Oops!!
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I made the same mistake as him, by finding that for the first half he had 105, and the second half he had 120, so our answer is 105+120=225. But, why is it the harmonic mean that works? I'm still so confused! Also, maybe there is an overlap in the first half and the second? Also, for anyone who doesn't know what the harmonic mean is, it is
$$\text{harmonic mean}=\frac{1}{\frac{2}{x}+\frac{2}{y}} $$where x and y are the speeds for a half of the roundtrip right?
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@professionalbronco That's close! The harmonic mean is actually equal to 2/(1/x + 1/y), or 1/(1/(2x) + 1/(2y)).
To make it a bit clearer why the average speed is the harmonic mean, we can derive it from the beginning...
So again, we go half the distance at x km/hr and half at y km/hr. Let's say that the total distance is k km. Then, we go (k/2) km at the speed x km/hr and (k/2) km at the speed y km/hr.
And we want to find the average speed-- well, that's just the (total distance)/(total amount of time). We already know (total distance) = k, so let's find (total amount of time): it's equal to (total amount of time going the first k/2 km) + (total amount of time going the second k/2 km).
Well, for the first k/2 km, we see that it takes (k/2) km * hr/x km = k/(2x) hr. And the second k/2 km takes k/(2y) hr. Meaning, we have a total time being k/(2x) + k/(2y).
Finally, the average speed is k/(k/(2x) + k/(2y)) = 1/(1/(2x) + 1/(2y)) = 2/(1/x + 1/y), or the harmonic mean of x and y.
So that's another way to figure it out, besides the faster but maybe a little less intuitive way Prof. Loh brings up in the video. The good thing is, once you understand how it's derived, then you can apply it without deriving every single time
Hope this helped!
(Also, as a general sidenote: you can take the harmonic mean of more than 2 numbers-- in which case, the harmonic mean equals (the # of #'s)/(the sum of all the reciprocals of the numbers), which does come out to be 2/(1/x + 1/y) for 2 terms)
