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Difficulty: HardAlgebraic Equations and Systems in Data Sufficiency

If aa and bb are non-zero real numbers, what is the value of a2+9b2ab\frac{a^2 + 9b^2}{ab}?

(1) a2−3ab−18b2=0a^2 - 3ab - 18b^2 = 0
(2) a>0a > 0 and b>0b > 0

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

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
The correct option is the choice stating that both statements together are sufficient, but neither statement alone is sufficient. Statement (1) produces two valid roots for the ratio of the variables, which lead to two different target values. Statement (2) specifies that both variables are positive, which eliminates the negative root and uniquely identifies the ratio and target value when combined with Statement (1).

Step-by-Step Solution

1
Rephrase the target expression in terms of the ratio k=abk = \frac{a}{b}.
a2+9b2ab=a2ab+9b2ab=ab+9(ba)=k+9k\frac{a^2 + 9b^2}{ab} = \frac{a^2}{ab} + \frac{9b^2}{ab} = \frac{a}{b} + 9\left(\frac{b}{a}\right) = k + \frac{9}{k}. Finding the ratio k=abk = \frac{a}{b} is sufficient to answer the question.
Dividing the numerator by the denominator simplifies the expression into a function of a single ratio variable.
2
Evaluate Statement (1) independently.
Divide a2−3ab−18b2=0a^2 - 3ab - 18b^2 = 0 by b2b^2 to get (ab)2−3(ab)−18=0\left(\frac{a}{b}\right)^2 - 3\left(\frac{a}{b}\right) - 18 = 0, or k2−3k−18=0k^2 - 3k - 18 = 0. Factoring yields (k−6)(k+3)=0(k - 6)(k + 3) = 0, so k=6k = 6 or k=−3k = -3. If k=6k = 6, target value is 6+96=7.56 + \frac{9}{6} = 7.5. If k=−3k = -3, target value is −3+9−3=−6-3 + \frac{9}{-3} = -6.
Since two distinct numerical outcomes are possible for the target expression, Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) independently.
a>0a > 0 and b>0b > 0 implies k=ab>0k = \frac{a}{b} > 0, but gives no numerical constraint on kk.
Without an equation linking aa and bb, Statement (2) alone is NOT sufficient.
4
Evaluate Statement (1) and Statement (2) combined.
Statement (1) gives k=6k = 6 or k=−3k = -3. Statement (2) requires k>0k > 0, eliminating k=−3k = -3 and establishing k=6k = 6 uniquely. Thus, the target value is uniquely 7.57.5.
Combining both statements uniquely determines the ratio kk and therefore the target expression.

Key Concept

Rephrasing homogeneous algebraic expressions into single ratio variables and analyzing degree constraints in quadratic systems.
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Topics

Algebraic Equations and Systems in Data Sufficiency

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