Solution:

If he runs up 8 steps, then he needs 30.0 seconds to reach the top.
If he runs up 13 steps, then he needs 22.5 seconds to reach the top.

The 5 additional steps take 7.5 seconds.
Therefore, each step takes 1.5 seconds.

Total steps in escalator = 8 + 30.0 / 1.5 = 28 or Total steps in escalator = 13 + 22.5 / 1.5 = 28.

If Darren did not run up any steps at all,
he would reach the top of the escalator in 42.0 seconds (i.e., 28 steps × 1.5 seconds/step).

Alternative Solution through Equations:

Let the total number of steps in the escalator be x.
The escalator moves at a constant speed given by

Speed of escalator = (x − 8)/30.0 = (x − 13)/22.5

The above equation may be solved as follows.

22.5 (x − 8) = 30.0 (x − 13); or
x = (13 × 30.0 − 8 × 22.5) / (30.0 − 22.5) = 28.

Now, Speed of escalator = (28 − 8)/30.0 = (28 − 13)/22.5 = 1/1.5 steps/second.
Time to reach top = Total Steps / Speed = 42.0 seconds.

Food for thought:

Does it matter whether the escalator is moving up or down?