Solution:

If he runs up 8 steps, then he needs 52.5 seconds to reach the top.
If he runs up 15 steps, then he needs 35.0 seconds to reach the top.

The 7 additional steps take 17.5 seconds.
Therefore, each step takes 2.5 seconds.

Total steps in escalator = 8 + 52.5 / 2.5 = 29 or Total steps in escalator = 15 + 35.0 / 2.5 = 29.

If Brian did not run up any steps at all,
he would reach the top of the escalator in 72.5 seconds (i.e., 29 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 − 8)/52.5 = (x − 15)/35.0

The above equation may be solved as follows.

35.0 (x − 8) = 52.5 (x − 15); or
x = (15 × 52.5 − 8 × 35.0) / (52.5 − 35.0) = 29.

Now, Speed of escalator = (29 − 8)/52.5 = (29 − 15)/35.0 = 1/2.5 steps/second.
Time to reach top = Total Steps / Speed = 72.5 seconds.

Food for thought:

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