Question

Two cars run at constant speeds around a one-mile racetrack. If the cars circle the track in the same direction, the faster car passes the slower car every 10 minutes. If the cars circle the track in opposite directions, the cars meet every 30 seconds. Find the speed of the faster car in miles per hour.

Note:
The racetrack is 'circular' in the sense that the start and end points are the same, and the cars are running laps around the racetrack at a constant speed.
If the track has circumference c miles, then and the car's speeds are x and y mi/s, then since time = distance/speed,

In the same direction, the slow car has a "lead" of 1 mile which the faster car has to make up.

1/(y-x) = 120
1/(x+y) = 30

120(y-x) = 30(x+y)
120y-120x = 30x+30y
90y = 150x
y = 5/3 x


1/(x + 5/3 x) = 30
8/3 x * 30 = 1
80x = 1
x = 1/80 mi/sec = 45 mph
y = 75 mph

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