Since the distance of the race doesn't matter--it's not mentioned in the question or answer choices--keep it simple and say that the race is 100 miles long. That means the first x percent of the race is x miles long, and the remainder of the race is 100 - x miles.
To find an average speed, we need total distance and total time. We've just decided that total distance is 100 miles. Total time is the sum of the time spent running at each speed. For the x miles run at 6 miles per hour:
time = distance/rate = x/6
For the 100 - x miles run at 8 miles per hour:
time = distance/rate = (100 - x)/8
Now we can set up the fraction for average speed:
total distance / total time = 100 / [(x/6) + (100-x)/8]
Start by simplifying the denominator:
(x/6) + (100-x)/8
= 4x/24 + (300-3x)/24
= (4x + 300 - 3x)/24
= (x + 300)/24
Put that back in the fraction:
100 / [(x + 300)/24]
= 2400/(x + 300), choice (E).