Question

How many steps would have to be took is a esulader was standing still is she counted 60 steps going up n 90 going down how many would she of counted total if the steps were standing still?

Answer 1

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:

`\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}`

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.

`w\cdot tu=60\\w\cdot td=90\\d=(w+e)tu\\d=(w-e)td`

Dividing the second by the first, we have

`(w\cdot td)/(w\cdot tu)=(90)/(60)\\\\ (td)/(tu)=(3)/(2)`

Equating the third and 4th equations, and substituting for td, we have

`(w+e)tu=d=(w-e)(3)/(2)tu\\\\e\left(1+(3)/(2)\right)=w\left((3)/(2)-1\right)\qquad\text{divide by $tu$,rearrange}\\\\5e=w\qquad\text{divide by $(1)/(2)$}`

Solving the first equation for tu and substituting into the third equation, we get

`d=(w+e)(60)/(w)\\\\d=((5e+e)60)/(5e)=(360e)/(5e)\\\\d=72`

The number of steps between floors is 72, which is the number she would count if the escalator were not moving.