Average speed of a man walking up a hill.

Hello everyone, this problem was broadcast on BBC 4's prog 'More Or Less' last Christmas & has been puzzling me ever since:

"A man goes for a walk after his Christmas dinner up a hill behind his house at an average speed of 2 miles/hour. He then returns by exactly the same route at an average speed of 3 miles/hour. What is his overall average speed?"

It is *not* (2 + 3) / 2. If it *is* the square root of (2 x 3) = 2.45 mph, could you please explain the vectors involved & why, in spite of the mutually cancelling slopes, it is not simply a horizontal linear average of 2.5 mph.

This was supposed to be done in the head in 5 seconds after a Christmas dinner! Many thanks, Mike. (Aged 66). (Worried)

Reply & thanks to Samer guirguis 2000

Thanks very much Samer for your prompt & sucinct answer to my problem. I really appreciate it, I would never have cracked it for all my messing about. Regards Mike. (Nod)