Almost all the serious mock takers would have been stumped sometime or the other by escalator based questions. While many simply shirk it off saying that it is nothing but a version of upstream downstream questions, few get the logic behind the same. In this series, we try to explore the logic behind escalator based questions.
In this article, I would be explaining the motion of one person along an escalator before we move to slightly more confusing scenarios involving multiple people.
The general question types are:
1) Finding the number of steps on a stationary escalator
2) Finding the speed of the escalator
3) Finding the number of steps taken by an individual or the escalator
The basic things that you need to remember are:
1) The angle of inclination, the height of each step do not matter. It is as good as saying that you are walking along or against a moving walkway
2) The distance covered is not in terms of meters or kilometers as is seen commonly in time-speed-distance problems but in terms of steps
3) A step taken by the escalator and a step taken by the person are the same
4) Whatever be the proportion of the number of steps taken by an individual and the number of steps traveled by the escalator, the total distance traveled by both the individual and the escalator is always equal to the number of visible steps on the escalator
5) Time taken by an individual to climb down the steps is the same amount of time for which the escalator is moving
Let’s go through a bit of theory first to understand how exactly an escalator functions. Say there are 10 steps that you need to climb to get to the top of an escalator. If the escalator is stationary, you would need to climb the 10 steps all by yourself. If the escalator were moving upwards, you would save some time as the speed would add up, the distance remaining the same. As this would be in inverse variance, the product of speed and time in both cases would be constant and the resultant would be the distance traveled which is nothing but the number of steps on the escalator. Also, almost all the questions will ask you the number of steps that are visible and so, you need to find values in terms of distance. Many aspirants fall into the trap of calculating speeds and then end up in a tangle because they have nowhere to go post that. There will be very few data points given in the questions which makes it all the more difficult to put them into a predefined structure.
Once you understand the above paragraph, most of the job is done. You have to simply put pen to paper and solve the rest of the part. Let’s see an example of a simple question from single person on an escalator.
Sriram takes 15 seconds to climb up a stationary escalator but 12 seconds when the escalator is moving. If the speed of the escalator is 2 steps/second, how many steps can be seen on the stationary escalator?
Let speed of Sriram be x steps/sec, speed of escalator is 2 steps/sec. Time taken in the first case is 15 seconds, time taken in the second case is 12 seconds. So, the equation formed would be:
15x = 12(x+2)
So, distance will be 15*8 = 120 steps.
In any question, you would have data pertaining to three out of the four things – speed of individual, speed of escalator, time taken by the individual, time taken by the escalator and the person together. Let’s see another question wherein the information is not direct.
Prasad can climb an escalator in 15 seconds and can come down the same escalator in 25 seconds. How much time would he take to climb up and then climb down the same escalator when it is stationary?
Let speed of Prasad be S, speed of the escalator be x. We can simply put the above formula and find the ratios of speeds:
Now, there is another bit of information given in the question which few might not realize right away. The escalator is stationary and so, x will be equal to 0 in the final case. So, the speed will be S. If the total distance is found in terms of S, we can easily get the amount of time taken.
Substituting x=S/4 in the earlier expression, we get:
75S/4=number of steps on the escalator
So, time taken to climb these steps at the speed of S steps/sec would be 75/4=18.75 seconds. Coming down would also take the same time as we are neglecting the role of gravity and so, we get the total time to be 37.5 seconds.
A person walking down an escalator takes 26 steps and it takes him 30 seconds to reach the bottom. The same person while running takes 18 seconds and 34 steps to reach the bottom. How many visible steps are there on the escalator?
Let the speed of the escalator be x. Bear in mind that the speed of the person is different when he is walking or running and so, you cannot take a single variable for the speed of the person. Also, you need to know that distance covered is the question here and so, distance covered by the person and distance covered by the escalator would be equal to the number of steps on the escalator.
In the first case, we know that the person takes 26 steps and travels for a total of 30 seconds. So, the number of steps covered by the escalator in 30 seconds will be 30x. Total number of visible steps = 26+30x
In the second case, we will have the total number of visible steps to be 34+18x
As these are equal, we get
26+30x = 34+18x
Number of steps will be 46.
Questions can be twisted easily in this topic and even if they are easy to do, they appear especially cryptic if one gets overwhelmed by the excessive data. Let’s see an example of one of the tougher problems from the topic.
A woman is walking down a downward-moving escalator and steps down 10 steps to reach the bottom. Just as she reaches the bottom of the escalator, a sale commences on the floor above. She runs back up the downward moving escalator at a speed five times that which she walked down. She covers 25 steps in reaching the top. How many steps are visible on the escalator when it is switched off?
In this case, we know the distance covered by the woman in both the cases and the relation between the speeds in the two cases. What we do not know is the time for which she travels and the speed of the escalator. Let the initial speed of the woman be S and that of the escalator be x.
The first part says that she has already reached the bottom along a downward-moving escalator. The time taken for this will be nothing but the distance covered/speed. So, we get the number of visible steps to be:
In the second part, we know that the direction is opposite to that of the escalator and so, we will subtract the speeds. The new speed of the woman becomes 5S while the remaining data remains the same. So, the number of visible steps would be:
As the distance is the same in both the cases, we get:
And so, number of visible steps would be 20 through reverse substitution.
What you have to understand is that, time taken would be the same by both the individual and the escalator. It is because of their speeds that the distance covered might vary. So, when the question says that the woman travelled for 10 steps while moving downwards, the assumption here is that she took these steps spaced equally throughout the available time period. A lot of people fail to see this and so, end up missing out on this part of the question.
With this much of conceptual knowledge, solving single person escalator problems should not be difficult. I will be covering multiple entity based escalator problems in the next article.
Update: You can read the second and the final part of the article here.