## Escalator Problems with Solutions in Time, Speed and Distance

Problems related to escalators baffle a lot of students preparing for CAT. Probably because not much time has been devoted to explaining this simple concept. Although we keep telling in our CAT classrooms continuously that escalator questions are similar to ‘upstream’ ‘downstream’ questions that students solve in time, speed and distance, the future MBAs still get confused because of the way that the questions are put. Today, we are going to see the theory behind escalator questions and solve them through both equations and ratios. In order to benefit the most, please solve these questions on your own before looking at the solutions.

An escalator is a moving stair, i.e. it moves continuously up or down. At any given point in time, **the total number of stairs are fixed in an escalator**. For example, for an escalator going up that is showing 50 steps, if 2 stairs disappear at the top, 2 stairs come out at the bottom. Most of the escalator questions involve people moving on an escalator. Remember that the speed of climbing stairs for a person does not vary whether the escalator is moving or still. For example, if Bantu can climb stairs at 2 stairs per second, his speed of climbing stairs on a moving escalator would stay the same.

### Common Problem Scenarios in Escalators

1. A person moving on a descending escalator and/or an ascending escalator and questions related to total number of steps, speed of the escalator etc.

2. Two persons moving in the same directions on an ascending/a descending escalator and questions related to total number of steps, speed of the escalator, speeds of either person etc.

3. Two persons moving in opposite directions on an ascending/a descending escalator and questions related to total number of steps, speed of the escalator, speeds of either person etc.

### Theory of Escalators in Time Speed and Distance

The basic thing to understand is that

1. if you’re moving ‘with’ the escalator, you’d have to climb less steps on your own because the escalator also will ‘push you forward’ some number of steps on its own. For example, if you climb 70 stairs on your own, and escalator pushes out 30 stairs in your favor in the same time, you’d have climbed 100 stairs in total.

2. If you’re moving ‘against’ the escalator, you’d have to climb more steps on your own because the escalator will try to ‘pull you back’ some number of steps on its own. For example, if you climb 70 stairs on your own, and escalator pushes out 30 stairs against you in the same time, you’d have climbed 40 stairs in total.

Remember!

**With the escalator = Less steps to climb than the total
Against the escalator = More steps to climb than the total**

### Moving With the Escalator

Suppose a person can climb P stairs/second and the escalator is **churning out** E stairs/second. Let the total number of stairs in the escalator is L. Let the person climb D stairs on his own on the escalator. The time taken to do this would be seconds. In this time, the escalator will churn out stairs.

Total number of steps = steps climbed oneself + steps produced by escalator OR

### Moving Against the Escalator

Given the same values as above, i.e. P, E, L, and D, the person is climbing more stairs than the total as he is working against the escalator. Therefore, the equation will be.

Total number of steps = steps climbed oneself – steps produced by escalator OR

Now let’s look at some problems!

**Problem 1:** A man and his wife walk up a moving escalator. The man walks twice as fast as his wife. When he arrives at the top, he has taken 28 steps. When she arrives at the top, she has taken 21 steps. How many steps are visible in the escalator at any one time.

**Problem 2:**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?

**Problem 3:**A famous mathematician who is always in a hurry walks up an up-going escalator at the rate of one step per second. Twenty steps bring him to the top. The next day he goes up at two steps per second, reaching the top in 32 steps. How many steps are there in the escalator?

**Problem 4:** A man can walk up a moving ‘up’ escalator in 30s. The same man can walk down this moving ‘up’ escalator in 90s. Assume that his walking speed is same upwards and downwards. How much time will he take to walk up the escalator, when it’s not moving?

**Problem 5:** Ram always walks down on a moving escalator to save time. He takes 50 steps while he goes down. One day due to power failure of 10 secs (when the escalator comes to a halt) he takes 9 secs more than usual time to get down. Find the visible steps of the escalator.

**Problem 6** Rohan walked down a descending escalator and took 40 steps to reach the bottom. Sohan started simultaneously from the bottom, taking 2 steps to every 1 step taken by Rohan. Time taken by Rohan to reach the bottom from the top is the same as time taken by Sohan to reach the top from the bottom.

**1.** How many steps more than Rohan did Sohan take before they crossed each other?

**2.** If Rohan were to walk at the speed of Sohan, what % of the initial time would he able to save?

**Problem 7:** Jordan wanted to climb down from the first floor to the ground floor of a shopping mall, whereas Karina wanted to climb up from the ground floor to the first floor. Both use the same escalator which was ascending from the ground floor to the first floor and both walked their respective destinations. Both of them started simultaneously from the top and the bottom of the escalator respectively and crossed each other after exactly 21 seconds. If instead, Karina had walked at of his speed while Jordan maintained his speed, they would have crossed each other after exactly 28 seconds from the start. Further if both Jordan and Karina had climbed up from the ground floor to the first floor using the same ascending escalator, the number of steps taken by Karina to reach the first floor would be 20% less than the number of steps taken by Jordan for the same.

**1.** If Jordan were to stand still on the same escalator, how long would it take for the escalator to take him from the ground floor to the first floor?

a. 42 sec b. 63 sec c. 84 sec d.105 sec e. None

**2.** Jordan walked down from the first floor to the ground floor using the same escalator. However, after some time the escalator stopped moving due to a power failure. Find the total time taken by Jordan to reach the ground floor, given that the time for which he walked on the moving escalator was the same as that for which he walked on the stationary escalator.

1.

2.

3.

4.

5.

**Problem 8:** Ann walked up a descending escalator and took 154 steps in 100 seconds to reach the top. Karen started simultaneously from the top, taking 3 steps for every 4 steps of Ann, and reached the bottom in 40 seconds.

**1.** How many steps from the bottom were they, when they crossed each other?

A. 22 B. 25 C. 30 D. 33 E. 36

**2.** Find the number of steps between Ann and Karen exactly seconds after they started walking on the escalator.

A. 21 B. 49 C. 28 D. 54 E. 42

**Problem 9:** On an upward moving escalator, Amar, Akbar and Anthony take 10 steps, 8 steps, and 5 steps, respectively, to reach the top. On the same upward moving escalator Amar takes 30 steps to come down from the top. Find the ratio of the time taken by Akbar and Anthony to reach the top.

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