The number x is the average (arithmetic mean) of the positiv

Question

The number x is the average (arithmetic mean) of the positive numbers a and b, and the reciprocal of the number y is the average (arithmetic mean) of the reciprocals of a and b. In terms of a and b, x – y = ?

A.  \frac{{a^2 + b^2}}{{2(a + b)}}

B.  \frac{{(a - b)^2}}{{2(a + b)}}

C.  \frac{{a + b}}{{2}}

D.  \frac{{a - b}}{{2}}

E. 0

Bunuel · Accepted Answer

The number x is the average (arithmetic mean) of the positive numbers a and b:  x=\frac{a+b}{2} ;

The reciprocal of the number y is the average (arithmetic mean) of the reciprocals of a and b:  \frac{1}{y}= \frac{\frac{1}{a}+\frac{1}{b}}{2}=\frac{a+b}{2ab}  -->  y=\frac{2ab}{a+b} ;

x-y=\frac{a+b}{2}-\frac{2ab}{a+b}=\frac{(a+b)(a+b)}{2(a+b)}-\frac{2*2ab}{2(a+b)}=\frac{a^2+2ab+b^2-4ab}{2(a+b)}=\frac{a^2-2ab+b^2}{2(a+b)}=\frac{(a-b)^2}{2(a+b)}

mau5 · Answer

Assume a=1,b=3 Thus x = 2. Again,  \frac{2}{y} = \frac{1}{1}+\frac{1}{3}

Thus, y = \frac{3}{2} 	o x-y = 2-\frac{3}{2} = \frac{1}{2} . Only option B satisfies.

Asifpirlo · Answer

x-y= (a+b)/2 - 2ab/(a+b) = (a-b)^2/(2(a+b)

GMAT01 · Answer

"the reciprocal of the number y is the average (arithmetic mean) of the reciprocals of a and b"

Why is it that the translation for this is not 1/Y =  /2 ?

Would someone be able to clarify this?

Bunuel · Answer

The reciprocal of the number y is the average (arithmetic mean) of the reciprocals of a and b:

\frac{1}{y}= \frac{\frac{1}{a}+\frac{1}{b}}{2}=\frac{a+b}{2ab}  -->  y=\frac{2ab}{a+b}

GMAT01 · Answer

Got it so next it would be:

/ 2 - 2ab/  = > but how would this not become (a+b)^2 - 2ab/ ??

stonecold · Answer

Great Question.
Here is my solution to this one =>

Using =>

Mean = \frac{Sum}{#}

x=\frac{a+b}{2}

\frac{2}{y}={\frac{1}{a}+\frac{1}{b}}

y=\frac{2ab}{a+b}

x-y=\frac{(a+b)^2-4ab}{2(a+b)} 
=>  \frac{(a-b)^2}{2(a+b)}

Hence B

EMPOWERgmatRichC · Answer

Hi All,

While this is a "thick" question, it can be solved by TESTing Values.

We're told that… 
X = average of A and B 
The reciprocal of Y = the average of the reciprocals of A and B

A = 2 
B = 4

X = (2+4)/2 = 3

1/Y = (1/2 + 1/4)/2 = (3/4)/2 = 3/8 
1/Y = 3/8 
Y = 8/3

We're asked for the value of X - Y: 
X - Y = 
3 - 8/3 = 
9/3 - 8/3 = 
1/3

So we're looking for an answer that equals 1/3 when we use X = 2 and Y = 4

From the answer choices, we can quickly eliminate C, D and E (none of them have a denominator that is (or could become) a "3"; so we're down to the first 2 answers.

Answer A: (4 + 16)/12 = 20/12 = 5/3 
Answer B: (-2)^2/12 = 4/12 = 1/3

Final Answer:  B

GMAT assassins aren't born, they're made, 
Rich

bumpbot · Answer

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The number x is the average (arithmetic mean) of the positiv

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