Puzzle #139: Ice, Slush, and Mud Rally

The weather is somewhat sloppy, but some people decided to have a rally in the ice, slush, and mud.

There were:

1) four types of vehicles: four-, three-, two-, and one-wheeled vehicles,
2) 400 wheels where the rubber meets the road (or whatever it was),
3) three times as many three-wheeled vehicles as one-wheeled vehicles,
4) four times as many four-wheeled vehicles as two-wheeled vehicles, and
5) a single-digit number of one-wheeled vehicles.

How vehicles were there of each type?

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, February 3, 2016 at noon Pacific Time. I will post the answer shortly after.