Manhattan GMAT Challenge Problem of the Week – 27 July 2010

by Manhattan Prep, Jul 27, 2010

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As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 mins and go!

Question

The harmonic mean of two numbers x and y, symbolized as h(x, y), is defined as 2 divided by the sum of the reciprocals of x and y, whereas the geometric mean g(x, y) is defined as the square root of the product of x and y (when this square root exists), and the arithmetic mean m(x, y) is defined as (x + y)/2. For which of the following pairs of values for x and y is g(x, y) equal to the arithmetic mean of h(x, y) and m(x, y)?

A. x = -2, y = -1

B. x = -1, y = 2

C. x = 2, y = 8

D. x = 8, y = 8

E. x = 8, y = 64

Answer

We should be organized as we try to make sense of all the given definitions. First, translate the definitions into algebraic symbols:

[pmath]h(x, y) = 2/(1/x + 1/y)[/pmath]

[pmath]g(x, y) = sqrt(xy)[/pmath]

[pmath]m(x, y) [/pmath]is the normal arithmetic mean, [pmath](x + y)/2[/pmath]

Now, we are asked for a special pair of values for which the following is true: once we calculate these three means, we'll find that g is the normal average (arithmetic mean) of h and m. This seems like a lot of work, so we should look for a shortcut. One way is to look among the answer choices for "easy" pairs, for which h, g, and m are easy to calculate. We should also recognize that the question's statement can only be true for one pair; it must be different from the others, so if we spot two easy pairs, we should first compute h, g, and m for the "more different-looking" of the two candidate pairs. Scanning the answer choices, looking for an easy pair to calculate, our eye should be drawn to (D), since the two values are equal. If both x and y equal 8, then m is super easy to calculate: m also equals 8. Let's now figure out g and h. Since g is defined as the square root of xy, in this case g equals the square root of 64, so g = 8 as well. Finally, h equals 2/(1/8 + 1/8) = 2/(2/8) = 8. The arithmetic mean of h (= 8 ) and m (= 8 ) is also 8, which equals g. We can stop right now: there can only be one right answer.

The correct answer is (D).

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