Cross-Multiplication Problem

[Coyosso]

I'm super confused as to why my solution path to a comparison problem didn't work. Any help would be appreciated. 

Problem 41, Chapter 8 (Algebra), GRE 5 lb. book of problems

Here are the constraints:

X > Y

XY =/= 0

 

Here is the problem:

   A:                                                      B:

  X²/ (Y + 1/Y)                   Y²/ (X + 1/X)

 

My strategy was to cross-multiply to make this more manageable when using test cases. 

So:

    A:                                            B:

   X² (X + 1/X)                        Y²/ (Y + 1/Y

    X³ + X                            Y³ + Y

 

(I could have factored out, but didn't feel a need to as far as helping me calculate test cases.)

Because X > Y, the test cases I used, respectively, were +/+, +/-, -/-, choosing simple numbers.

 

(+/+)     X=2, Y=1

     A: 10      B: 2

 

(+/-)      X= 1, Y=  -1

  A:   2      B;  -2

 

(-/- )   X= -1, Y= -2

   A: -2     B: -10

 

In all test cases, A is bigger using the equation I derived using cross-multiplication.

The book the answer is "D." Their strategy was not to try and simplify like I did, but instead, directly use test cases off the original formulas. So either, my derivation of the simpler formula was wrong, or my calculation of test cases was wrong. I don't think my use of test cases was incorrect, so where did I go wrong in cross-multiplying?  I thought that such an operation was perfectly valid in examples like these? 

I'm super confused. Any help would be greatly apprecaited. Thank you, in advance!

 

[Coyosso]

Oh wait, I think i understand why this didn't work. Cross-multiplying assumes there is an equality between the two expressions, when the whole point of this type of problem is determine the nature of the relationship between the two expressions. By cross-multiplying, equality is preserved, but the value of each expression is altered, meaning that I am no longer evaluating the original problem. Is this correct?

[TakeruK]

10 hours ago, Coyosso said:

Oh wait, I think i understand why this didn't work. Cross-multiplying assumes there is an equality between the two expressions, when the whole point of this type of problem is determine the nature of the relationship between the two expressions. By cross-multiplying, equality is preserved, but the value of each expression is altered, meaning that I am no longer evaluating the original problem. Is this correct?

In short, yes, that is where you made the mistake in your answer above. For this particular question and questions like it, i.e. no relationship given between A and B, you can manipulate the expressions if you want but only within each expression. e.g. if A was written as X^2 (1 + X), you could rewrite A to be X^2 + X^3 if that makes it easier for you to plug in the numbers for test cases.

Some extra information that might be helpful for other questions:

Note that you're not quite correct that you can cross-multiply only when there is equality between the two expressions. You can also cross-multiply inequalities, for example, if you have the expression:

3A / (B+C) > 4        (assume B+C > 0 so that this is a valid expression)

It is still correct to rewrite this as:

3A > 4(B+C)

However, if you are multiplying by a negative number, you must flip the inequality, so the above action was only valid if B+C is greater than zero. If you know B+C is less than zero, then the expression becomes 3A < 4(B+C).