## (DS) if 0<x<y, what is the value of . . .

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### (DS) if 0<x<y, what is the value of . . .

If 0<x<y, what is the value of (x+y)^2 / (x-y)^2

(1) x^2 + y^2 = 3xy

(2) xy = 3

Thank you ~
RonPurewal
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### Re: (DS) if 0<x<y, what is the value of . . .

Anonymous wrote:If 0<x<y, what is the value of (x+y)^2 / (x-y)^2

(1) x^2 + y^2 = 3xy

(2) xy = 3

Thank you ~

you should either (a) memorize the expansions of (x + y)^2 and (x - y)^2, or (b) be able to generate those expansions within a few seconds on demand.

therefore, you should immediately be able to rephrase the given question to "what is (x^2 + 2xy + y^2) / (x^2 - 2xy + y^2) ?"
note that you don't have to use the rephrased version once you've generated it; as with all other rephrases, you're free to go back and use the original version of the question.

--

statement (1):
use the expanded (rephrased) version of the fraction.
note that both the top and the bottom of the fraction contain the expression (x^2 + y^2), which, per statement (1), you can extract and replace with the expression 3xy.
therefore,
(x^2 + 2xy + y^2) / (x^2 - 2xy + y^2)
= (3xy + 2xy) / 3xy - 2xy)
= 5xy / xy
= 5
sufficient.

--

statement (2):
attempt to substitute into the expanded (rephrased) version of the given fraction.
(x^2 + 2xy + y^2) / (x^2 - 2xy + y^2)
= (x^2 + 6 + y^2) / (x^2 - 6 + y^2)
this doesn't simplify further in any obvious (or non-obvious) way.

if you have time remaining, and you want to make sure that the value of this expression really does vary depending on x and y, then substitute in values accordingly.
just make sure that you obey BOTH conditions: 0 < x < y (from the question prompt) and xy = 3 (from statement 2).
try x = 1, y = 3: (1 + 6 + 9) / (1 - 6 + 9) = 16/4 = 4.
note that, in conjunction with statement 1, this already guarantees insufficiency, because the fraction must be able to assume the value 5 (else the two statements would contradict each other). therefore, the fraction could be either 4 or 5; insufficient.
if you're using this statement first, try another set of numbers: x = 1/2, y = 6. that gives (1/4 + 6 + 36) / (1/4 - 6 + 36) = (42 1/4) / (30 1/4), which is most definitely not 4. there's no reason to actually calculate the value, since it's definitely different from the last value calculated (= 4); insufficient.

--

ans = (a)
Linh
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### Re: (DS) if 0<x<y, what is the value of . . .

Wow, this is an interesting question! I "confidently" slipped into a hidden trap of this problem on TWR recently.
I wondered why I didn’t see the similarity among terms? Maybe I didn’t face many questions of this type, so I was not cautious (of course I should be cautious at a level of “feeling”, as Ron said, feeling of something “specific”). But that makes me love DS anyway. It’s always surprising!

So I looked at Quant review 2nd edition and OG 16, I saw few problems of this “grouped” type— in which the similarity terms are grouped together and there is a way to “reconcile the difference”--saying it in a poetic way, he-- to make a statement sufficient.
I reproduce them here (I modified them a bit to follow a rule of the forum)

[redacted]
RonPurewal
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### Re: (DS) if 0<x<y, what is the value of . . .

the forum rule of "don't reproduce OG problems" extends to slightly modified versions of the problems, too. (:

basically, the point there was to get you thinking about certain kinds of patterns that might be exposed in the problems. we can't have this kind of discussion of the OG problems here on the forum, but, it's good that you are thinking about them in the way you are.
Linh
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Joined: Fri May 30, 2014 6:10 am

### Re: (DS) if 0<x<y, what is the value of . . .

Yes, Ron, patterns.
(he, I don't quite understand about patterns)

You have the perfect score, 800. You must see something new on the test. How did you deal with that something?

Also, as a question maker, what does pattern mean to the makers? What techniques they use to create a bunch of questions that follow the same pattern? (unless you don't want to reveal…)
RonPurewal
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Posts: 19747
Joined: Tue Aug 14, 2007 8:23 am

### Re: (DS) if 0<x<y, what is the value of . . .

to the test taker, EVERYTHING on the test is "new". that's actually the whole point of this exam—it's impossible to beat this test by memorizing things.

beyond that, i don't really understand what you are trying to ask.