Donna drove half the distance of a trip at 40 mih At what sp

Donna drove half the distance of a trip at 40 mi/h. At what speed would she have to drive for the rest of the distance so that the average speed for the entire trip would be 45 mi/h?

Solution

The obvious answer is 50, since (40+50)/2 = 45.
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In other words, she drove 1/2d * 40 + 1/2d * 50 = 20 + 25 = 45.
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The problem is that time has been ignored with this solution.
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Assume the distance to be traveled is 80 miles.
At 40 mph, the first half of the trip will take 1 hr @ 40 mph.
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To average 45 mph for the whole 80 miles, the total time is:
80 = 45t
45t = 80
t = 80/45 = 16/9 = 1 + 7/9 hrs.
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Since the first hour is gone, the remaining distance must be traveled in 7/9 hr.
40 = r * t = r * 7/9
360 = 7r
7r = 360
r = 51.428