How Many Steps on the Escalator?

Date: Mon, 1 Jul 1996 18:00:39 -0400 (EDT)
From: Anonymous
Subject: walking up the up escalator

Dear Dr. Math,

A famous mathematician who is always in a hurry walks up an 
up-going escalator at the rate of one step per second. Twenty steps 
bring him to the top. The next day he goes up at two steps per 
second, reaching the top in 32 steps. How many steps are there in 
the escalator?

I know that the speed of the escalator is the same on both days, but 
I am having trouble because the numbers don't seem to relate. If he 
is traveling two steps per second and the elevator is moving at the 
same speed, shouldn't it take him 40 steps because he is traveling 
twice as fast?

Help!

Nancy Geldermann

Date: Tue, 2 Jul 1996 06:15:08 -0400 (EDT) 
From: Dr. Anthony
Subject: Re: walking up the up escalator

You need to introduce some letters to represent unknown quantities 
for this type of question. 

We let n = number of steps on a stationary escalator to go from 
bottom to top. u = speed of escalator, s = step size. 

Day(1) speed of man relative to stair = s 
Time to travel total length = 20 secs
Total speed = u+s

We can write down the equation for total distance ns: 

ns = (u+s)*20

Day(2) speed of man relative to stair = 2s 
Time to travel total length = 16 secs
Total speed = u+2s

Equation for total distance is then:

ns = (u+2s)*16

We could divide both equations through by s to give: 

n = (u/s + 1)*20	n/20 = u/s + 1
n = (u/s + 2)*16	n/16 = u/s + 2
------------------
Subtract n(1/20 - 1/16) = 0 - 1

n(-1/80) = -1

n = 80

So the escalator has 80 steps.

-Doctor Anthony, The Math Forum