Thank you for your question.
For the purpose of this question, it's not super important that you know why we use the word "harmonic." What's important is that you know what the harmonic mean is.
A mean, normally, is when we add up all the numbers, and divide them by how many numbers there are. For example, if we have the numbers 1,2, and 4, we add them up, and divide them by 3:
$$ \frac{1 + 2 + 4}{3} = \frac{7}{3} $$
When we talk about the harmonic mean we do something a little different. First, we take one over all the numbers (we "flip" them upside down). Then, we take that mean:
$$ \frac{\frac{1}{1} + \frac{1}{2} + \frac{1}{3}}{3} $$
Finally, we flip the whole thing again. A good way to think about this is that because we flipped them to start with, we have to flip the answer back, after we take the mean. This give us:
$$ \frac{1}{\frac{\frac{1}{1} + \frac{1}{2} + \frac{1}{3}}{3}} = \frac{3}{\frac{1}{1} + \frac{1}{2} + \frac{1}{3}} $$
This may look a little bit weird, because it is strange compared to the mean you may be used to. But as Po describes, this is actually really helpful for many problems in math, including this one!
About your question on why this works, try picking two letters instead of numbers, like "a" and "b," as speeds up and down the hill. Then try the problem again. You'll see you get the same formula! In the problem, we did this with a specific example (4 mph and 12 mph), but it didn't have to be those speeds, it could have been anything!
I hope this helps! You're asking great questions. Feel free to reach out if you ever get stuck, and keep doing a great job learning!
Happy Learning!
The Daily Challenge Team