ \) \(\ds S\) \(=\) \(\ds 46\)

$\blacksquare$


Sources

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