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Difficulty: MediumAlgebraic Equations and Systems in Data Sufficiency

If mm and nn are real numbers, what is the value of m2−n2m^2 - n^2?

(1) (m+n)2=36(m + n)^2 = 36
(2) m2+n2=20m^2 + n^2 = 20

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

Answer

Statements (1) and (2) TOGETHER are NOT sufficient.
The correct response identifies that statements (1) and (2) together leave sign ambiguity in the difference of squares because mm and nn can swap values while maintaining the same sum and sum of squares.

Step-by-Step Solution

1
Rephrase the target question
The target expression is m2−n2=(m+n)(m−n)m^2 - n^2 = (m + n)(m - n). To find a unique value, we need a unique numerical value for (m+n)(m−n)(m + n)(m - n).
Factoring the difference of squares clarifies what information is necessary.
2
Evaluate Statement (1) alone
(m+n)2=36  ⟹  m+n=6(m + n)^2 = 36 \implies m + n = 6 or m+n=−6m + n = -6. We have no information about m−nm - n.
Since m−nm - n can take infinitely many values, m2−n2m^2 - n^2 is not uniquely determined. Statement (1) is insufficient.
3
Evaluate Statement (2) alone
m2+n2=20m^2 + n^2 = 20.
Different pairs of numbers like (m,n)=(20,0)(m, n) = (\sqrt{20}, 0) give m2−n2=20m^2 - n^2 = 20, while (m,n)=(0,20)(m, n) = (0, \sqrt{20}) give m2−n2=−20m^2 - n^2 = -20. Statement (2) is insufficient.
4
Evaluate Statements (1) and (2) combined
Expand Statement (1): m2+2mn+n2=36m^2 + 2mn + n^2 = 36. Substitute Statement (2): 20+2mn=36  ⟹  2mn=16  ⟹  mn=820 + 2mn = 36 \implies 2mn = 16 \implies mn = 8.
Now test values satisfying m+n=6m + n = 6 and mn=8mn = 8:
Case 1: m=4,n=2  ⟹  m2−n2=42−22=16−4=12m = 4, n = 2 \implies m^2 - n^2 = 4^2 - 2^2 = 16 - 4 = 12.
Case 2: m=2,n=4  ⟹  m2−n2=22−42=4−16=−12m = 2, n = 4 \implies m^2 - n^2 = 2^2 - 4^2 = 4 - 16 = -12.
Since m2−n2m^2 - n^2 yields two distinct values (1212 and −12-12), the statements together are insufficient.
Symmetry between variables in a non-linear system creates multiple possible values for non-symmetric target expressions.

Key Concept

Symmetry and Degree Ambiguity in Algebraic Data Sufficiency
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Topics

Algebraic Equations and Systems in Data Sufficiency

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