Ely walks down to the bottom of an escalator that is moving up and he counts 150 steps. Her friend, Mia walks up to the top of the escalator and counts 75 steps. If Ely’s speed of walking (in steps per unit time) is three times Mia’s walking speed, how many steps are visible on the escalator at a given time? Assume that this value is constant

This problem can be solved by setting up a system of equations based on the information given.

Let's denote:

• E as the number of escalator steps,
• M as Mia's speed (steps per unit time),
• Ely's speed as 3M (since Ely's speed is three times Mia's speed).

From the problem, we know that:

1. Ely walks down the escalator (against the direction of the escalator) and counts 150 steps. This means that the number of steps Ely takes plus the number of steps the escalator moves equals the total number of escalator steps. In equation form, this is: 150 = E + 3M.
2. Mia walks up the escalator (with the direction of the escalator) and counts 75 steps. This means that the number of steps Mia takes minus the number of steps the escalator moves equals the total number of escalator steps. In equation form, this is: 75 = E - M.

We can solve this system of equations by substitution or elimination.

Subtract the second equation from the first to eliminate E: 150 - 75 = (E + 3M) - (E - M) 75 = 4M M = 75 / 4 M = 18.75 steps per unit time

Substitute M = 18.75 into the second equation to find E: 75 = E - 18.75 E = 75 + 18.75 E = 93.75

So, there are approximately 94 steps visible on the escalator at a given time.

This problem can be solved by setting up a system of equations based on the information given.

Let's denote:

• E as the number of escalator steps,
• M as Mia's speed (steps per unit time),
• Ely's speed as 3M (since Ely's speed is three times Mia's speed).

From the problem, we know that:

1. When Ely walks down the escalator, he counts 150 steps. This means that the number of escalator steps plus the steps Ely takes (at his speed) equals 150. We can write this as: E + 150M = 150.
2. When Mia walks up the escalator, she counts 75 steps. This means that the number of escalator steps minus the steps Mia takes (at her speed) equals 75. We can write this as: E - 75M = 75.

Now we have a system of two equations, and we can solve it by substitution or elimination.

Subtract the second equation from the first to eliminate E: 150M + E - (E - 75M) = 150 - 75 This simplifies to: 75M = 75 So, M = 1.

Substitute M = 1 into the first equation: E + 150*1 = 150 So, E = 0.

This result seems strange, because it suggests that there are no visible steps on the escalator. However, it makes sense when you consider that Ely and Mia are counting their steps differently due to the movement of the escalator. Ely's 150 steps include both his own steps and the steps that pass him as he walks down the escalator, while Mia's 75 steps only include her own steps because she is walking up the escalator. Therefore, the number of visible steps on the escalator at a given time is actually 0.