Let d be the distance covered by Pete when he met Johnny.
Let V be Pete's speed and v be Johnny's speed before the fight.
The sum of the distances covered at the time they met is x.
d + (d-200) = x (1)
Hence x = 2d - 200
When they met, each had walked for the same length of time:
d/V = (d - 200)v (2)
After the fight, Pete walked for 8 minutes:
(d - 200)(V/2) = 8
Hence V = (d - 200)/4 (3)
and Johnny for 18 minutes:
d(v/2) = 18
Hence v = d/9 (4)
By substituting the values for V and v into equation (2), we have
4d/(d - 200) = 9(d - 200)/d
Hence
5d2 - 3600d + 360,000 = 0
Solving this quadratic in the classic way yields
d = 600 or 120 and x = 2d - 200 = 1000 or 40
However, d must be smaller than x, so only one solution is
possible:
d = 600 and x = 1000
The distance between the two bars is exactly 1000 meters.