Bob, Ted, Carol, and Alice are all running in a relay race together. Their total time for the race is 1 hour. Bob ran at a pace of 8 miles per hour for exactly one-quarter of the time. Ted ran for 5 minutes more than Bob did. Carol ran the same distance as Ted but at twice the speed. If the total distance the team ran was 5 miles, how fast was Alice running, in miles per hour?

Question

A. 3 mph               B. 4 mph  
C. 8 mph              D. 15 mph

Bot · Answer

If there were 1 hour total and Bob ran for exactly one-quarter of that, then Bob ran for 0.25*60 minutes = <<1*60*0.25=15>>15 minutes.
So Ted ran for 15+5 = <<15+5=20>>20 minutes.
There are 60 minutes total, so Carol and Alice ran for 60-15-20 = <<60-15-20=25>>25 minutes total.
Carol ran twice as fast as Ted, for the same distance. This means Carol took half the amount of time as Ted.
Thus, Carol ran for 20/3 = <<20/3=6.666666666666667>>6.67 minutes, approximately.
Thus, Alice ran for 25-6.67 = <<25-6.67=18.33>>18.33 minutes, approximately.
Bob was running at 8 miles per hour for 15 minutes, or 0.25 hours. So, Bob ran 8*0.25 = <<8*0.25=2>>2 miles.
Let x be Ted's speed in miles per hour. If he ran for 20 minutes, or 20/60 = 1/3 hours, then he ran x*(1/3) miles.
Since Carol ran the same distance as Ted, but at twice the speed, she ran at 2x miles per hour.
The distance Carol ran is the same as the distance Ted ran: x*(1/3).
Alice had not run yet, so the distance Bob, Ted, and Carol ran together is 2 + x*(1/3) + x*(1/3) = 5 miles.
Combining terms, we get 2+(2/3)*x = 5.
Subtracting 2 from both sides, we get (2/3)*x = 3.
Multiplying both sides by 3/2 to solve for x, we get x = 4.5.
Thus, Ted ran 4.5 miles per hour.
Since Carol ran twice as fast as Ted, her speed was 2*(4.5)=9 miles per hour.
Finally, we need to find out how fast Alice was running. We know she ran for 18.33 minutes, or 18.33/60≈0.306 hours.
Let y be Alice's speed in miles per hour. Then, she ran y*(0.306) miles.
Since Bob, Ted, Carol, and Alice ran 5 miles in total, the equation is 2 + 4.5*(1/3) + 9*(1/6) + 0.306*y = 5.
2 + 1.5 + 1.5 + 0.306*y = 5
Subtracting 3 from both sides, we get 0.306*y = 2.
Dividing both sides by 0.306, we get y = 2/0.306 ≈ 6.54.
Thus, Alice was running at approximately 6.54 miles per hour. There is no exact answer choice, but 6.54 is closer to 8 miles per hour than any other choice. Therefore, the answer is C. 8 mph.