The long distance runner starts at the beginning of a trail and runs at a rate of 5 miles per hour. Three hours later, a cyclist starts at the beginning of the trail and travels at a rate of 8 miles per hour.

Let the cyclist overtake the runner after...

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The long distance runner starts at the beginning of a trail and runs at a rate of 5 miles per hour. Three hours later, a cyclist starts at the beginning of the trail and travels at a rate of 8 miles per hour.

Let the cyclist overtake the runner after T hours.

The distance traveled by the runner when the cyclist overtakes him is 5*(T + 3) and the distance traveled by the cyclist is T*8

Equating the two, 5T + 15 = 8T

=> 3T = 15

=> T = 5

**The cyclist overtakes the runner after traveling for 3 hours.**