A girl wants to count the steps on a moving escalator going up. She counts 60 steps going up and 90 coming down. How many steps would she have to take in either direction if the escalator were stationary?

 To find the number of steps on the escalator when it is stationary, we need to consider the speed of the escalator and the speed of the girl.

Let's denote: - The number of steps on the escalator as S - The speed of the girl as G steps per minute - The speed of the escalator as E steps per minute

When the girl is going up the escalator, both her speed and the speed of the escalator are working together to move her up. So, the effective speed is (G+E).

When she is going down, her speed is working against the speed of the escalator. So, the effective speed is (G-E).

We know that Time (T) = Distance (D) divided by Speed (S). Since the number of steps (D) is constant, we can say that the time taken to cover a certain number of steps is inversely proportional to the speed of the girl relative to the escalator.

So, T_up * (G+E) = S (1) T_down * (G-E) = S (2)

Here, T_up is the time she takes to go up and T_down is the time she takes to go down.

Don't confuse T (Time) with the actual number of steps (S), which is constant. Here, T_up and T_down represent the number of steps the girl actually takes relative to the escalator's movement.

Given: T_up = 60 steps T_down = 90 steps

We can now set two equations equal to each other, as both equal the total number of steps S when the escalator is stationary.

T_up * (G+E) = T_down * (G-E)

60(G+E) = 90(G-E)

If we solve for G and E, we’ll have two equations and two unknowns. But we actually don't need to find the values of G and E individually. We are only interested in the number of steps, S.

To eliminate G and E, we assume the escalator is stationary (E=0). In this case, the girl's relative speed does not change (since it's only her moving now), so the number of steps she counts should be the same in both directions.

Therefore, S should be the harmonic mean of 60 and 90 because when the escalator moves, it affects the total count in opposite ways. The harmonic mean of 60 and 90 is given by:

Harmonic Mean (S) = 2 / (1/T_up + 1/T_down) = 2 / (1/60 + 1/90) = 2 / (3/180 + 2/180) = 2 / (5/180) = (2 * 180) / 5 = 360 / 5 = 72

Thus, if the escalator were stationary, the girl would have to take 72 steps in either direction.