(x^(-1) - y^(-1))/((xy)^(-1)(x - y)) =

Question

\frac{x^{-1} - y^{-1} }{ (xy)^{-1} (x-y)}  =

(A) - 1

(B) 1

(C) - xy

(D) xy

(E)  \frac{1}{xy}

Source: GMAT Quantum

BrentGMATPrepNow · Accepted Answer

Rather than deal with performing a variety of operations involving many fractions, a fast approach is to plug some values into the original expression.

Let's see what happens when x = 2 and y = 4

We get:  \frac{x^{-1} - y^{-1} }{ (xy)^{-1} (x-y)}=\frac{2^{-1} - 4^{-1} }{ (2	imes4)^{-1} (2-4)}

Aside:  2^{-1}=\frac{1}{2}=0.5 ,  4^{-1}=\frac{1}{4}=0.25  and  8^{-1}=\frac{1}{8}=0.125

So we get:  =\frac{0.5 - 0.25 }{ (8)^{-1} (-2)}

=\frac{0.5 - 0.25 }{ (0.125)(-2)}

=\frac{0.25 }{ -0.25}=-1

Since answer choice A is the only answer choice that evaluates to be -1 when x = 2 and y = 4, the correct answer is A

Cheers,
Brent

BrentGMATPrepNow · Answer

A student asked me to solve the question without testing values. So here's an algebraic solution...

Let's examine and simplify the numerator and denominator  individually

NUMERATOR:  x^{-1} - y^{-1} = \frac{1}{x}-\frac{1}{y} = \frac{y}{xy}-\frac{x}{xy}=\frac{y-x}{xy}

DENOMINATOR:  (xy)^{-1} (x-y)=(\frac{1}{xy})(x-y)=\frac{x-y}{xy}=\frac{(-1)(-x+y)}{xy}=\frac{(-1)(y-x)}{xy}

Replace numerator and denominator with equivalent values to get:  \frac{x^{-1} - y^{-1} }{ (xy)^{-1} (x-y)}=\frac{\frac{y-x}{xy}}{\frac{(-1)(y-x)}{xy}}

Rewrite as:  (\frac{y-x}{xy})(\frac{xy}{(-1)(y-x)})

Combine to get:  \frac{(y-x)(xy)}{(xy)(-1)(y-x)}

Simplify to get:  \frac{1}{-1} , which equals  -1

Answer: A

Cheers,
Brent

Logo · Answer

Hi  BrentGMATPrepNow

Can you explain how you got from (1/x)-(1/y) to (y/xy)-(x/xy) in the numerator?

Thanks in advance!

BrentGMATPrepNow · Answer

Take:  \frac{1}{x}

Create an equivalent fraction my multiplying top and bottom by  y  to get:  \frac{y}{xy}

Likewise, we can take:  \frac{1}{y}

And create an equivalent fraction my multiplying top and bottom by  x  to get:  \frac{x}{xy}

So,  \frac{1}{x}-\frac{1}{y} = \frac{y}{xy}-\frac{x}{xy}

Logo · Answer

Got it. So essentially you manipulated the expression by multiplying the variables by xy/xy in order to have a common denominator.

Thank you!

BlueRocketAsh · Answer

x^−1 − y^−1 / (xy)^−1(x−y) =   -  
 = y/(x−y) - x/(x−y)
 =(y-x)/(x-y)
 = -1

jeanslew1 · Answer

Here's the approach I took for future users if they find current solutions confusing:

\frac{x^{-1}y^{-1}}{(xy)^{-1}(x-y)}

=\frac{x^{-1}(1-\frac{y^{-1}}{x^{-1}})}{(xy)^{-1}(x-y)}  --> Factor out  x^{-1}

=\frac{x^{-1}(1-xy^{-1})}{(xy)^{-1}(x-y)}  --> This part becomes  xy^{-1}

=\frac{x^{-1}(1-xy^{-1})}{(xy)^{-1}(x-y)}  --> Take the same approach to factor out  y^{-1}

=\frac{x^{-1}y^{-1}(y-x)}{(xy)^{-1}(x-y)}

=\frac{(xy)^{-1}(y-x)}{(xy)^{-1}(x-y)}  --> Exponent rules allow us to make this change and then cancel num and denom

=\frac{(y-x)}{(x-y)}

=\frac{-(x-y)}{(x-y)}

=-1

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(x^(-1) - y^(-1))/((xy)^(-1)(x - y)) =

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