Average speeds of back and forth trip

1MileCrash

1. The problem statement, all variables and given/known data

You drive on Interstate 10 from San Antonio to Houston. Half the time at 55 km/h, and the other half at 90 km/h. On the way back, you travel half the distance at 55 km/h and the other half at 90km/h.

What is the average speed of the trip from San Antonio to Houston?
What is the average speed of the return trip from Houston to San Antonio?
What is the average speed of the whole trip?


3. The attempt at a solution

From San Antonio to Houston, half the time is traveled between the two speeds.

So:

d / (1/2)(d/55) + (1/2)(d/90)
68.27586 km/h average speed.

From Houston to San Antonio, half the distance is traveled at each speed.

((1/2)55t + (1/2)90t) / t
=(1/2)55 + (1/2)90
=
72.5 average speed.


My book seems to have the two answers reversed. Did I do these correctly?

SammyS

You said, "From San Antonio to Houston, half the time is traveled between the two speeds."

I added the extra -- necessary -- parentheses in the following.
d / ((1/2)(d/55) + (1/2)(d/90))
What is (d/2)/55 ? ... It's the time required to travel a distance d/2 at a speed of 55...
etc ...

d/(total time) is average speed.

So this is not the case of: "half the time is traveled between the two speeds".

1MileCrash

I am not sure what you are saying..

Does my work not agree with total distance / total time? Why are you telling me that?

Also, why isn't it a case of traveling half the time at each speed? That's explicitly what the problem says.

SammyS

Because d/2 is half the distance !

So you are finding the time for half the distance at 55 + half the distance at 90.

1MileCrash

Now I'm really confused. If time is (d/r), (1/2)(d/r) or (d/2r) is half of the time is it not??

Wait, I think I'm starting to get it. d/r is the time spent traveling d at r, so half of it is not half of the total time, only half of the time spent traveling d at r.

Thanks!