GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 15 Oct 2019, 16:51 GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  S96-14

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 58340

Show Tags 00:00

Difficulty:   25% (medium)

Question Stats: 78% (02:13) correct 22% (02:54) wrong based on 41 sessions

HideShow timer Statistics

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. $$x = 8$$, $$y = 64$$

_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 58340

Show Tags

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.

_________________
Intern  B
Joined: 31 Dec 2012
Posts: 11

Show Tags

We can summarise that if X=Y , then Harmonic mean, arithmetic mean and geometric mean will be X or Y.
Intern  B
Joined: 26 Jun 2016
Posts: 20
Location: Viet Nam
Concentration: Finance, Entrepreneurship
GMAT 1: 580 Q48 V23 GPA: 3.25

Show Tags

1
Another way is finding the relation between x & y:

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

Intern  Joined: 02 Aug 2017
Posts: 2

Show Tags

1
I think this is a poor-quality question and I agree with explanation. The "of" in the last line just before h(x,y) should be omitted.
Other wise the meaning of the question changes.
It should be: For which of the following pairs of values for x and y is g(x,y) equal to THE ARITHMETIC MEAN h(x,y) and m(x,y). Re S96-14   [#permalink] 26 Sep 2018, 10:51
Display posts from previous: Sort by

S96-14

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

Moderators: chetan2u, Bunuel

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne  