Given
A person counted 60 steps going up an up-escalator, and 90 steps going down the same up-escalator.
Find
The number of steps she would count if the escalator were standing still.
Solution
This is perhaps the long way around, but we get there.
Define the following variables:
[tex] \begin{array}{rl}d&\text{steps between floors}\\w&\text{walking rate, steps per minute}\\e&\text{escalator rate, steps per minute}\\tu&\text{time it takes to go up the escalator, minutes}\\td&\text{time it takes to go down the escalator, minutes}\end{array} [/tex]
The rate at which the steps of the distance d are traversed is w+e (going up) or w-e (going down). The number of steps counted is the rate at which steps are walked (w) multiplied by the time spent going up or down. We can write 4 equations in the 5 unknowns.
[tex]w\cdot tu=60\\w\cdot td=90\\d=(w+e)tu\\d=(w-e)td[/tex]
Dividing the second by the first, we have
[tex]\dfrac{w\cdot td}{w\cdot tu}=\dfrac{90}{60}\\\\ \dfrac{td}{tu}=\dfrac{3}{2}[/tex]
Equating the third and 4th equations, and substituting for td, we have
[tex](w+e)tu=d=(w-e)\dfrac{3}{2}tu\\\\e\left(1+\dfrac{3}{2}\right)=w\left(\dfrac{3}{2}-1\right)\qquad\text{divide by $tu$,rearrange}\\\\5e=w\qquad\text{divide by $\frac{1}{2}$}[/tex]
Solving the first equation for tu and substituting into the third equation, we get
[tex]d=(w+e)\dfrac{60}{w}\\\\d=\dfrac{(5e+e)60}{5e}=\dfrac{360e}{5e}\\\\d=72[/tex]
The number of steps between floors is 72, which is the number she would count if the escalator were not moving.
