Alex’s Escalator Puzzle: Solving a Real-World Speed and Distance Problem
In a bustling mall, Alex frequently uses the escalator to move between floors. One day, he wonders about the number of steps on the escalator and the speed at which it moves. Given two scenarios, Alex walks up the escalator at different speeds and observes how long it takes him to reach the top. Let's explore how to calculate the number of steps and the speed of the escalator using mathematical reasoning.
Problem Statement
Alex walks up the escalator at a constant speed. If Alex walks at 120 steps per minute, he reaches the top in 30 seconds. If Alex walks at 80 steps per minute, he takes 40 seconds to reach the top. We need to find the total number of steps on the escalator and the speed of the escalator.
Mathematical Formulation
To solve this problem, let's introduce the following variables:
v speed of the escalator in steps per minute N total number of steps on the escalatorWhen Alex walks up at 120 steps per minute, the total time taken is 30 seconds (0.5 minutes). Thus, the total distance (number of steps) can be expressed as:
120v N/0.5 2NSimilarly, when Alex walks at 80 steps per minute, the total time taken is 40 seconds (2/3 minutes). Thus, the total distance (number of steps) can be expressed as:
80v N/(2/3) 3N/2By subtracting the second equation from the first, we can solve for N:
120v - 80v 2N - 3N/2 40v N/2 N 80Substituting N back into one of the original equations, we find the speed of the escalator:
120v 2N 120v 2(80) 120v 160 v 40Thus, the escalator has 80 steps, and it moves at a speed of 40 steps per minute.
Conclusion
By using mathematical reasoning, we have calculated the number of steps on the escalator and the speed at which it moves. This problem demonstrates the application of speed and distance concepts in real-world scenarios, making it a valuable exercise for understanding fundamental mathematical principles.