@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!

]]>Similarly, in the challenge problem, since the speeds up and down are in the ratio 4:12, and the distances are the same, the times traveled should be in the "flipped" ratio 12:4. That's why it's reasonable to see that to get the final answer, a lot of "flipping" is involved!

Why do we flip again at the end, then?

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