Consider this word problem: John can shovel snow from a walk in 4 minutes. Mary can shovel the same walk in 3 minutes. How long will it take them to shovel the walk together?
Ignore real-world assumptions (i.e. crashing into each other, etc.), how would you solve this problem?
An incorrect approach: Their average time is (3+4)/2 = 3.5 minutes, so working together, it would take (1/2)(3.5) = 1.75 minutes or 105 seconds.
A correct approach: 1/4 + 1/3 = 1/x, or 1/x = (3+4)/12 which implies x = 12/(3+4) = 1 5/7 minutes or 102 6/7 seconds.
First, would you even notice (or care about the small time differences?
Second, why does the correct approach produce a smaller time than the incorrect approach?
Third, if given other shoveling times for John and Mary, will the time difference between the two approaches always remain small?
And fourth, for given shoveling times, would the two approachs ever produce the same result?
Somehow, these types of rate problems have always fascinated me, while frustrating students...and the root lies in the distinctions between the arithmetic mean, the geometric mean, and possibly even the harmonic mean.
Not sure what they are...you have some reading to do!
Source: Based on Z. Usiskin's article in Math. Teacher, Sept. 1980