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S96-14

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Math Expert
Joined: 02 Sep 2009
Posts: 42618

Kudos [?]: 135771 [0], given: 12708

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16 Sep 2014, 00:51
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Difficulty:

45% (medium)

Question Stats:

69% (01:54) correct 31% (02:54) wrong based on 29 sessions

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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 $$\frac{x + y}{2}$$. 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. $$x = 8$$, $$y = 64$$
[Reveal] Spoiler: OA

_________________

Kudos [?]: 135771 [0], given: 12708

Math Expert
Joined: 02 Sep 2009
Posts: 42618

Kudos [?]: 135771 [0], given: 12708

Show Tags

16 Sep 2014, 00:51
Official Solution:

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 $$\frac{x + y}{2}$$. 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. $$x = 8$$, $$y = 64$$

We should be organized as we try to make sense of all the given definitions. First, translate the definitions into algebraic symbols:
$$h(x, y) = \frac{2}{\frac{1}{x} + \frac{1}{y}}$$
$$g(x, y) = \sqrt{xy}$$

$$m(x, y)$$ is the normal arithmetic mean, $$\frac{x + y}{2}$$

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 $$\frac{2}{\frac{1}{8} + \frac{1}{8}} = \frac{2}{\frac{2}{8}} = 8$$. 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.

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Kudos [?]: 135771 [0], given: 12708

Intern
Joined: 31 Dec 2012
Posts: 12

Kudos [?]: 2 [0], given: 135

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05 Jul 2017, 07:49
We can summarise that if X=Y , then Harmonic mean, arithmetic mean and geometric mean will be X or Y.

Kudos [?]: 2 [0], given: 135

Intern
Joined: 26 Jun 2016
Posts: 24

Kudos [?]: 5 [0], given: 275

Location: Viet Nam
Concentration: Finance, Entrepreneurship
GMAT 1: 580 Q48 V23
GPA: 3.25

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22 Nov 2017, 19:56
Another way is finding the relation between x & y:

g(x,y) = m(x,y)
-> square both sides we have: $$xy = (x+y)^2/4$$
-> $$4xy = x^2+y^2 + 2xy$$
-> $$x^2+y^2-2xy = 0$$
->$$(x-y)^2 = 0$$
-> $$x=y$$

Kudos [?]: 5 [0], given: 275

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