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Difficulty: HardValue vs. Yes/No Data Sufficiency Decision Logic

If xx and yy are non-zero real numbers, is xy>0\frac{x}{y} > 0?

(1) ∣x+y∣<∣x−y∣|x + y| < |x - y|
(2) x2y+xy2<0x^2 y + x y^2 < 0

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Answer
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. C
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The option stating that Statement (1) alone is sufficient while Statement (2) alone is not sufficient is correct. Statement (1) simplifies to xy<0xy < 0, which proves conclusively that xy\frac{x}{y} is negative. Thus, the answer to the question 'Is xy>0\frac{x}{y} > 0?' is a definitive 'No', which makes Statement (1) sufficient. Statement (2) simplifies to xy(x+y)<0xy(x + y) < 0, which permits scenarios where xy>0xy > 0 as well as scenarios where xy<0xy < 0, rendering Statement (2) insufficient.

Step-by-Step Solution

1
Rephrase the question stem target.
The expression xy>0\frac{x}{y} > 0 is true if and only if xx and yy have the same sign, which means xy>0xy > 0. The question asks: Is xy>0xy > 0?
Converting division of signed variables into product sign simplifies algebraic evaluation.
2
Evaluate Statement (1): ∣x+y∣<∣x−y∣|x + y| < |x - y|.
Squaring both sides yields (x+y)2<(x−y)2  ⟹  x2+2xy+y2<x2−2xy+y2  ⟹  4xy<0  ⟹  xy<0(x + y)^2 < (x - y)^2 \implies x^2 + 2xy + y^2 < x^2 - 2xy + y^2 \implies 4xy < 0 \implies xy < 0.
Since xy<0xy < 0, xy\frac{x}{y} MUST be negative, so the answer to 'Is xy>0\frac{x}{y} > 0?' is a definitive 'NO'.
3
Apply Yes/No Data Sufficiency logic to Statement (1).
A statement that yields a definitive 'No' to a Yes/No question is SUFFICIENT.
Sufficiency in a Yes/No question requires a conclusive YES or a conclusive NO; ambiguity is the only cause for insufficiency.
4
Evaluate Statement (2): x2y+xy2<0x^2 y + x y^2 < 0.
Factor out xyxy to get xy(x+y)<0xy(x + y) < 0. Case A: If x=−2x = -2 and y=−3y = -3, then xy=6>0xy = 6 > 0 and x+y=−5<0x+y = -5 < 0, giving 6(−5)=−30<06(-5) = -30 < 0 (Answer: YES). Case B: If x=5x = 5 and y=−2y = -2, then xy=−10<0xy = -10 < 0 and x+y=3>0x+y = 3 > 0, giving −10(3)=−30<0-10(3) = -30 < 0 (Answer: NO).
Because Statement (2) allows both a YES and a NO response, Statement (2) alone is NOT sufficient.

Key Concept

Definitive Yes/No Decision Logic in Data Sufficiency
Estimated Time:2m 0s
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Topics

Value vs. Yes/No Data Sufficiency Decision Logic

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