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# Is x<1/y?

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Math Revolution GMAT Instructor
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24 Aug 2016, 18:21
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Question Stats:

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Is x<1/y?
1) y>0
2) xy<1

*An answer will be posted in 2 days.
[Reveal] Spoiler: OA

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24 Aug 2016, 18:38
Top Contributor
There is no information about x or y so nothing can be done to reduce the initial statement...at first.

Statement 1: y>0, explains nothing about x. NSUF
Statement 2: xy<1, x and y could be different signs or x and y could be fractions or a combination of the two. NSUF

Statement 1 & 2: if y>0 and xy<1 then divide the equation by y and you have the statement. SUF

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25 Aug 2016, 14:36
S1)$$y>0$$

Pick $$y=1$$ and $$x=2$$;
=>$$2<\frac{1}{1}$$
=>No

Pick $$y=2$$ and $$x=-2$$
=>$$-2<\frac{1}{2}$$
=>Yes

So S1 is insufficient

BCE

S2)$$xy<1$$

Pick $$x=\frac{1}{2}$$ and $$y=\frac{1}{2}$$
=>$$\frac{1}{2}<2$$
=> Yes

Pick $$x=\frac{1}{2}$$ and $$y=\frac{-1}{2}$$
=>$$\frac{1}{2}<-2$$
=>No

So S2 is insufficient (Note: This statement could be a trap if inequality in question stem is just cross multiplied without checking possible values that $$y$$ could take.)

CE

S1 and S2)

Pick $$x=\frac{1}{2}$$ and $$y=\frac{1}{2}$$;This meets both the statements
=>$$\frac{1}{2}<2$$
=>Yes

Pick $$x=-2$$ and $$y=1$$; This meets both the statements
=> $$-2<\frac{1}{1}$$
=>Yes

Here $$y>0$$ is satisfied from statement 1; both positive and negative values (integer and fraction as well) for $$x$$ are tried and that should cover all possible values for $$x$$ and the output is an YES in both the cases. Hence Statements 1 and 2 taken together is sufficient. Answer is C.

(S1 says that $$y>0$$; So if $$y$$, which is greater than 0, is divided both sides of statement 2, then it will not change the sign of the inequality and we get $$x<\frac{1}{y}$$ which is exactly what the question stem is asking. Hence C)

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26 Aug 2016, 05:08
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MathRevolution wrote:
Is x < 1/y?

1) y > 0
2) xy < 1

Target question: Is x < 1/y?

Statement 1: y > 0
No information about x, so statement 1 is NOT SUFFICIENT

Statement 2: xy < 1
IMPORTANT: some students will want to divide both sides by y to get x < 1/y, HOWEVER doing so is incorrect, because we don't know whether y is POSITIVE or NEGATIVE.
If y is POSITIVE, then dividing both sides by y gives us x < 1/y
If y is NEGATIVE, then dividing both sides by y gives us x > 1/y [when we divide both sides by a negative value, we must REVERSE the direction of the inequality]
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that y is POSITIVE
Statement 2 tells us that xy < 1
Since we know that y is POSITIVE, we can safely divide both sides by y to get the inequality x < 1/y.

Since we can answer the target question with certainty, the combined statements are SUFFICIENT

[Reveal] Spoiler:
C

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Math Revolution GMAT Instructor
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27 Aug 2016, 04:21
If we modify the original condition and the question, in case of inequality, square is very important. If we multiply y^2 to both sides, we get xy^2<y?, xy^2-y<0?, y(xy-1)<0?. There are 2 variables (x and y) and there is a high chance that C is the correct answer. Using 1) & 2), the answer is always yes and the condition is sufficient. Thus, the correct answer is C.

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12 Dec 2017, 11:28
$$x < 1/y$$ ?
Re-arranging,
=> $$x - (1/y) < 0$$ ?
=> $$(xy - 1)/y < 0$$ ?
For this term to be negative, ($$xy < 1$$ AND $$y > 0$$) OR ($$xy > 1$$ AND $$y < 0$$) ?
Question becomes => ($$xy < 1$$ AND $$y > 0$$) OR ($$xy > 1$$ AND $$y < 0$$) ?

Statement 1: $$y > 0$$, but we don't know if $$xy < 1$$ => InSufficient
Statement 2: $$xy < 1$$, but we don't know if $$y > 0$$ => InSufficient

Statement 1 + 2: $$y > 0$$ AND $$xy < 1$$ => Answers our question

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Is x<1/y?   [#permalink] 12 Dec 2017, 11:28
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