Problems on Escalators - Time, Speed & Distance


  • QA/DILR Mentor | Be Legend


    Authored by Nitin Gupta, Founder, Director at AlphaNumeric.

    Escalator is similar to Boats & Streams
    Distance = no. of steps in escalator (when escalator is not moving)
    Here speed is usually given as no. of steps per second
    Let speed of escalator is e steps/sec
    Let speed of man/woman = m steps per second
    Case 1: when escalator & man are moving in same direction, effective speed = (m + e) steps/sec
    Case 2: when escalator & man are moving in opposite direction, effective speed = (m - e) steps/sec

    Another important concept:
    Case 1: when escalator & man are moving in same direction, --- No. of steps covered by man is always less than actual no. of steps in escalator.
    Case 2: when escalator & man are moving in opposite direction, --- No. of steps covered by man is always more than actual no. of steps in escalator.

    You walk upwards on an escalator, with a speed of 1 step per second. After 50 steps you are at the end. You turn around and run downwards with a speed of 5 steps per second. After 125 steps you are back at the beginning of the escalator. How many steps do you need if the escalator stands still?

    Approach 1:
    Let N = no. of steps in escalator (when escalator is not moving)
    Case 1: walk upwards on an escalator, with a speed of 1 step per second. After 50 steps you are at the end
    N /(e+1) = 50/1-------------(1)
    Case 2: run downwards with a speed of 5 steps per second. After 125 steps you are back at the beginning of the escalator
    N/(5-e) = 125/5 --------(2)
    Dividing eq 1 by eq 2 , you will get e= 1
    Putting e = 1 you will get N= 100 steps.

    Approach 2:
    Remember
    Case 1: when escalator & man are moving in same direction, --- No. of steps covered by man is always less than actual no. of steps in escalator.
    Case 2: when escalator & man are moving in opposite direction, --- No. of steps covered by man is always more than actual no. of steps in escalator.
    say escalator speed x steps/sec.
    so total steps = 50 + 50x (from upward condition, in 50 sec escalator will cover 50x).
    Total time to reach up is 50 sec.
    total time to reach down = 25 sec.(125 steps, 5 steps/sec)
    total steps = 125 - 25x (in 25 sec escalator will cover 25 x)
    50 + 50x = 125 - 25x
    75 x = 75 => x =1,
    so total steps = 50 + 50 * 1= 100

    You all must know both approach but use approach 2 alwyas as it is easy!

    A walks down an up-escalator and counts 150 steps. B walks up the same escalator and counts 75 steps. A takes three times as many steps in a given time as B. How many steps are visible on the escalator?

    Approach 1:
    Let N = no. of steps in escalator (when escalator is not moving)
    Speed of a / speed of b = 3:2 , let speed of a = 3x & speed of b = 2x
    Case 1: “A “ walk down on up- escalator,
    N /(3x-e) =1 50/3x-------------(1)
    Case 2: “B“ walk up on up- escalator,
    N/(2x+e) = 75/2x --------(2)
    Dividing eq 1 by eq 2 , you will get e= x
    Putting e = x you will get N= 120 steps.

    Approach 2:
    Remember
    Case 1: when escalator & man are moving in same direction, --- No. of steps covered by man is always less than actual no. of steps in escalator.
    Case 2: when escalator & man are moving in opposite direction, --- No. of steps covered by man is always more than actual no. of steps in escalator.
    Let T be time B takes to make 25 steps. Then B takes 3T to make 75, and A takes 2T to make 150. Suppose the escalator has N steps visible and moves n steps in time T. Then A covers N + 2n = 150, N - 3n = 75.
    Hence N = 120, n = 15.

    Colin takes the underground train to work and uses an escalator at the railway station. If Colin runs up 8 steps of the escalator, then it takes him 37.5 seconds to reach the top of the escalator. If he runs up 14 steps of the escalator, then it takes him only 28.5 seconds to reach the top. How many seconds would it take Colin to reach the top if he did not run up any steps of the escalator at all?

    Approach 1:
    If he runs up 8 steps, then he needs 37.5 seconds to reach the top.
    If he runs up 14 steps, then he needs 28.5 seconds to reach the top.
    The 6 additional steps take 9.0 seconds.
    Therefore, each step takes 1.5 seconds.
    Total steps in escalator = 8 + 37.5 / 1.5 = 33 or Total steps in escalator = 14 + 28.5 / 1.5 = 33.
    If Colin did not run up any steps at all,
    he would reach the top of the escalator in 49.5 seconds (i.e., 33 steps × 1.5 seconds/step).

    Approach 2 : Alternative Solution through Equations:
    Let the total number of steps in the escalator be x.
    The escalator moves at a constant speed given by
    Speed of escalator = (x − 8)/37.5 = (x − 14)/28.5
    The above equation may be solved as follows.
    28.5 (x − 8) = 37.5 (x − 14); or
    x = (14 × 37.5 − 8 × 28.5) / (37.5 − 28.5) = 33.
    Now, Speed of escalator = (33 − 8)/37.5 = (33 − 14)/28.5 = 1/1.5 steps/second.
    Time to reach top = Total Steps / Speed = 49.5 seconds.

    Approach 3 :
    Let escalator speed = e steps/sec
    No. of steps = 8+37.5x = 14+28.5x
    X = 2/3
    No. of steps = 8 + 37.5*2/3 = 33 steps
    Now, Speed of escalator = (33 − 8)/37.5 = (33 − 14)/28.5 = 1/1.5 steps/second.
    Time to reach top = Total Steps / Speed = 49.5 seconds.

    Shyama and Vyom walk up an escalator (moving stairway). The escalator moves at a constant speed. Shyama takes three steps for every two of Vyom’s steps. Shyama gets to the top of the escalator after having taken 25 steps, while Vyom (because his slower pace lets the escalator do a little more of the work) takes only 20 steps to reach the top. If the escalator were turned off, how many steps would they have to take to walk up? (cat 2001)
    a. 40
    b. 50
    c. 60
    d. 80

    25 + 25E/3S = 20 + 20E/2S
    take E/S = k
    25 + 25k/3 = 20 + 10k
    k = 3
    Thus steps = 25 + 25 * 3/3 = 50

    2 kids, John and Jim, are running on an escalator (a moving stairway). John is running three times as fast as Jim, and by the time they are off the escalator, John has stepped on 75 stairs while Jim has stepped on 50 stairs. How many stairs does the escalator have? How is its speed related to the speed of the boys? Were they running with or against the escalator?

    The answers are: the length is 100 stairs, the boys were running along the escalator which was moving with the same speed as the slow boy. Solution: in the time the fast boy stepped on 75 stairs, the slow one could step on only 25, so, since he stepped on 50, he spent twice as much time on the escalator as the fast one. Therefore his speed relative to the ground was half that of the fast boy, therefore the escalator's speed was the same as the speed of the slow boy, and he counted exactly half the stairs. Another way is to use algebra (omitted).

    Approach 2:
    Assume A takes 1 step per unit time. Then B will take 3 steps per same unit time. Also, assume the the escalator is moving at E steps per unit time.
    Let T be the total number of steps.
    Let ta be time taken by A on the escalator, tb = time taken by B on the escalator.
    Since A takes 50 steps - therefore we have:
    50 = T/(1+E) units of time.
    similarly,
    75 /3= T/(3+E) units of time.
    Solving for T, E we get E = 1 step per unit time; T = 100 steps

    An escalator is descending at constant speed. A walks down and takes 50 steps to reach the bottom. B runs down faster and takes 90 steps to reach the bottom.If B takes 90 steps in the same time as A takes 10 steps then how many steps are visible when the escalator is not operating?

    Method 1:
    B = 9
    A = 1
    50 + 50x = 90 + 10x
    => x = 1
    total steps = 100

    Method 2:
    There are 100 steps in the escalator.
    Case 1 : When A & B walk on unmoving surface
    10 Steps of A = 90 Steps of B .................... [Eq.1]
    => Speed of A : Speed of B :: 1 : 9
    => Time of A : Time of B :: 9 : 1
    Case 2 : When A & B walk on escalator
    Steps taken by B : 90
    Time taken by B : T
    Steps taken by A : 50
    Time taken by A : 5T [From [Eq.1]]
    Time of A : Time of B :: 5 : 1
    let e be the number of steps moved by the escalator when A takes 1 step
    So
    (1+ e) / (9 + e) = 1/5
    => e = 1 step for every step of A
    A takes 50 steps
    => Escalator has moved by 50 steps
    Total number of steps = 50 + 50 = 100

    On an upward moving escalator Amit, Sanjeev and Vicky take 10 steps, 8 steps and 5 steps respectively to reach the top. On the same upward moving escalator Amit takes 30 steps to come down from the top.Find the ratio of the time taken by Sanjeev and Vicky to reach the top

    no. of steps = 10+x = 30 - 3x ===> 4x = 20 ===> x = 5; so total no. of steps in escalator is 15; now sanjeev takes 8 steps (remaining 7 are taken by escalator on his behalf); amit--5 steps (remaining 10 are taken by escalator on his behalf); so required ratio is 7:10

    pavan walked up a descending escalator and took 154 steps in 100 sec to reach the top. Rishabh started simultaneously from the top taking 3 steps for every 4 steps taken by pavan, reach the bottom in 40sec. How many steps from bottom were they when they crossed each other?

    pavan's speed = 4x m/s
    escalator speed = y m/s
    400x - 100y = D
    rishabh's speed = 3x m/s
    120x + 40y = D
    400x - 100y = 120x + 40y
    280x = 140y
    ratio = 2:1
    it means P has to cover twice of step
    P's speed = 4x
    E's speed = 2x
    R's speed = 3x
    so , distance = 154/2 = 77
    77/(2x+5x) = 11/x sec
    so they will meet after 11/x secs that means (11/x) * 2x = 22 steps from bottom


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