Solution:

If he runs up 6 steps, then he needs 45.0 seconds to reach the top.
If he runs up 14 steps, then he needs 25.0 seconds to reach the top.

The 8 additional steps take 20.0 seconds.
Therefore, each step takes 2.5 seconds.

Total steps in escalator = 6 + 45.0 / 2.5 = 24 or Total steps in escalator = 14 + 25.0 / 2.5 = 24.

If Michael did not run up any steps at all,
he would reach the top of the escalator in 60.0 seconds (i.e., 24 steps × 2.5 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 − 6)/45.0 = (x − 14)/25.0

The above equation may be solved as follows.

25.0 (x − 6) = 45.0 (x − 14); or
x = (14 × 45.0 − 6 × 25.0) / (45.0 − 25.0) = 24.

Now, Speed of escalator = (24 − 6)/45.0 = (24 − 14)/25.0 = 1/2.5 steps/second.
Time to reach top = Total Steps / Speed = 60.0 seconds.

Food for thought:

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