You know the relationship between speed and distance and time, because it is written on every speed limit sign you pass:
... speed = miles/hour = distance/time
You don't know the distance, but you know speed and time. Solving the above equation for time, we get
... time = distance/speed
You don't know the distance, but you can write an equation based on what you do know.
... time going = distance/460
... time returning = distance/500
... 4.8 = time going + time returning = distance/460 + distance/500
Performing the addition, you get
... 4.8 = distance(500 + 460)/(500·460)
And solving for distance, you get
... distance = 4.8·500·460/(500+460) = 1150
Putting this into our expressions for time going and returning, we have
... time going = 1150/460 = 2.5 . . . hours
... time returning = 1150/500 = 2.3 . . . hours
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Once you realize the travel times are inversely proportional to the speeds, you can see that ...
... time going : time returning = 500 : 460 = 25 : 23 . . . . . the reverse of the speed ratio
That is, the time going is 25/(25+23) = 25/48 of the total time, so is
... (25/48)·4.8 hours = 2.5 hours
Of course, the return time is (23/48)·4.8 = 2.3 hours, or the difference
... 4.8-2.5 = 2.3 hours.
