Expert: Paul Klarreich - 11/22/2005

Question
How exactly does one find the average of two speeds travelled over equal distance.  For example, if I travel from point A to point B at a rate of 80 km/h and then travel back to point A at a rate of 100 km/h, what is my average speed for the whole trip?  The obvious, but incorrect, thing to do would be to add 100 and 80 and then divide by 2, giving the result 90.  How then does one go about solving a problem of this nature where only the speeds are known, but the time and distance are not?

Answer
Hi, Jeremy,

--------- YOUR QUESTION ---------------
How exactly does one find the average of two speeds travelled over equal distance. For example, if I travel from point A to point B at a rate of 80 km/h and then travel back to point A at a rate of 100 km/h, what is my average speed for the whole trip? The obvious, but incorrect, thing to do would be to add 100 and 80 and then divide by 2, giving the result 90. How then does one go about solving a problem of this nature where only the speeds are known, but the time and distance are not?
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There is an old 'trick' question that used to make the rounds:  Johnny travels the 100 miles to Albany at 25 miles/hour.  (In those days, there were no interstates.) How fast must he drive the return trip to average 50 mph for the round trip?  

Average speed is simply total distance divided by total time.  There really is no difficulty here, even though we don't know the distance.  (Actually there is something called the 'harmonic mean' that we can use.  More on this later.)

Since D = vt, we can always write  t = D/v, or v = D/t, depending on what we need.

Just assume the distance from A to B is D.  Then:

From A to B:  v1 = 80, so t1 = D/80.
From B to A:  v2 = 100, so t2 = D/100

Total distance = 2D.
Total time = t1 + t2 = D/80 + D/100
               total distance         2D
Average speed = --------------- = ------------
                 total time      D/80 + D/100

Now clear fractions -- multiply by the LCD of 80 and 100, which is 400.
                 800D    800D
Average speed = ------- = ---- = 800/9 = 88 1/9 kph.
               5D + 4D    9D

Nice of those D's to cancel, isn't it?  So we really didn't need the distance after all.  In fact, we could have just assumed the distance to be 1 km, and written:

               total distance         2
Average speed = --------------- = ------------
                 total time      1/80 + 1/100

That expression is called the Harmonic Mean of 80 and 100.  You can see that the 'average' value of 90, which is the Arithmetic Mean, gives the wrong answer.

And about that drive to Albany.  It is impossible for Johnny to average 50 mph.  On a basic level, 200 miles at 50 mph would require making the trip in 4 hours.  But he already took 4 hours to do 100 miles at 25 hours, so he would have to be 'beamed' back home to average 50 mph.

Or you can try solving for v2 in the harmonic mean and find it must be infinite.