Two non zero real numbers, x and y, satisfy xy = x-y. which of the fol

Question

Two non zero real numbers, x and y, satisfy  xy = x-y . which of the following is a possible value of  \frac{x}{y}  +  \frac{y}{x}   - xy  ?

A) -2
B) -1/2
C) 1/3
D) 1/2
E) 2

BassanioGratiano · Accepted Answer

\frac{x}{y}  +  \frac{y}{x}   - xy

\frac{(x^2+y^2)}{xy}   - (x-y)

\frac{(x^2+y^2)}{(x-y)}   -\frac{(x-y)^2}{(x-y)}

\frac{(x^2+y^2)}{(x-y)}   -\frac{(x^2-2xy+y^2)}{(x-y)}

\frac{2xy}{(x-y)}

\frac{2xy}{xy}

2

E.

EMPOWERgmatRichC · Answer

Hi manpreetsingh86,

BassanioGratiano has already presented a great "Algebra" explanation for this question, so I won't rehash that math here. Instead, here's how a combination of TESTing VALUES and some Number Property logic can be used:

We're told that X and Y CANNOT be 0, so the equation XY = X-Y is interesting....What types of numbers have a product that is equal to their DIFFERENCE??? The rest of the prompt and the answer choices strongly *hint* that fractions may be involved.

The first/simplest example I could come up with was...

X = 1
Y = 1/2
1(1/2) = 1 - 1/2
1/2 = 1/2

With those 2 numbers, let's see what result we get in the question...

X/Y + Y/X - XY
1/(1/2) + (1/2)/1 - 1(1/2)
2 + 1/2 - 1/2
2

That answer APPEARS in the answer choices, so we're done.

Final Answer:  D

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

manpreetsingh86 · Answer

here is an another solution

xy = x-y --------------------------1)
dividing both sides by xy we have;
1 = 1/y - 1/x ---------------------------2)

now coming to the original equation.

\frac{x}{y}  +  \frac{y}{x}   - xy

substitute the value of xy , from 1 we have

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

\frac{x}{y}  +  \frac{y}{x}   -x+y

\frac{x}{y}  -x  +  \frac{y}{x}   +y

x(\frac{1}{y}-1)  + y(\frac{1}{x}+1)

from 2) we know that (1/y-1) =1/x and (1/x+1) = 1/y

thus we have

x(1/x)+y(1/y) =2

jfranciscocuencag · Answer

Hello!

Could someone please explain to me why do we have to square the second term?

\frac{(x^2+y^2)}{(x-y)}   -\frac{(x-y)^2}{(x-y)}

Kind regards!

Pksri · Answer

Hi Manpreet,

Sorry for responding to the very old topic!

If it is non zero real number it can be both positive or negative. If you choose x = 1 and y = -1/2, the answer we obtain here is -2, which is also available in the option. In this case we cant use the approach that you have explained. The only way is algebraic I feel.

-Pksri

EMPOWERgmatRichC · Answer

Hi Pksri,

When adding or subtracting NEGATIVE numbers, you have to be really careful about keeping track of the "signs."

The example you listed: X = 1, Y = -1/2 does NOT actually fit what we're told.

(X)(Y) = X - Y

With the values you listed, we have
(X)(Y) = (1)(-1/2) = -1/2
X - Y = 1 - (-1/2) = +3/2

Those totals do NOT equal one another, so this is not a proper TEST - and thus, it tells us nothing about what the correct answer is. 
 
GMAT assassins aren't born, they're made,
Rich

Pksri · Answer

My bad, so true! Dumb mistake from me

EMPOWERgmatRichC · Answer

Hi Pksri,

It's better to make those little mistakes now (during practice) than when everything matters the most (on Test Day). As you continue to study, here are a couple of ideas to keep in mind:

1) Most GMAT questions are written so that they can be approached in more than one way, so just because you can get to the correct answer doesn't necessarily mean that "your way" is fastest or easiest. Thus, developing multiple skills for each question type should help you to improve your Score AND your pacing.

2) Little mistakes (especially on "gettable" questions) tend to cost many GMATers some serious points on Test Day. The only way to properly keep those silly/little mistakes from happening is to do ALL of your work on your note-pad (NEVER "in your head"). Thankfully, the work is almost always straight-forward, but to maximize your overall Score, you have to be committed to doing that type of organized note-taking throughout the course of the Exam.

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

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Two non zero real numbers, x and y, satisfy xy = x-y. which of the fol

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