Tackling Escalator Problems - Saquib Hasnain

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    Saquib Hasnain did his MBA from IIFT, Delhi and has rich experience in mentoring CAT aspirants in Quant and Data Interpretation & Logical Reasoning sections. He is currently working as the Academic head at Vistamind, Kolkata.

    The concept of escalators is similar to upstream and downstream. Think of the river as your stairs and the length of the river as the length of the stairs, the only difference is that in river we measure it in terms of KM/Meters and in escalators we measure it in terms of Number of steps.

    So in upstream, the relative speed is Boat - River

    Think of the escalator as your river and the person is trying to move up the escalator (upstream), so even in this case, the relative speed would be
    (Speed of man - speed of escalator)

    In case of downstream, the relative speed is B + R
    Even in escalators, the person is trying to go down and the escalator is moving downwards, so the relative speed would be (Speed of man + speed of escalator)

    Now let us take a simple example -

    Escalator Type 1 - (where the LENGTH of the Escalator is asked)

    P and Q walk up a moving up escalator at constant speeds. For every 5 steps that P takes Q takes 3 steps. How many steps would each have to climb when the escalator is switched off, given that P takes 50 and Q takes 40 steps to climb up the moving up escalator respectively?
    (1) 80
    (2) 90
    (3) 100
    (4) None of these

    As I said, this is same as upstream and downstream.
    So both the person and the escalator are moving in the same direction. In this case, the relative speed would be Speed of (Man + Stairs)

    We dont know the speed of the escalator, so assume that to be E
    Now think, both for P and Q the length of the escalator should be same, right?
    and P speed is 5 steps/sec (say) and Q's is 3 steps/sec
    P is walking 50 steps and he does it in 10 sec..but in this 10 sec, even the escalator must have moved with speed "E". The length moved by it would be 10E
    So we can say the total length is 50 + 10 E ( step moved by P and the escalator helps P in reaching the top.
    Similarly for Q we can write 40 + 40/3 * E

    Both lengths must be same, so -
    50 + 10 E = 40 + 40E/3
    So E = 3
    So total length is 50 + 10 E = 80

    Escalators Type 2 - (Where the time taken to reach by only one person is involved)

    So say the question is something like this -

    Anand takes 90 secs to move up an escalator which is moving downward and he takes 30 secs to move down the same escalator. How much time will he take to go up or down, when the escalator is switched off?

    Method 1-

    Take the length of the stairs LCM(90,30) = 90
    Time taken when going upstream is 90 sec, so we can say
    M - E = 90/90 = 1
    M + E = 90/30 = 3
    So speed of Man = 4/2 = 2
    Time taken is 90/2 = 45 sec

    Method 2 - Shortcut

    When moving upstream speed is M + E
    when moving when escalator is stopped is M
    and when moving upstream speed is M - E
    speed is in AP, so time will be in HP, since they are inversely proportional.
    So time taken would be -
    2 x 90 x 30 / (90+30) = 60 x 90/120 = 45 secs

    Escalator Type 3 - (When both person are moving in opposite directions)

    A can take 10 steps per second and B can take 7 steps per second. A mischievously starts climbing down an escalator that is moving upwards and at the same instant B starts climbing up the same escalator. They meet after 5 seconds. If the escalator works at a steady rate of 4 steps per second then how many steps are visible when the escalator is stopped?
    (1) 90
    (2) 80
    (3) 85
    (4) None of these

    A and B take 10 and 7 steps per second. Escalator takes 4 steps per second.
    A is climbing down so effective speed is 10 – 4 =6 steps/sec
    B is climbing up so effective speed is 7 + 4 = 11 steps/sec
    They are moving in opp directions so rel speed is 17 steps/sec.
    As they meet in 5 sec, total steps = 17 * 5 = 85 steps.

    Saquib is climbing on a moving escalator that is going up and takes 30 steps to reach the top. Atreya on the other hand is coming down on the same escalator. For every 5 steps that Atreya takes, Saquib takes only 3 steps. Both of them take the same amount of time to reach the other end.
    a) What is the total number of steps in the escalator?
    b) What is the difference in the number of steps that both of them had taken when they crossed each other?

    (Question Source : Ravi Handa sir)

    Let us assume their speeds are 5s and 3s, and the speed of the escalator is 'x'
    Since both of them take the same time for the same distance, their effective speed is the same.
    => 5s - x = 3s + x
    => x = s
    Speed of Saquib : Speed of escalator = 3s : s = 3 : 1
    => When Saquib takes 30 steps, the escalator takes 10 steps.
    => Total number of steps = 30 + 10 = 40 steps.
    Both of them would have covered 20 steps when they crossed each other.
    Saquib going up would have taken 15 steps, whereas the escalator would have taken 5 steps for him.
    Atreya coming down would have taken 25 steps, out of which the escalator would have nullified the movement of 5 steps for him.
    Difference in the number of steps = 25 - 15 = 10 steps.

    Hope you are clear with the concepts. Will solve some other questions from TSD as a bonus :)

    A boat-man while moving downstream dropped his straw hat at a point X. He further travelled a distance downstream to a point Y in 2 hrs. Upon reaching Y he turned back. For how long must the boat travel (time being calculated from the instant he turned round) upstream so that he might be able to retrieve his hat once again?
    (1) 2 hrs
    (2) 3 hrs
    (3) 5 hrs
    (4) Cannot be determined.

    When boat is moving downstream speed is b + r
    and straw is r.
    relative speed is (b + r - r) {since both are moving in same direction}
    so relative speed is b
    When boat is moving upstream..speed is b- r
    and straw is r again..
    so relative speed is (b - r +r ) {since bother are moving in opposite direction) = b
    so in both cases relative speed is b..
    so we can consider the straw to be static and the boat to be moving at speed of B.
    Now when a static object travels for 2 hours with a speed of B it takes same 2 hours to return with speed of B
    so doesn't matter whether upstream or downstream..answer will remain same

    There is a tunnel A-B. Jerry enters the tunnel and after travelling 1/3rd distance of the tunnel, looks back and sees Tom approaching the entrance of the tunnel from a certain distance.
    Now Jerry has two options -
    a. Either he can walk towards the entrance of the tunnel with the double the initial speed OR
    b. He can walk towards the exit of the tunnel at the usual speed.
    In the both these cases, he just manages to meet Tom.
    Find out the ratio of speed of Tom : Jerry

    Let the tunnel length be 3 and Tom is at a distance X from the entrance of the tunnel.
    In the first case, we can write
    Speed of Tom/ 2 x Speed of Jerry = (x/ 1)
    (Note : Speed and distance are proportional, as time is constant)
    So T/J = 2x.....(i)
    Also in second case we can write
    T/J = (3 + x)/2........(ii)
    equating both of them we get
    x = 1
    So Tom / Jerry speed ration is 2 : 1

    A train started from city A to city B and travelled a certain distance as usual speed. It then developed an engine problem because of which its speed reduced by 10% and it reached it's destination 1 hour late. Has the engine problem happened after it has travelled 20% more distance, the train would have been 50 mins late. For how long did the train travel before it developed the engine problem?
    a. 4.5
    b. 6
    c. 7.5
    d. 9

    If the engine problem happened 20% further..so if it has traveled before engine problem is D..then 20% is D/5..now it is saving 10mins bcz this D/5 now it has traveled with original speed..instead of reduced..
    So speed ratio
    Original : Reduced = 10 : 9
    Time ratio
    Original : Reduced will be 9 : 10
    Differnece of 1 is equivalent to 10mins..so 9 is 90 mins.
    D/5 he is traveling in 90mins.
    So D he will travel in 450mins = 7.5 hours.

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