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Math Expert V
Joined: 02 Sep 2009
Posts: 56376
The harmonic mean of two numbers x and y, symbolized as h(x, y), is de  [#permalink]

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Difficulty:   15% (low)

Question Stats: 80% (02:24) correct 20% (02:58) wrong based on 135 sessions

### HideShow timer Statistics Tough and Tricky questions: Number Properties.

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

Kudos for a correct solution.

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Manager  Joined: 30 Mar 2013
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Re: The harmonic mean of two numbers x and y, symbolized as h(x, y), is de  [#permalink]

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1
d?
took me four mins and 9 seconds....I didn't realize we may be tested on this (harmonic mean) on the exam....
A) g(x.y)= $$\sqrt{2}$$
h(x,y)= -1/2 -1 =-3/2 so h(x,y) is 2/(-3/2) = -4/2.
m(x,y)= (-2-1)/2= -3/2

average of h(x,y) and m(x,y) = (-4/2 -3/2)/2 = -7/4
Not equal to the g(x,y)

Tried them all except B, where g(x'y) would have been $$\sqrt{-2}$$ which doesn't exist.

D: g(x,y) =8 (I took only the positive root, please correct me here if this is wrong)
h(x,y)= 2/(1/8+1/8) = 2/(2/8)= 16/2 =8
m(x,y)= 16/2 =8

average of h(x,y) and m(x,y) = (8+8) /2 =16/2 =8.

D

Although in retrospect should have started with D since it seems to be the easiest to solve.
Intern  Joined: 29 Sep 2014
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Re: The harmonic mean of two numbers x and y, symbolized as h(x, y), is de  [#permalink]

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1
Hi there,

If we assume x=y then g(x,x) = 1/2 * [m(x,x) + h(x,x)] implies x = abs(x). So in other words the equation is true for any x positive.
So we hold a solution where x=y with x and y are positive numbers.

Looks like D is a winner.
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Re: The harmonic mean of two numbers x and y, symbolized as h(x, y), is de  [#permalink]

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1
$$h(x,y) = \frac{2xy}{x+y}$$

$$g(x,y) = \sqrt{xy}$$

$$m(x,y) = \frac{x+y}{2}$$

We require to satisfy below condition

$$g(x,y) = \frac{m(x,y) + h(x,y)}{2}$$

$$\sqrt{xy} = \frac{2xy}{2(x+y)} + \frac{x+y}{4}$$

$$\sqrt{xy} = \frac{xy}{x+y} + \frac{x+y}{4}$$

Just observe the above formed equation (Don't solve)

RHS >> Addition of 2 fractions would result in another fraction or an integer; but NOT a square root

LHS >> To get a proper integer, only options C, D & E can be considered as they will produce a perfect square root

Option C fails: $$\frac{2*8}{10} + \frac{10}{4}$$ >> Will produce fraction

Option D success

$$8 = \frac{8*8}{16} + \frac{16}{4}$$

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Joined: 02 Sep 2009
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Re: The harmonic mean of two numbers x and y, symbolized as h(x, y), is de  [#permalink]

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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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Re: The harmonic mean of two numbers x and y, symbolized as h(x, y), is de  [#permalink]

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Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer.

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+y2 . 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

The arithmetic mean, the geometric mean and the harmonic mean of two equal numbers are equal((x+x)/2=x, sqrt(x*x)=x, (2x*x)/(x+x)=x).

So without tedious calculation, (D) should be answer.
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Re: The harmonic mean of two numbers x and y, symbolized as h(x, y), is de  [#permalink]

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2
h(x,y) = 2/(1/x +1/y) = 2/(x+y/xy) = 2xy/x+y
g(x,y) = \sqrt{'xy'}
m(x,y) = (x+y)/2

to find value of (x,y) when g(x,y) = m(x,y)
4xy = (x+y)^2
as (x+y)^2 = x^2 + y^2 + 2xy

so x^2 + y^2 = 2xy
let's now plug in all answers to satisfy this equation.
x=8, y=8 only satisfies this equation

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"When you wanna succeed as bad as you wanna breathe, then you will be successful." -Eric Thomas Re: The harmonic mean of two numbers x and y, symbolized as h(x, y), is de   [#permalink] 18 Aug 2018, 09:20
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