Lesson Moving escalator problems

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Moving escalator problems


Problem 1

An escalator is moving downward from  2nd floor to the  1st floor.  Emily walks down from the  2nd  floor to the first floor,
and she walks  40 steps.  Albert walks up from the first floor to the second floor,  and he walks  80 steps.
If  Albert walks twice as fast as  Emily,  find the number of steps of the escalator when it is stationary.

Solution 

Let x be the number of steps of the escalator when it is stationary (the number under the problem's question).


During the time Emily makes her walk down, the escalator moves (x-40) steps.


During the time Albert makes his walk up, escalator moves (x-80) steps.


Albert moves twice as fast as Emily, which means that Emily's time is twice the Albert's time.


But the escalator speed is a constant, so


    x - 40 = 2*(x-80).


It is your basic equation from the condition to find x.


From the equation

    x - 40   = 2x - 160

    160 - 40 = 2x - x

     x       = 120.


ANSWER.  The number of steps of the escalator when it is stationary is 120 (in one direction).

Problem 2

An escalator is moving up.  If  Zach walks down from the top to the bottom,  he walks  120  steps.
If he walks up from the bottom,  he walks  90  steps.  He walks down  2  times as fast as he walks up.
Find the number of steps of the escalator when it is not moving.

Solution 

Let n be the number of steps of the escalator when it is not moving  (the unknown value under the problem's question).


Let T be the time for Zack walking down from the top to the bottom of the escalator.

Then the time for Zack walking up from the bottom of the escalator is 2T, according to the condition.


For Zack going down, the speed of the escalator is  %28n-120%29%2FT  steps per unit of time.


For Zack going up,   the speed of the escalator is  %28n-90%29%2F%282T%29  steps per unit of time.


The speed is the same, which gives you an equation

    %28n-120%29%2FT = %28n-90%29%2F%282T%29.


To solve it, cancel factor T in both denominators, then cross multiply and simplify.  You will get

    2*(n-120) = n-90

    2n - 240  = n - 90

    2n - n    = 240 - 90

     n        = 150.


ANSWER.  The number of steps of the escalator when it is not moving is 150.

Problem 3

Bob steps onto an escalator which is moving up to the  2nd floor.  If he walks one step per second on it,
he walks  20 steps to reach to the  2nd floor.  If he walks  2  steps per seconds,  he walks  30  steps to reach to the  2nd floor.
Find the number of steps of the escalator when it is stationary.

Solution 

Let x be the number of steps of the escalator when it is stationary (on one its side).


In the first scenario, Bob makes 20 steps on the escalator in 20 seconds --- hence, the escalator moves x - 20 steps in 20 seconds.


In the second scenario, Bob makes 30 steps on the escalator in 15 seconds --- hence, the escalator moves x - 30 steps in 15 seconds.


Escalator moves uniformly with the same speed/(rate) in both cases.


Hence, the ratio %28x-20%29%2F%28x-30%29  is the same as ratio of times  20%2F15 :


    %28x-20%29%2F%28x-30%29 = 20%2F15,  or

    %28x-20%29%2F%28x-30%29 = 4%2F3.


Cross-multiply and solve for x


    3*(x-20) = 4*(x-30)

    3x - 60 = 4x - 120

    120 - 60 = 4x - 3x

    60       = x.


ANSWER.  There are 60 steps of the escalator (on one its side).


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Use this file/link  ALGEBRA-I - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-I.

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