Manhattan GMAT Challenge Problem of the Week – 27 July 2010

by on July 27th, 2010

Here is a new Challenge Problem! If you want to win prizes, try entering our Challenge Problem Showdown. The more people enter our challenge, the better the prizes.

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:

h(x, y) = 2/(1/x + 1/y)

g(x, y) = sqrt(xy)

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

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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3 comments

  • Hi,
    can u please share the name of winners?

  • Good that you asked from Manhattan folk.

    BTW, What is the difficulty in this problem. I can't see any.

    If suppose, two numbers are same, then obviously Arithmetic Mean, Geometric Mean and Harmonic are all same. One can solve in a 2 minutes time. No need to do any calculation.

    • A typo in above post
      -----
      Good that you asked from Manhattan folk.

      BTW, What is the difficulty in this problem. I can't see any.
      If suppose, two numbers are same, then obviously Arithmetic Mean, Geometric Mean and Harmonic are all same.

      One can solve in a 2 secs time. No need to do any calculation.

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