Given:
Quantity A: 5/2x - 2/2y
Quantity B: (5y-2x)/(2x-2y)

Let's see what happens if we rewrite Quantity A as one big fraction.
First get common denominators...
Quantity A: 5y/2xy - 2x/2yx
Quantity B: (5y-2x)/(2x-2y)

Combine to get:
Quantity A: (5y-2x)/2yx
Quantity B: (5y-2x)/(2x-2y)

NICE!! This numerators are the same for both fractions.
This gives me the opportunity to use a strategy I call "Looking for Equality" (see our video below for more on this)
I can see that if I choose an x- and y-value that makes the numerators both zero, this will speed up my solution. You'll see why shortly.

I can see that, if x = 5 and y = 2, then we can plug these values into the quantities to get:
Quantity A: [5(2)-2(5)]/(2)(2)(5)
Quantity B: [5(2)-2(5)]/[2(5)-2(2)]

Evaluate:
Quantity A: 0
Quantity B: 0

So, when x = 5 and y = 2 the quantities are EQUAL.
The correct answer is EITHER C or D

At this point, every pair of x- and y-values will either yield equal quantities or not.
Let's plug in super-easy numbers.
We'll plug in x = 1 and y = 1
We get:
Quantity A: [5(1)-2(1)]/(2)(1)(1)
Quantity B: [5(1)-2(1)]/[2(1)-2(1)]

Evaluate:
Quantity A: 3/2
Quantity B: 3/0
These quantities are definitely NOT EQUAL.

Answer:

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I've tried for x=1, y=2, x=1, y=3, and x=-1, y=-2, and getting A as the answer. What to do now? How can I chose sensible numbers?

It was a good one.
Answer is Choice D.
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This is actually time consuming, is it an official GRE question?