Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the 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).