Posted by: cochinblogger | December 27, 2014

Intuition and Math: A Feel for Averages

I found the following problem in my son’s (he was in the eighth standard) textbook: A man covers the distance from his house to his office at an average speed of 20 kmph and after work, he return home via the same route at an average speed of 30 kmph. What was his average speed over the entire journey?

My son’s answer was instantaneous: 25 kmph, the average of 20 and 30. And I unthinkingly nodded my assent.

But the correct answer is not 25 mph. In fact, my father (who happens to be a mathematician) remembers an irate letter to the editor in the New York Times in the 1950s or 1960s. The letter writer complained that the average speed the magazine had published (naturally, the figures in the magazine were different, but the same principle applies, so let’s pretend the numbers in the magazine were also 20 and 30) was wrong on grounds of elementary logic: The average of 20 and 30 is 25, and so the overall average is 25, and not the erroneous value (which I will soon unveil) the newspaper had published. In its carefully worded reply, the newspaper editor explained that the overall average was not 25, even though 25 was certainly the average of 20 and 30. I don’t know if this reply convinced the letter writer; I have my doubts about this. Some internalized notions are hard to eradicate.

So, what then is the overall average speed for the entire journey (house to office and back)? Let us assume the distance from the house to the office is x km. The time taken for this leg of the journey is the distance divided by the speed, x/20 hours. And the time taken for the return leg of the journey is x/30 hours. The total journey time is therefore (x/20 + x/30) = 5x/60 = x/12 hours. The average speed for the entire journey is the total distance divided by the total time: 2x/(x/12) = 2 X 12 = 24 kmph.

Our intuition is a wonderful faculty, but it can deceive. This simple speed averaging problem is a case in point. Those who want to explore this apparently paradoxical result in greater depth should use as the starting point the observation that 24 is the harmonic mean of 20 and 30. The arithmetic mean of 20 and 30, on the other hand, is 25.

I explained this as best as I could to my son, but he did not look very convinced. I’m sure many adults too will find this result hard to digest, as did the irate letter writer to the New York Times more than half a century ago.

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