S96-14

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 · 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

JTR · Answer

We can summarise that if X=Y , then Harmonic mean, arithmetic mean and geometric mean will be X or Y.

Hientran48 · Answer

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

-> answer D

Harvarditis · Answer

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).

KsNw0w · Answer

Hi, the question asks (x,y) for which: g(x,y) = m(h(x,y),m(x,y)).
What you have solved algebraically is g(x,y) = m(x,y).

Using the process of elimination we can clearly remove A & E as they have \sqrt{2} in the g(x,y) whereas RHS will never have a square root as its simple addition of integers and their reciprocals.
B is eliminated because square root is not defined.
Of the remaining two, just plug any value and you'll have the answer as D.

sujoykrdatta · Answer

Geometric mean of x and y = root(xy)
Arithmetic mean of x and y = (x+y)/2
Harmonic mean of x and y = 2xy/(x+y)

We can try to solve this using algebra (very complicated) or try to solve using the options.

Let us now use a simple piece of information:

If the numbers x and y are equal, the geometric mean, arithmetic mean and harmonic mean are equal to one another and also equal to the numbers themselves.  
(Note: If x and y are unequal, we have: AM > GM > HM)

Keeping this is in mind, if we check the options, we see that Option D has both numbers equal (= 8)
Thus, AM, GM and HM for the numbers x = y = 8 would also be 8
Thus, the Arithmetic Mean of the Geometric mean and Harmonic mean would also be 8

Thus, the correct answer is  OPTION D

Aalto700 · Answer

Since the equation is too complicated with square root and reciprocal, try the most simple pair of x and y, which is x=y=8. that we can simplify the equation. => answer D

bumpbot · Answer

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S96-14

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