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)

No sir

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

Kudos

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