Solution:

If he runs up 7 steps, then he needs 48.0 seconds to reach the top.
If he runs up 13 steps, then he needs 39.0 seconds to reach the top.

The 6 additional steps take 9.0 seconds.
Therefore, each step takes 1.5 seconds.

Total steps in escalator = 7 + 48.0 / 1.5 = 39 or Total steps in escalator = 13 + 39.0 / 1.5 = 39.

If Michael did not run up any steps at all,
he would reach the top of the escalator in 58.5 seconds (i.e., 39 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 − 7)/48.0 = (x − 13)/39.0

The above equation may be solved as follows.

39.0 (x − 7) = 48.0 (x − 13); or
x = (13 × 48.0 − 7 × 39.0) / (48.0 − 39.0) = 39.

Now, Speed of escalator = (39 − 7)/48.0 = (39 − 13)/39.0 = 1/1.5 steps/second.
Time to reach top = Total Steps / Speed = 58.5 seconds.

Food for thought:

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