Ya, OK. Not terribly difficult. But let's look at two answer choices in particular:
(A) (X+Y)/2
(B) (X+Y)/(2*X*Y)
(C) (2*X*Y)/(X+Y)
(D) (X+Y)/(X*Y)
(E) (X*Y)/(X+Y)
Answer choice A & C.
Answer choice A should be recognizable as the
arithmetic mean of X and Y. It does not apply to this question, however. Explanation below..
Now, how about answer choice C? This is known as the
harmonic mean of X and Y, and it does apply to this question.
Rates: Arithmetic Mean vs Harmonic Mean
What's the application of each?
Well first off, when dealing with average speeds as in the question at the top of this thread, in either case the following
concept applies:
- Average speed = (Total distance) / (Total time)
In this particular problem, the harmonic mean came in to play for calculating the average speed because
distances were equal, and the equation for harmonic mean was correctly derived in the solutions given by rijul007 and others in the thread.
The harmonic mean is also sometimes expressed in the format 2/((1/X)+(1/Y)) which is equivalent to answer choice C above, and sometimes
described as "The number of observations divided by the reciprocal of each number in the series." It can apply to any number of rates, e.g. 3 different rates (X,Y,Z) as: 3/((1/X)+(1/Y)+(1/Z))
How about arithmetic mean, then? Well, consider the following question:
Jeff rides his bike up a hill for H hours at a rate of X miles per hour, and then rides his bike back down the hill for H hours at a rate of Y miles per hour. What is his average speed for the entire trip, in terms of X and Y?
Notice now that
time is equal for both segments of the trip. So, in calculating average speed with the
concept above, we would have:
Average speed = (X*H + Y*H)/(2*H) = (X+Y)/2 = arithmetic mean of X and Y. Note that the H drops off.
So, in general, when calculating an average rate...
Arithmetic means yield an average of multiple rates that occur for the same duration.
Harmonic means yield an average of multiple rates over the same unit of measure (e.g. distance).
And, of course, the rates can be of any units the GMAT chooses. Miles per hour, kilometers per hour, gallons per hour, questions per minute, etc. See also:
https://www.beatthegmat.com/average-rate ... 00817.html
