I've been trying to crack this, but I've just given up at this point. The problem sounds like this:
The escalator is always going down. When a boy runs up it, he counts 150 steps, and when he runs down it, he counts 30 steps. The escalator and the boy do not change their speed. How many steps would the boy count if the escalator were not to move?
I've tried many ways but i always seem to get that the discriminant is less than 0. Does anyone have an idea on how to solve this?
Let $n$ be the number you are looking for, it is the "number of steps of the static escalator."
Assume that the boy runs $b$ steps per second (or any other time period of your choice) and the escalator moves down $e$ steps per second. Then the boy needs $\frac{150}b$ seconds to reach the top when running up. During this time, the escalator moved down $150 \frac eb$ steps, so
$$150-150\frac eb = n.$$
Analogously,
$$30+30\frac eb = n.$$
This can be seen as a linear system of equations in the unknowns $\frac eb, n$. Your favorite way of solving linear systems of equations will then give the solution
$$\frac eb = \frac23, \quad n = 50,$$
so that the answer you are looking for is $n=50$.