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Difficulty: MediumAlgebraic Equations and Systems in Data Sufficiency

If uu and vv are non-zero real numbers, what is the value of u−2vu+v\frac{u - 2v}{u + v}?

(1) 3u2−5uv−2v2=03u^2 - 5uv - 2v^2 = 0
(2) u>v>0u > v > 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 answer identifies that neither statement alone provides a unique value for the expression, but combining them eliminates the negative root case, leaving a unique value of 0 for the target expression.

Step-by-Step Solution

1
Rephrase the target expression
Dividing numerator and denominator by vv, the target expression is uv−2uv+1\frac{\frac{u}{v} - 2}{\frac{u}{v} + 1}. Finding a unique value for uv\frac{u}{v} determines the target expression.
Simplifying the question target clarifies what information is necessary to achieve sufficiency.
2
Evaluate Statement (1) independently
Factor 3u2−5uv−2v2=03u^2 - 5uv - 2v^2 = 0 as (3u+v)(u−2v)=0(3u + v)(u - 2v) = 0. This gives 3u+v=0  ⟹  u=−v33u + v = 0 \implies u = -\frac{v}{3} or u−2v=0  ⟹  u=2vu - 2v = 0 \implies u = 2v. If u=2vu = 2v, then u−2vu+v=03v=0\frac{u - 2v}{u + v} = \frac{0}{3v} = 0. If u=−v3u = -\frac{v}{3}, then u−2vu+v=−v3−2v−v3+v=−73v23v=−72\frac{u - 2v}{u + v} = \frac{-\frac{v}{3} - 2v}{-\frac{v}{3} + v} = \frac{-\frac{7}{3}v}{\frac{2}{3}v} = -\frac{7}{2}. Two different values are possible.
Since Statement (1) produces two distinct numerical outcomes, it is not sufficient alone.
3
Evaluate Statement (2) independently
Statement (2) states u>v>0u > v > 0. This indicates that both uu and vv are positive, but gives no fixed algebraic equality for uv\frac{u}{v}.
Infinitely many positive pairs (u,v)(u, v) satisfy u>v>0u > v > 0 while yielding different values for the expression. Statement (2) is not sufficient alone.
4
Evaluate Statements (1) and (2) combined
From Statement (2), u>0u > 0 and v>0v > 0, so 3u+v>03u + v > 0. Therefore, the factor 3u+v=03u + v = 0 is impossible. This leaves u−2v=0u - 2v = 0 as the only valid relation, so u=2vu = 2v. Substituting u=2vu = 2v yields 2v−2v2v+v=0\frac{2v - 2v}{2v + v} = 0.
Combining the inequality constraint with the quadratic factorization uniquely determines the value of the target expression.

Key Concept

Algebraic Equations and Systems in Data Sufficiency
Estimated Time:2m 0s
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Algebraic Equations and Systems in Data Sufficiency

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