A long-distance runner started on a course at an average speed of 8 mph. Half an hour later, a second runner began the same course at an average speed of 10 mph. How long after the second runner starts will the second runner overtake the first runner?

Number:

Using the Distance Formula, D = RT where D is the distance, R is the rate of motion, and T is the time.
Hence, Distance for runner 1 is D1 and distance for runner 2 is D2.
Since the two runners are said to travel at the "same course", distance is equal; D1 = D2.
So these are the derived equations:
D1 = RT
D1 = 8 (x + 1/2)
D2 = RT
D2 = 10 (x) ; since runner 2 travelled the least amount of time (since he started late) we call his T as x.
Since D1 = D2 it gives;
8 (x + 1/2) = 10x
Solving for X:
8x + 4 = 10x
4 = 10x - 8x
4 = 2x
2 = x
Therefore, after 2 hours that the second runner starts, the second runner overtakes the first runner.
CHECK:
Distance of first runner:
D1 = 8(x + 1/2)
D1 = 8x + 4
D1 = 8(2)+4
D1 = 20 miles
D2 = 10x
D2 = 10(2)
D2 = 20 miles
This means that after 2 hours that the second runner starts, he will be on equal distance (20 miles) with the first runner. Since the second runner is running at a faster rate, he will start to overtake the first runner.

Using the Distance Formula, D = RT where D is the distance, R is the rate of motion, and T is the time.
Hence, Distance for runner 1 is D1 and distance for runner 2 is D2.
Since the two runners are said to travel at the "same course", distance is equal; D1 = D2.
So these are the derived equations:
D1 = RT
D1 = 8 (x + 1/2)
D2 = RT
D2 = 10 (x) ; since runner 2 travelled the least amount of time (since he started late) we call his T as x.
Since D1 = D2 it gives;
8 (x + 1/2) = 10x
Solving for X:
8x + 4 = 10x
4 = 10x - 8x
4 = 2x
2 = x
Therefore, after 2 hours that the second runner starts, the second runner overtakes the first runner.
CHECK:
Distance of first runner:
D1 = 8(x + 1/2)
D1 = 8x + 4
D1 = 8(2)+4
D1 = 20 miles
D2 = 10x
D2 = 10(2)
D2 = 20 miles
This means that after 2 hours that the second runner starts, he will be on equal distance (20 miles) with the first runner. Since the second runner is running at a faster rate, he will start to overtake the first runner.

## Using the Distance Formula, D

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