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The harmonic mean of two numbers x and y, symbolized as h(x, y), is de

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

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New post 11 Nov 2014, 09:32
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A
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E

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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. \(x = 8\), \(y = 64\)

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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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New post 11 Nov 2014, 10:05
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.
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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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New post 11 Nov 2014, 17:23
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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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New post 12 Nov 2014, 00:07
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\(h(x,y) = \frac{2xy}{x+y}\)

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

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

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

\(8 = \frac{8*8}{16} + \frac{16}{4}\)

Answer = D
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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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New post 12 Nov 2014, 04:10
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.


Answer: D.
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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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New post 07 Dec 2015, 01:25
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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New post 18 Aug 2018, 09:20
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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

So Answer is D
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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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