Examkin

Question

Difficulty: Very hardValue vs. Yes/No Data Sufficiency Decision Logic

If aa and bb are non-zero real numbers, is a3b+ab3>0a^3 b + a b^3 > 0?

(1) a2−5ab+6b2=0a^2 - 5ab + 6b^2 = 0
(2) ∣a+b∣<∣a−b∣|a + b| < |a - b|

Which of the following choices correctly describes the sufficiency of the statements?

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  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. EACH statement ALONE is sufficient.Answer
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

EACH statement ALONE is sufficient.
Rephrasing the stem shows that a3b+ab3=ab(a2+b2)>0a^3b + ab^3 = ab(a^2 + b^2) > 0 reduces to asking whether ab>0ab > 0. Statement (1) yields a=2ba = 2b or a=3ba = 3b, both of which make ab>0ab > 0 (a definitive 'Yes'). Statement (2) simplifies to 4ab<0  ⟹  ab<04ab < 0 \implies ab < 0, which yields a definitive 'No'. Because each statement independently produces a definitive, consistent Yes/No answer, each statement alone is sufficient.

Step-by-Step Solution

1
Rephrase the question stem target.
The target expression a3b+ab3a^3b + ab^3 factors into ab(a2+b2)ab(a^2 + b^2). Because aa and bb are non-zero real numbers, a2+b2>0a^2 + b^2 > 0 always. Thus, a3b+ab3>0a^3b + ab^3 > 0 if and only if ab>0ab > 0. The question reduces to a Yes/No query: 'Is ab>0ab > 0?'
Simplifying the stem isolates the essential logical condition needed to evaluate sufficiency.
2
Evaluate Statement (1) alone.
Factoring a2−5ab+6b2=0a^2 - 5ab + 6b^2 = 0 yields (a−2b)(a−3b)=0(a - 2b)(a - 3b) = 0, meaning a=2ba = 2b or a=3ba = 3b. If a=2ba = 2b, then ab=2b2>0ab = 2b^2 > 0 (since b≠0b \neq 0). If a=3ba = 3b, then ab=3b2>0ab = 3b^2 > 0 (since b≠0b \neq 0). In both cases, ab>0ab > 0 is true, giving a definitive 'Yes'.
Although Statement (1) yields two possible values for the ratio a/ba/b, both values guarantee that ab>0ab > 0. In Yes/No Data Sufficiency, a consistent 'Yes' is sufficient.
3
Evaluate Statement (2) alone.
Squaring both non-negative sides of ∣a+b∣<∣a−b∣|a + b| < |a - b| gives (a+b)2<(a−b)2(a + b)^2 < (a - b)^2, which simplifies to a2+2ab+b2<a2−2ab+b2  ⟹  4ab<0  ⟹  ab<0a^2 + 2ab + b^2 < a^2 - 2ab + b^2 \implies 4ab < 0 \implies ab < 0. Since ab<0ab < 0, the answer to 'Is ab>0ab > 0?' is a definitive 'No'.
A statement that yields a consistent, definitive 'No' is sufficient in a Yes/No Data Sufficiency question.
4
Synthesize results.
Since Statement (1) alone produces a definitive 'Yes' and Statement (2) alone produces a definitive 'No', each statement alone is sufficient to answer the question.
Both statements independently satisfy the decision criteria for Data Sufficiency.

Key Concept

Value vs. Yes/No Data Sufficiency Decision Logic
Rate this question

Topics

Value vs. Yes/No Data Sufficiency Decision Logic

More questions from this topic