Questions...

Re: Mysteries  Who invented the harmonic mean and what is its purpose and why does it work?
@thomas This was a really good explanation! I really like how you explained why this is "weird", and why the usual arithmetic mean doesn't work  that because 12 mph is greater than 8 mph, so when traveling the same distance, you spend less time going at 12 mph, so the answer should be closer to 8 mph instead of right in between 8 and 12. Still, I don't really get why the harmonic mean works. Why do I want to find the average time it takes ((1/x+1/y)/2) to get the average speed?
In the last problem about the houses you included, is the answer supposed to be 1/(1/2+1/3)=6/5 hours? I don't understand why here the harmonic mean is not applied. Thanks!