Question

A girl wants to count the steps of a moving escalator which is going up. If she is going up on it, she counts 60 steps. If she is walking down, taking the same time per step, then she counts 90 steps. How many steps would she have to take in either direction, if the escalator were standing still?

Answer 1

Let the speed of the girl be "s", and the speed of the escalator be "e".

While going up the escalator is making the girl go faster so the actual speed going up is: s+e

Going down, the escalator is going against her or slowing her down. The actual speed going down is: s - e

Let N be the amount of steps when escalator is not moving.
The number of steps can be found by multiplying speed by time.
The time is takes to go up will be less than time to go down because N is constant and speed is faster going up.

$N = (s+e) T_{u} \\ N = (s-e) T_{d}$
Now the 60 steps she counts going up represent the number of steps going speed "s" in time T_up. This is because N can be broken into 2 parts, the steps the girl does and the steps the escalator does. Likewise for going down.

$N = s T_{u}+e T_u = 60 + e(\frac{60}{s}) \\ \\ N = s T_{d}-e T_d = 90 - e(\frac{90}{s})$

Set 2 equations equal since both equal constant N. Solve for s:
$60 + e(\frac{60}{s}) = 90 - e(\frac{90}{s}) \\ \\ \frac{150}{s} = \frac{30}{e} \\ \\ s = 5e$

Substitute back into one of the equations:
$N = 60 + e(\frac{60}{5e}) \\ \\ N = 60 +12 \\ \\ N = 72$

Therefore if escalator is not moving, the girl would count 72 steps.