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A kid is walking up a downward-moving escalator with a constant speed and counts 72 stairs. Then the kid walks down with the same speed and now counts 36 stairs. Being confused, the kid waits until the mall closes and the escalator is still, so he now walks up the escalator with half the speed he had before. How many stairs did he count?

Total number of steps = steps climbed oneself + steps produced by escalator, so:

Let the kid's speed be $v$. Let the speed of the escalator be $u$. So in the two cases: $$72-72\dfrac{u}{v} = 36+36\dfrac{u}{v} $$ so $$\dfrac{u}{v} = \dfrac{1}{3}$$

So the total number of stairs at any given moment are 48, right?

Now how do we answer the last question?

How does his speed affect the number of stairs he counted?

Parcly Taxel
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Marius Stephant
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1 Answers1

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This is a trick question. The escalator is still, so the number of steps he counts is 48, no matter how fast he walks.

A (perhaps) simpler derivation of 48 steps counted when the escalator is still is as follows. Let unit time be the time the kid takes to traverse a step, $u$ the distance the kid traverses on a still escalator per unit time and $v$ the escalator's speed per unit time. Then $$\frac1{72}=u-v$$ $$\frac1{36}=u+v$$ from which we may derive $u=\frac1{48}$. On a still escalator, halving the speed only increases the time he takes to traverse it; he counts the same number of steps.

Parcly Taxel
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