How to know whether a question asks for an average rate from a time or distance perspective?

Question

I have a sample problem: Bob travels up a hill at $X$ miles an hour and down the hill at $Y$ miles an hour (You can assume that the length of the trail going up is the length of the trail going down in terms of distance).

If a question were to ask:

What is Bob's average speed in miles per hour?

How would I know whether the question is talking about the average miles per hour from a time perspective (where the answer would be the harmonic mean of $X$ and $Y$)?

$$\frac{2XY}{X+Y}$$

Or if the question is asking about the average miles per hour from a distance perspective (where the arithmetic mean would be the answer)

$$\frac{X+Y}{2}$$

To further clarify time perspective:

If $X = 3$, $Y = 12$ and the length of the hill is 3 miles:

The time it would take to go up the hill is one hour while the time it takes to go down the hill is 15 minutes. From a time perspective $X$ is four times more weighted than $Y$, not equally as weighted (they are equally weighted in the distance perspective since the distances up and down the hill are the same).

How would I go about a problem like the one "What is Bob's average speed in miles per hour?" My reason for this interest is due to my observation of a question that had the answer developed from a time perspective. I have also seen questions with the answers derived from a distance perspective.

Thank you.

Answer 1

The generally accepted meaning is:

$$ \text{average speed} = \frac{\text{total distance}}{\text{total time}} $$

Besides, the arithmetic mean of the two speeds has no particular meaning of interest in this scenario. For example, if Bob ran up the hill at X mph, then circled the top of the hill at Y mph for 30 seconds, the arithmetic mean of the two speeds would still be the same.

---

[EDIT] P.S. To clear up any confusion about harmonic mean possibly having anything to do with average rates in general, that's just an artifact of the given scenario - where Bob runs equal distances up and down the hill.

Suppose instead that Bob ran at $X$ mph for a time $T_1$, then at $Y$ mph for $T_2$. The total distance covered would be $X T_1 + Y T_2$, and the average speed is $(X T_1 + Y T_2) / (T_1 + T_2)$.

If - and only if - the distances are equal $X T_1 = Y T_2 = L$ then it so happens that the average speed calculates to $(L + L) / (L / X + L / Y) = 2 X Y / (X + Y)$ the harmonic mean.