1. ## [solved]"Down the Escalator"

27) Down the Escalator
Recently, while in London, I decided to walk down the escalator of a tube station. I did some quick calculation in my mind, found that if I walk down twenty-six steps, I require thirty seconds to reach the bottom. However, if I am able to step down thirty-four stairs I would only require eighteen seconds to get to the bottom.
If the time is measured from the moment the top step begins to descend to the time I step off the last step at the bottom, can you tell the height of the stairway in steps?

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who can solve this? (explain the solution plz)

2. Originally Posted by zenith20
27) Down the Escalator
Recently, while in London, I decided to walk down the escalator of a tube station. I did some quick calculation in my mind, found that if I walk down twenty-six steps, I require thirty seconds to reach the bottom. However, if I am able to step down thirty-four stairs I would only require eighteen seconds to get to the bottom.
If the time is measured from the moment the top step begins to descend to the time I step off the last step at the bottom, can you tell the height of the stairway in steps?

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who can solve this? (explain the solution plz)

let $r$ = rate the escalator moves in steps per second

$h$ = height of the escalator in steps

$h = \left(r + \frac{26}{30}\right) \cdot 30$

$h = \left(r + \frac{34}{18}\right) \cdot 18
$

setting each expression for $h$ equal ...

$\left(r + \frac{26}{30}\right) \cdot 30 = \left(r + \frac{34}{18}\right) \cdot 18
$

$30r + 26 = 18r + 34$

$12r = 8$

$r = \frac{2}{3}$ steps/sec

substitute this value for $r$ into either expression for $h$ ...

$h = \left(\frac{2}{3} + \frac{26}{30}\right) \cdot 30$

$h = 20 + 26 = 46$ steps

3. Dear Skeeter, thanks for your solution really appreciated.