Question 320373: Jim and Ryan run laps around a track which is 1 /4 of a mile long. If Jim runs h laps per hour, Ryan runs half as fast as Jim, and Jim starts running half an hour before Ryan, how many miles farther than Ryan has Jim run two hours after Jim starts?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 1 lap equals 1/4 of a mile.
Jim runs h laps per hour.
Ryan runs half as fast as Jim.
This means that Ryan runs .5*h laps per hour.
If Jim runs 5 laps, then Ryan has run 2.5 laps which is .5 * 5.
Jim starts running 1/2 hour before Ryan.
Total duration of Jim running is 2 hours.
This means that Jim has run 2 hours and Ryan has run 1.5 hours.
Jim runs for 2 hours at h laps per hour.
Ryan runs for 1.5 hours at .5*h laps per hour.
Total number of laps that Jim runs is 2 * h.
Total number of laps that Ryan runs is 1.5 * .5 * h = .75 * h.
Since each lap is 1/4 of a mile, this means that:
Total distance that Jim runs is 2 * h * .25 miles.
Total distance that Ryan runs is .75 * h * .25 miles.
Total distance that Jim runs simplifies to .5 * h miles.
Total distance that Ryan runs simplifies to .1875 * h miles.
Total distance that Jim runs farther than Ryan would be equal to:
(.5 * h) - (.1875 * h) = .3125 * h.
Jim runs .3125 * h miles farther than Ryan.
To understand what this means, substitute any value for h and see if it makes sense.
Let's assume that Jim runs 40 laps per hour.
That makes h = 40.
Jim runs for 2 hours.
Jim runs a total of 2 * 40 * 1/4 = 20 miles.
If Jim runs 40 laps per hour, then Ryan runs 20 laps per hour.
Ryan runs for 1.5 hours.
Ryan runs a total of 1.5 * 20 * 1/4 = 7.5 miles.
The formula says that Jim runs .3125 * h more miles than Ryan.
If h = 40, that equates to .3125 * 40 which equals 12.5 miles more than Ryan.
Since Jim runs 20 miles and Ryan runs 7.5 miles, that means that Jim has run 12.5 miles more than Ryan.
The formula holds true and so the answer is correct.
The answer is that Jim runs .3125 * h more miles than Ryan.
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