What does the word average signify? Did you know Wikipedia lists at least 14 different kinds of averages? (http://en.wikipedia.org/wiki/Average)

The most common average is the arithmetic mean, but it is not always appropriate. Basil runs 800 metres at 8 metres per second and another 800 m at 7 metres per second. What is his average speed? Many ‘good’ students blithely add 8 and 7 and then divide by 2 to get 7.5. wrong!

Let’s take an extreme case. Extreme cases are an essential part of your mathematical toolbox. Suppose Basil runs his first 800 m at 8 m/s and his second 800 m at the glacially slow speed of 0. 0000…1 m/s. (Imagine as many zeros as you like before the one). Is the average speed (8 + 0)/2 = 4 metres per second? Certainly not!

Imagine another case. I am driving my car at 100 km per hour for 0.99999 …9 hours (or 59.999…9 minutes) and for a brief fleeting instant at 200 km per h. Is my average speed (100+200)/2 = 150 km/h? Certainly not!

Instead of the ordinary mean (a + b)/2, what the student needs is a weighted mean (w

_{1}× a + w_{2}× b)/( w_{1}+ w_{2}). But, what are the appropriate weights? The obvious (and correct answer) are the times spent at each speed. Then we get (t_{1}× s_{1}+ t_{2}× s_{2})/( t_{1}+ t_{2}) = (d_{1}+ d_{2})/( t_{1}+ t_{2}) = total distance/total time, as taught by the physics teachers.
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