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Difficulty: EasyAlgebraic Equations and Systems in Data Sufficiency

If pp and qq are numbers, what is the value of the product pqpq?

(1) (p+q)2=49(p + q)^2 = 49
(2) p2+q2=25p^2 + q^2 = 25

  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 to determine that pq=12pq = 12, but neither statement alone is sufficient.
The correct option states that both statements together are sufficient, but neither alone is sufficient. Expanding (p+q)2=p2+2pq+q2(p+q)^2 = p^2 + 2pq + q^2 allows substituting (p+q)2=49(p+q)^2 = 49 from statement (1) and p2+q2=25p^2 + q^2 = 25 from statement (2), yielding 49=25+2pq49 = 25 + 2pq, which uniquely determines pq=12pq = 12. Neither statement alone isolates pqpq.

Step-by-Step Solution

1
Rephrase the target question using algebraic identities
Recall the identity (p+q)2=p2+2pq+q2(p + q)^2 = p^2 + 2pq + q^2. Rearranging gives 2pq=(p+q)2−(p2+q2)2pq = (p + q)^2 - (p^2 + q^2). Thus, knowing both (p+q)2(p + q)^2 and (p2+q2)(p^2 + q^2) will determine pqpq.
Target rephrasing simplifies evaluating statement sufficiency.
2
Evaluate Statement (1) independently
Statement (1) gives (p+q)2=49(p + q)^2 = 49. If p=7p = 7 and q=0q = 0, then pq=0pq = 0. If p=4p = 4 and q=3q = 3, then (4+3)2=49(4+3)^2 = 49 and pq=12pq = 12. Multiple values for pqpq exist.
A statement is sufficient only if it yields one unique value for the target expression.
3
Evaluate Statement (2) independently
Statement (2) gives p2+q2=25p^2 + q^2 = 25. If p=5p = 5 and q=0q = 0, then pq=0pq = 0. If p=4p = 4 and q=3q = 3, then 42+32=254^2 + 3^2 = 25 and pq=12pq = 12. Multiple values for pqpq exist.
Statement (2) alone does not yield a unique product.
4
Combine Statement (1) and Statement (2)
Substitute (p+q)2=49(p + q)^2 = 49 and p2+q2=25p^2 + q^2 = 25 into (p+q)2=p2+2pq+q2(p + q)^2 = p^2 + 2pq + q^2 to get 49=25+2pq2pq=24pq=1249 = 25 + 2pq 2pq = 24 pq = 12. A single, unique value is found.
Combining the statements provides enough information to determine the value of the target expression pqpq uniquely.

Key Concept

Algebraic Expression Manipulation via Quadratic Identities
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Topics

Algebraic Equations and Systems in Data Sufficiency

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