Calculating the Number of Steps on a Stationary Escalator

In this article, we will explore a unique problem involving an escalator and how to calculate the number of steps on a stationary escalator using a step-by-step approach. This problem involves understanding the relationship between the movement of the escalator and the steps taken by individuals moving up or down it.

Introduction to the Problem

Consider an escalator on which two individuals, A and B, are moving. Individual A takes 120 steps to reach the bottom of the escalator, which is moving upwards. Simultaneously, individual B takes 60 steps to reach the top of the escalator. Interestingly, A takes two steps for every step taken by B. Our task is to find the number of steps visible on the escalator when it is stationary.

Setting Up the Problem with Variables

To solve this problem, we will set up a series of relationships between the given variables and unknowns. Let's define the following variables:

N: The total number of steps visible when the escalator is stationary. v_e: The speed of the escalator in steps per second. v_a: The speed of individual A in steps per second. v_b: The speed of individual B in steps per second.

Given Information and Relationships

The given information is:

A takes 120 steps to reach the bottom of the escalator moving against it. B takes 60 steps to reach the top of the escalator moving with it. A takes two steps for every step of B, i.e., v_a 2v_b.

Establishing Relationships

Effective Speeds

When A is moving down, the effective speed is v_a - v_e. When B is moving up, the effective speed is v_b v_e.

The time taken by A to reach the bottom can be expressed as:

t_a 120 / (v_a - v_e)

The time taken by B to reach the top can be expressed as:

t_b 60 / (v_b v_e)

Equate Times and Solve

Since A and B are moving simultaneously over the same duration, we can equate t_a t_b:

120 / (v_a - v_e) 60 / (v_b v_e)

Substituting v_a 2v_b into the equation:

120 / (2v_b - v_e) 60 / (v_b v_e)

Cross-Multiplication and Simplification

By cross-multiplying, we get:

120(v_b v_e) 60(2v_b - v_e)

Simplifying:

120v_b 120v_e 120v_b - 60v_e

Rearranging the terms:

180v_e 0

This indicates an error in our initial simplification. Let's re-evaluate the equation:

120v_b 120v_e 120v_b - 60v_e

Rearranging:

180v_e 0

This should be simplified to:

180v_e 120v_b - 60v_e

180v_e 60v_e 120v_b

240v_e 120v_b

v_e v_b / 2

Calculate Steps on the Escalator

The total number of steps on the escalator when it is stationary is N. We can derive N using the relationship:

N 120 / (v_a - v_e)

Substituting v_a 2v_b and v_e v_b / 2:

N 120 / (2v_b - v_b / 2)

N 120 / (1.5v_b - v_b / 2)

N 120 / (1.5v_b - 0.5v_b)

N 120 / v_b

Using the relationship v_b 60 / 60 (since B takes 60 steps to reach the top), we get:

N 120 / (60 / 60)

N 180

Conclusion

The total number of visible steps on the escalator when it is stationary is 180 steps. This solution demonstrates how to effectively use algebraic relationships and step-by-step calculations to solve a complex problem involving moving escalators and individuals.

Additional Insights

Understanding the dynamics of an escalator and the impact of movement can help in various real-world applications, such as building design and user experience optimization. By breaking down the problem into manageable steps and using algebraic relationships, we can solve complex problems involving motion and kinetics.

Feel free to explore more such problems and apply the same methodology to enhance your understanding and problem-solving skills.