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Difficulty: MediumQuestion Stem Simplification and Target Rephrasing

If aa and bb are non-zero real numbers such that a2≠b2a^2 \neq b^2, is a2b+ab2a3b−ab3>0\frac{a^2 b + a b^2}{a^3 b - a b^3} > 0?

(1) a>ba > b
(2) a+b>0a + b > 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 target question asks whether a2b+ab2a3b−ab3>0\frac{a^2 b + a b^2}{a^3 b - a b^3} > 0. Factoring the numerator as ab(a+b)ab(a+b) and the denominator as ab(a+b)(a−b)ab(a+b)(a-b) allows us to cancel common non-zero factors, simplifying the target question to 'Is 1a−b>0\frac{1}{a-b} > 0?', which is equivalent to 'Is a>ba > b?'. Statement (1) directly states that a>ba > b, answering the rephrased question with a definitive 'Yes'. Statement (2) states that a+b>0a+b > 0, which gives no information regarding whether a>ba > b. Therefore, Statement (1) alone is sufficient, while Statement (2) alone is not.

Step-by-Step Solution

1
Rephrase the target question stem algebraically by factoring numerator and denominator.
Numerator: a2b+ab2=ab(a+b)a^2 b + a b^2 = a b(a + b). Denominator: a3b−ab3=ab(a2−b2)=ab(a+b)(a−b)a^3 b - a b^3 = a b(a^2 - b^2) = a b(a + b)(a - b). The expression becomes ab(a+b)ab(a+b)(a−b)\frac{a b(a + b)}{a b(a + b)(a - b)}.
Simplifying complex rational expressions before evaluating statements prevents misinterpretation of necessary conditions.
2
Cancel non-zero common terms in the simplified fraction.
Since ab≠0a b \neq 0 and a+b≠0a + b \neq 0 (given a2≠b2a^2 \neq b^2), common terms cancel out to yield 1a−b>0\frac{1}{a - b} > 0.
The sign of 1a−b\frac{1}{a - b} is strictly positive if and only if a−b>0a - b > 0, which is equivalent to a>ba > b.
3
Evaluate Statement (1): a>ba > b.
This directly yields a−b>0a - b > 0, so 1a−b>0\frac{1}{a - b} > 0. This provides a definitive 'Yes' answer.
Statement (1) alone provides sufficient information.
4
Evaluate Statement (2): a+b>0a + b > 0.
Knowing a+b>0a + b > 0 gives no information about whether a>ba > b or a<ba < b (e.g., if a=5,b=2a=5, b=2, a>ba > b; if a=2,b=5a=2, b=5, a<ba < b).
Statement (2) alone is not sufficient to determine if a>ba > b.

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Question Stem Simplification and Target Rephrasing
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Question Stem Simplification and Target Rephrasing

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