1. ## Step by step

Walking slowly down the down-moving escalator, it requires 50 steps to reach the bottom.
Running up the escalator one step at a time, it takes 125 steps to reach the top.
Assuming that running pace is five times as fast as walking pace (that is, five steps are taken at running pace in the same time interval as one step at walking pace), and that each trip was made at uniform speed, how many steps would be visible if the escalator stopped running?

Alan

2. ## Re: Step by step

I calculate that there would be <span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>75 </font color=yellow></span hi> steps visible.

Solution:
<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>Let the velocity of of the excalator be Ve
Let the Time to walk the escalator = Tw
Let the Time to run the escalator = Tr
Let the walking velocity = Vw
Let the running velocity = Vr

We know that:
Vw = 50 /Tw
Vr = 125/ Tr

The length of the escalator is then:
50 + Ve*Tw = 125 - Ve*Tr

And we are given that
5*Vw = Vr
5 * 50 /Tw = 125 /Tr

Rearranging:
2 * Tw = Tr

Replacing in the other equation:
50 + Ve*Tw = 125 - 2*Ve*Tw

Rearranging:
Ve*Tw = 25
So length = 50 + 25 = 125 - 2*25 = 75</font color=yellow></span hi>

Steve

3. ## Re: Step by step

I calculate <span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>100</font color=yellow></span hi> steps. My solution:

<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>L = escalator length in steps
Ve = escalator speed
Vd = speed pacing down escalator, td = time taken for trip down
Vu = speed running up escalator, tu = time taken for trip up

Vu = 5 x Vd

Trip down:
L = (Ve + Vd)td, td = 50/Vd
=> L = 50(Ve/Vd + 1)
=> Ve = Vd(L/50 - 1) ...(A)

Trip up:
L = (Vu - Ve)tu = (5Vd - Ve)tu , tu = 125/Vu = 25/Vd
=> L = 125(1 - Ve/(5Vd))

... substitute (A) ->
=> L = 125(1 - (L/50 - 1)/5) = 25(5 - L/50 + 1) = 25(6 - L/50)
=> 50L = 25(300 - L)
=> 2L = 300 - L
=> L = 100</font color=yellow></span hi>

Alan

4. ## Re: Step by step

<P ID="edit" class=small>(Edited by KTYorke on 08-Sep-03 12:37. to fix the hidden answer... NOTE: you can now use the "hide" tag to hide answers instead of a combination of "highlight" and "yellow" :-))</P>OOPS. You are correct. I made a math error:
I went from
<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow> 5 * 50 /Tw = 125 /Tr
and Rearranging:
2 * Tw = Tr
It SHOULD have been:
2 * Tr = Tw !!

Giving by replacing in the other equation:
50 + 2* Ve*Tr = 125 - Ve*Tr
Rearrangement
Rearranging:
Ve*Tr = 25

So steps = 100 as you stated</font color=yellow></span hi>

God I hate to make math errors!
Steve

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