sums of ave not equaling ave of sums

Question

Curious. In a word problem I am asked for the ave speed of a man going to work 45 miles in 70 mins, then home 45 miles in 90 mins (due to traffic). Find ave speed. If I take the total distance / total time, I get 90 miles/160 mins = 0.5625 miles/min = 33.75 MPH, the "correct" answer.

However, if I average the two speeds by adding 45 miles/70 mins and 45 miles/90 mins and divide by 2, I get an answer of 34.28571428571429 MPH which would have caused me to miss the correct answer on the choices given on the test. Why are the two averages not the same, or at least a lot closer, as I am calculating them to 14 decimal places on my computer?

Answer 1

You spent a total of 160 minutes driving, but only 70/160 going to work and 90/160 coming back, so the return time is weighted more than the other.  Instead of adding the two speeds and dividing by 2, you should compute a weighted average:

 

(45)(70/160)/(70/60) + (45)(90/160)/(90/60) = 33.75 mph