(Photo: Boegh)

Quant problems on Time, Speed and Distance make frequent appearances in management entrance exams. To my collection of posts on PaGaLGuY (detailed list), I would like to add a few thoughts about solving escalator-related problems.

At the basic level, escalator related problems aren’t too different from boats and streams problem. Think of the escalator as just a replacement for a river, the only difference being that escalators move in both directions whereas rivers only flow downstream.

I categorise escalator problem into two categories,

1. When one person is moving

2. When two people are moving

The problems are related to each other and some of them use data from the previous question.

Type 1: When one person is moving

1. Ravi takes 40 seconds to walk up on an escalator which is moving upwards but he takes 60 seconds to walk up on an escalator which is moving downwards. How much time will he take to walk up if the escalator is not moving?

This is exactly like a boats and streams problem, and we are given the times taken by a person rowing downstream and upstream.

Let us assume that the speed of Ravi is ‘r’ and the speed of the escalator is ‘x’.

In the first case, when the escalator is moving up, Ravi’s effective speed is r + x as the motion of the escalator is assisting his movement.

In the second case, when the escalator is moving down, Ravi’s effective speed is ‘r – x’ as the motion of the escalator is a hindrance to his movement.

When the escalator is not moving, his speed would be ‘r’.

The distance covered in each case is constant.

The three speeds ‘r + x’, ‘r’ and ‘r – x’ are in an Arithmetic Progression.

=> The times taken will be in a Harmonic Progression.

=> Time taken when the escalator is not moving will be the Harmonic mean of the other two times given.

=> Time taken when escalator is not moving = 2*40*60/(40+60) = 48 seconds.

2. Ravi takes 60 seconds on an escalator which is moving down when he walks down but takes 40 seconds when he runs down. He takes 20 steps when he walking whereas he takes 30 steps when he is running. What is the total number of steps in the escalator?

Let us say that the speed of the escalator is ‘x’ steps per second.

Distance covered by Ravi is the same whether he is walking or running.

Distance when he is walking = 20 + 60x (60x is covered by escalator)

Distance when he is running = 30 + 40x (40x is covered by escalator)

=> 20 + 60x = 30 + 40x

=> x = 0.5

Total number of steps = 20 + 60(0.5) = 50 steps

Type 2: When two people are moving

1. Ravi and Rakesh are climbing on a moving escalator that is going up. Ravi takes 10 seconds to reach the top but Rakesh takes 8 seconds to reach the top. This happens because Rakesh is faster than Ravi. Rakesh takes 4 steps whereas Ravi can take only 3 steps in one second. What is the total number of steps in the escalator?

This question can be interpreted as two people rowing a boat downstream.

Let us assume that the escalator moves at the rate of x steps per second.

Distance covered by both of them is same

=> 10(3 + x) = 8(4 + x)

=> 10x – 8x = 32 – 30

=> x = 1 step per second

Number of steps in the escalator = 10(3 + 1) = 8 (4 + 1) = 40 steps

Another variation of this problem could be if we were not given the times taken by Rakesh and Ravi but the number of steps that they took to reach the top. Let us look at that problem.

2. Ravi and Rakesh are climbing on a moving escalator that is going up. Ravi takes 30 steps to reach the top whereas Rakesh takes 32 steps for the same. This happens because Rakesh is faster than Ravi. Rakesh takes 4 steps whereas Ravi can take only 3 steps in one second. What is the total number of steps in the escalator?

Before moving ahead, please notice the difference between previous question and this one.

The first thing that we will do is to figure out the time taken by Rakesh and Ravi to reach the top.

Rakesh takes 32/4 = 8 seconds whereas Ravi takes 30/3 = 10 seconds.

From now on, everything is same as the previous problem.

Let us assume that the escalator is moving with a speed of x steps per second.

Distance covered by both of them is same

=> 10(3 + x) = 8(4 + x)

=> 10x – 8x = 32 – 30

=> x = 1 step per second

Number of steps in the escalator = 10(3 + 1) = 8 (4 + 1) = 40 steps

A slightly more difficult version would be when we do not know the speeds of Rakesh and Ravi but only the ratio. Let us look at that in the next problem.

3. Ravi and Rakesh are climbing on a moving escalator that is going up. Ravi takes 30 steps to reach the top whereas Rakesh takes 32 steps for the same. This happens because Rakesh is faster than Ravi. For every 4 steps that Rakesh takes, Ravi takes only 3 steps. What is the total number of steps in the escalator?

If you notice the difference between the previous question and this one, the problem is that now we don’t have a definite amount of time taken so we have to solve this by ratios.

From the given data, we know that the ratio of speeds of Rakesh and Ravi is 4 : 3

To cover distance which is in the ratio 32 : 30 or 16:15, then will take times in the ratio = 16/4 : 15/3 = 4 : 5

If the total number of steps is n, in case of Rakesh the escalator covers ‘n – 32’ and in case of Ravi the escalator covers ‘n – 30’

These will be in the same ratio as the times taken by Rakesh and Ravi

=> (n – 32)/(n – 30) = 4/5

=> 5n – 160 = 4n – 120

=> n = 40 steps

Another variation of this type of problem could be when we are given the steps by only one person and they are moving in opposite directions. Let us look at that.

4. Ravi is climbing on a moving escalator that is going up and takes 30 steps to reach the top. Rakesh on the other hand is coming down on the same escalator. For every 5 steps that Rakesh takes, Ravi 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?

The extra information in this question is “Both of them take the same amount of time to reach the other end” and that is really the key to solving this question.

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 Ravi : Speed of escalator = 3s : s = 3 : 1

=> When Ravi 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.

Ravi going up would have taken 15 steps, whereas the escalator would have taken 5 steps for him.

Rakesh 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.

Another way of solving this question is,

Ravi has taken 30 steps in the full journey, whereas Rakesh has taken 30*(5/3) = 50 steps.

The difference in the number of steps in full journey = 50 – 30 = 20 steps.

The difference in the number of steps when they cross each other, which is exactly half of the journey = 20 / 2 = 10 steps.

I hope in this post I have covered most of the type of escalator problems which have been asked and using the above ideas and concepts you will be able to solve such problems easily in future.

Ravi Handa, an alumnus of IIT Kharagpur, has been teaching for CAT and various other competitive exams for around a decade. He currently runs an online CAT coaching and CAT Preparation course on his website http://www.handakafunda.com