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

Question

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

Bunuel · Accepted Answer

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.

usre123 · Answer

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.

gillesalex07 · Answer

Hi there,

If we assume x=y then g(x,x) = 1/2 *   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.

PareshGmat · Answer

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

MathRevolution · Answer

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.

hasnain3047 · Answer

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

IanStewart · Answer

This question is problematic for several reasons. In real math, the "geometric mean" is only defined for positive numbers. This question is anti-mathematically attempting to define it for negatives. It says the geometric mean should be "the square root of the product". But there's no such thing as "the square root" of a positive number; 16, say, has two square roots, 4 and -4 (it's true that √16 = 4, and never -4, but that's because the the radical "√" symbol means "the non-negative square root", and not because 16 has only one square root). You can't correctly define a geometric mean for negative values anyway, but any mathematician who did would insist that it equal the negative square root, not the positive root, because a mean should always lie in between the smallest and largest values in a set. From the wording "the square root" here, I'd have to guess they intend us to take the positive square root, but then the "geometric mean" doesn't obey any of the basic properties of a mean. So that whole part of the question is just mathematically nonsensical, and it's completely unclear what we're meant to even do with the negative values in answers A or B if we decide to compute a geometric mean using them.

One of the most famous results in all of mathematics is the Harmonic Mean-Geometric Mean-Arithmetic Mean inequality, which says for sets of positive numbers,

HM < GM < AM

and HM = GM = AM when all values in a set are equal. Anyone familiar with that inequality will be able to see very quickly that GM = (HM + AM)/2 when our two values are identical, and so will get the answer D in a matter of seconds. Since you don't need to know a thing about harmonic means or geometric means for the GMAT, and since the GMAT is not supposed to be a test of whether you have an advanced math degree (and so is not designed to reward test takers with specialized advanced knowledge), this question is unrealistic for yet another reason.

One of the most famous results in all of mathematics is the Harmonic Mean-Geometric Mean-Arithmetic Mean inequality, which says for sets of positive numbers,

HM  <  GM  <  AM

and HM = GM = AM when all values in a set are equal. Anyone familiar with that inequality will be able to see very quickly that GM = (HM + AM)/2 when our two values are identical, and so will get the answer D in a matter of seconds. Since you don't need to know a thing about harmonic means or geometric means for the GMAT, and since the GMAT is not supposed to be a test of whether you have an advanced math degree (and so is not designed to reward test takers with specialized advanced knowledge), this question is unrealistic for yet another reason.

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

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