Examkin

Question

Difficulty: MediumAlgebraic Equations and Systems in Data Sufficiency

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

(1) a+b=3aba + b = 3ab
(2) a−b=aba - b = ab

  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.
Evaluating each statement independently shows that Statement (1) reduces the expression to 9ab−29ab - 2 and Statement (2) reduces it to ab+2ab + 2, both of which depend on the value of abab. Combining both statements yields a system of linear equations in aa and bb, giving a=1a = 1 and b=1/2b = 1/2. This uniquely determines the value of the target expression as 5/25/2. Therefore, both statements together are sufficient, but neither statement alone is sufficient.

Step-by-Step Solution

1
Rephrase the target expression.
a2+b2ab=ab+ba\frac{a^2 + b^2}{ab} = \frac{a}{b} + \frac{b}{a}. Alternatively, squaring identities give a2+b2ab=(a+b)2−2abab=(a+b)2ab−2\frac{a^2+b^2}{ab} = \frac{(a+b)^2 - 2ab}{ab} = \frac{(a+b)^2}{ab} - 2.
Simplifying the target expression shows what combination of variables is required.
2
Evaluate Statement (1) alone: a+b=3aba + b = 3ab.
Substituting a+b=3aba+b = 3ab into the target expression yields (3ab)2−2abab=9ab−2\frac{(3ab)^2 - 2ab}{ab} = 9ab - 2.
Since the value depends on abab, and abab can take multiple non-zero values (e.g., if a=1,b=1/2a=1, b=1/2, ab=1/2ab=1/2 and the expression is 2.52.5; if a=2,b=2/5a=2, b=2/5, ab=4/5ab=4/5 and the expression is 5.25.2), Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) alone: a−b=aba - b = ab.
Expressing a2+b2=(a−b)2+2ab=(ab)2+2aba^2+b^2 = (a-b)^2 + 2ab = (ab)^2 + 2ab, the target expression becomes (ab)2+2abab=ab+2\frac{(ab)^2 + 2ab}{ab} = ab + 2.
Since the value depends on abab, which is not fixed by Statement (2) alone, Statement (2) alone is NOT sufficient.
4
Evaluate Statements (1) and (2) together.
Adding the two equations (a+b)+(a−b)=3ab+ab(a + b) + (a - b) = 3ab + ab gives 2a=4ab2a = 4ab. Since a≠0a \neq 0, dividing by 2a2a yields b=12b = \frac{1}{2}.
Substituting b=12b = \frac{1}{2} into a−b=aba - b = ab gives a−12=12a  ⟹  a=1a - \frac{1}{2} = \frac{1}{2}a \implies a = 1. With a=1a=1 and b=12b=\frac{1}{2}, ab=12ab = \frac{1}{2}, and a2+b2ab=1+1/41/2=52\frac{a^2+b^2}{ab} = \frac{1 + 1/4}{1/2} = \frac{5}{2}, giving a single unique value.

Key Concept

Algebraic System Reduction and Target Rephrasing in Data Sufficiency
Estimated Time:2m 0s
Rate this question

Topics

Algebraic Equations and Systems in Data Sufficiency

More questions from this topic