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Difficulty: MediumAlgebraic Equations and Systems in Data Sufficiency

If rr and ss are non-zero real numbers, what is the value of r2+2s2rs\frac{r^2 + 2s^2}{rs}?

(1) r2−3rs+2s2=0r^2 - 3rs + 2s^2 = 0
(2) r+s=6r + s = 6

  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 correct option states that Statement (1) alone is sufficient while Statement (2) is not. Rephrasing the target expression r2+2s2rs\frac{r^2 + 2s^2}{rs} as rs+2(sr)\frac{r}{s} + 2\left(\frac{s}{r}\right) shows that the target value depends exclusively on the ratio t=rst = \frac{r}{s}. Statement (1) factors into (r−s)(r−2s)=0(r - s)(r - 2s) = 0, giving t=1t = 1 or t=2t = 2. Testing t=1t = 1 yields 1+2=31 + 2 = 3, and testing t=2t = 2 yields 2+1=32 + 1 = 3. Because both possible values of tt evaluate to the same constant result (33), Statement (1) alone uniquely determines the target value and is sufficient. Statement (2) allows infinitely many ratios for r/sr/s, giving different outcomes, and is therefore insufficient.

Step-by-Step Solution

1
Rephrase the target expression algebraically.
Dividing each term in the numerator by the denominator gives r2+2s2rs=rs+2(sr)\frac{r^2 + 2s^2}{rs} = \frac{r}{s} + 2\left(\frac{s}{r}\right). Letting t=rst = \frac{r}{s}, the target expression is t+2tt + \frac{2}{t}.
Simplifying the question stem reveals that we only need to know the ratio rs\frac{r}{s}.
2
Evaluate Statement (1): r2−3rs+2s2=0r^2 - 3rs + 2s^2 = 0.
Divide the entire equation by s2s^2 (since s≠0s \neq 0) to get (rs)2−3(rs)+2=0\left(\frac{r}{s}\right)^2 - 3\left(\frac{r}{s}\right) + 2 = 0, or t2−3t+2=0t^2 - 3t + 2 = 0. Factoring yields (t−1)(t−2)=0(t - 1)(t - 2) = 0, so t=1t = 1 or t=2t = 2.
If t=1t = 1, t+2t=1+2=3t + \frac{2}{t} = 1 + 2 = 3.
If t=2t = 2, t+2t=2+22=3t + \frac{2}{t} = 2 + \frac{2}{2} = 3.
In both cases, the target expression equals 33.
Although there are two solutions for the ratio tt, both lead to the identical value of 33 for the target expression. Thus, Statement (1) alone is sufficient.
3
Evaluate Statement (2): r+s=6r + s = 6.
If r=2r = 2 and s=4s = 4, then rs=12\frac{r}{s} = \frac{1}{2}, giving a target value of 12+4=4.5\frac{1}{2} + 4 = 4.5.
If r=3r = 3 and s=3s = 3, then rs=1\frac{r}{s} = 1, giving a target value of 1+2=31 + 2 = 3.
Since different pairs of rr and ss yield different values for the expression, Statement (2) alone is not sufficient.
Knowing only the linear sum of two variables does not uniquely fix their ratio.

Key Concept

Question Stem Rephrasing and Symmetry in Homogeneous Quadratic Expressions
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Topics

Algebraic Equations and Systems in Data Sufficiency

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