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x/y+y/x=8 [#permalink]
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Carcass wrote:
\(\frac{x}{y}+\frac{y}{x}=8\). What is the value of \(\frac{x+y}{\frac{1}{x}+\frac{1}{y}}\)


(A) 5
(B) 8
(C) 10
(D) 12
(E) 16


Unless I'm missing something, this question is flawed. Here's why:

Given: \(\frac{x}{y}+\frac{y}{x}=8\)

Rewrite with common denominators: \(\frac{x^2}{xy}+\frac{y^2}{xy}=8\)

Combine fractions: \(\frac{x^2+y^2}{xy}=8\)

Multiply both sides of the equation by \(xy\) to get: \(x^2+y^2=8xy\)

Add \(2xy\) both sides of the equation: \(x^2+2xy+y^2=10xy\)

Factor the left side: \((x+y)^2=10xy\)

Finally, divide both sides by \(xy\) to get: \(\frac{(x+y)^2}{xy}=10\)

So, IF the given expression evaluates to be \(\frac{(x+y)^2}{xy}\), then the correct answer will be \(10\).


The given expression: \(\frac{x+y}{\frac{1}{x}+\frac{1}{y}}\)

Rewrite the denominator with common denominators: \(\frac{x+y}{\frac{y}{xy}+\frac{x}{xy}}\)

Combine terms in the denominator: \(\frac{x+y}{(\frac{x+y}{xy})}\)

Since we're dividing by a fraction, we'll multiply by the reciprocal to get: \((x+y)(\frac{xy}{x+y})\)

Unfortunately this simplifies to be \(xy\), which we don't know the value of.

Please let me know if I'm missing something (or if I made a mistake above)
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Re: x/y+y/x=8 [#permalink]
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No sir

Thank you for the explanation. Not GRE question. Simply is that

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Re: x/y+y/x=8 [#permalink]
Carcass, can you please advise what is the most efficient approach to solve this question?
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Re: x/y+y/x=8 [#permalink]
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tkorzhan18 wrote:
Carcass, can you please advise what is the most efficient approach to solve this question?


Brent above is perfectly correct. I agree. The question miss something because

x/y+y/x=8

x^2+y^2/xy=8

to have 8 in the end x and y must =4

and if x and y = 4 the second quantity for which we are looking for the value must be = 4 therefore at most the answer choice must be 4 OR a multiple of 4 and we do have C which is = to 10

Not a good question.

I hope this helps
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Re: x/y+y/x=8 [#permalink]
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