A driver averaged 50 km/h on the roundtrip between home and a city d kilometers away. The average speeds for going and returning were x and y km/h, respectively.
I know how you arrive at the equation in the title, but I do not know how it is supposed to be correct. From my understanding, the equation is supposed to map the going speed to the returning speed, but if I insert something like 60 or 70 in the function, then I get a value that does not give me the required average of 50km/h when added to x and divided by two ( average speed = (going speed + returning speed)/2 ).
I know that I am most likely missing something, and verily I hope that this is not one of those questions that makes you feel extremely dumb when you get the answer. However, I have been trying to understand this for about an hour and I just can't see how the function is correct.
Thanks for reading or answering!
P.S I don't know exactly what tags I should be adding to the question here. The problem is in a calculus textbook in the infinite limits section of the limits chapter. So, feel free to add the proper and tags remove the present tags if they are not the proper ones.
The average speed on a trip is the total distance divided by the total time. It is not the average of the speeds on the two legs. It is in fact the harmonic mean of the two speeds. It is also the mean of the two speeds weighted by the time spent at each speed.
As an example, if the trip is $150$ km each way and you do the outbound leg at $30$ km/hr it takes $5$ hours. To do $300$ km at $50$ km/hr you only have a total of $6$ hours, so must do the return trip in $1$ hour at an average of $150$ km/hr. Plugging into your formula, $$y=\frac {25x}{x-25}=\frac {25 \cdot 30}{30-25}=\frac {750}5=150$$