Let their rates are "u" and "v" ft per second, where "u" stands for the faster track-man.

Then the first equation is 

 = u + v.          (1)

This equation is for the case when they are running in opposite directions.
     (Paradoxically, moving in opposite directions means moving towards each other along the circumference in this case !)
The left side says that they together cover 300 ft in 10 seconds.
The right side is the rate of decreasing the distance between them measured along the circumference. 
   (Thinking on this equation, keep in mind that they meet for every circle counting after the first meeting).

So the first equation is 

u + v = 30.             (1')


The second equation is 

 = 300.    (2)

This equation is for the case they are running in one direction, and the equation says that for the faster track-man 
the path from one meeting point to the next meeting point is in one full circumference longer than for the slower track-man.
 
The equation (2) is equivalent to

u - v = 6.         (2')

The equations (1') and (2') are the governing equations, and we can easily solve them by adding. Doing so, you get

2u = 30 + 6  --->  2u = 36   --->  u =  = 18.

So the faster track-man speed is 18 ft/s.

Then the slower track-man speed is 30-18 = 12 ft/s.

Check.   =  = 10 seconds.

         =  = 50 seconds.

Answer.  18 ft/s  and  12 ft/s.