A car and a motorcycle leave at noon from the same location, heading in the same direction. The average speed of the car is 30 mph slower than twice the speed of the motorcycle. In two hours, the car is 20 miles ahead of the motorcycle. Find the speed of both the car and the motorcycle, in miles per hour.

Let's call the speed of the motorcycle ' M '.

Twice the speed of the motorcycle is 2M.

30 mph slower than that is (2M - 30) ... the car's speed.

Distance = (speed) x (time)

In 2 hours, the motorcycle covers 2 (M) miles.

In 2 hours, the car covers 2 (2M - 30) miles.

You said that the car is 20 miles ahead of the motorcycle,

so we can write

2 (2M - 30) = 2M + 20

Eliminate the parentheses

on the left: 4M - 60 = 2M + 20

Subtract 2M from each side: 2M - 60 = 20

Add 60 to each side: 2M = 80

Divide each side by 2 : M = 40

The motorcycle's speed = M = 40 miles per hour.

The car's speed = (2M - 30) = 50 miles per hour.