How many MPH does a man have to run on the way back to average 6MPH for the entire trip?

Question

A man runs a race that requires a roundtrip back to the starting line once he reaches the 'half-way' mark. He averages 3MPH to the half-way mark. How fast does he have to run on the way back to average 6MPH for the entire trip?

Hint: It's not 9MPH.

MPH=Miles per Hour (for non U.S. individuals)

Answer 1

It's not possible. In order to average 6mph for the entire race, he would have to run the entire race in the time it took just to run the first half.

For simplicity's sake, let's say the race is 6 miles long (3 miles out and 3 miles back). Running the entire race at an average speed of 6mph would take exactly one hour. If the man runs the first half of the race (3 miles) at 3mph, it would take him 1 hour just to get to the halfway mark. All of the time needed to average 6mph has already been used up.

This scenario holds up no matter how long the race is and no matter how long it takes to run the first half at 3mph.
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Answer 2

It is impossible for him to average 6 MPH if he goes 3 mph for the first half of the race.

If the halfway mark is 3 miles away, and it takes him 1 hour to get there, it will be impossible for him to average 6 mph because the entire race would be 6 miles and he already has spent an hour just going 3 miles.

Answer 3

Actually, it can be 9 mph. All you have to do is to take a distance average rather than a time average. It's a common way of winning money from wisacres who try to get a bet on this.