Two trains started from the same station at the same time and traveled in opposite directions. after traveling 4 hours, they were 480 miles apart. the rate of the fastest train exceeded the rate of the slow train by 10 mph. find the rate of each train.

The two trains are separating at the rate of 480 mi/(4 h) = 120 mi/h. Then their average speed is (120 mi/h)/2 = 60 mi/h. The faster train will have this speed plus half the difference. The slower train will have this speed less half the difference.

The fastest train is traveling 65 mph.
The slowest train is traveling 55 mph.


If you want an equation, you can call the speed of the slowest train "s". Then that of the fastest train is (s+10) and their sum is
  s +(s +10) = 120
  2s = 110
  s = 110/2 = 55
  s+10 = 65

Okay so it will be 1123