Question

Evaluate the problem, and the two statements - labelled (1) and (2) - that contain certain data or information. Using the given statements, decide whether the information provided is sufficient to answer the question.
Is \( (w + x)a<(y + z)a \)?
Statement 1: \( w<y \) and \( x<z \)
Statement 2: \( a<0 \)

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

When an inequality involves multiplying by a variable (like 'a' here), immediately recognize that you need to know the sign of that variable. Statement 2 provides the sign, and Statement 1 provides the relationship between the other terms. This structure often requires combining both statements for sufficiency.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a Data Sufficiency question involving inequalities. The key principle to remember is that when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. The question is a "Yes/No" question.
Step 2: Detailed Explanation:
The question asks if \( (w + x)a<(y + z)a \). The answer depends on two things: the relationship between \( (w+x) \) and \( (y+z) \), and the sign of \( a \).
Analyze Statement (1): \( w<y \) and \( x<z \).
We can add these two inequalities together: \[ w + x<y + z \] Now, let's return to the original question: Is \( (w + x)a<(y + z)a \)? This depends on the sign of \(a\).

If \(a\) is positive (e.g., \(a=2\)), then we can divide by \(a\) without changing the sign, and the inequality becomes \(w+x<y+z\), which we know is true. So the answer is "Yes".
If \(a\) is negative (e.g., \(a=-2\)), then we must reverse the sign when dividing. The inequality becomes \(w+x>y+z\), which we know is false. So the answer is "No".
Since we can get both "Yes" and "No", Statement (1) is not sufficient.
Analyze Statement (2): \( a<0 \).
This tells us that \(a\) is negative. We can simplify the original inequality by dividing both sides by \(a\) and reversing the inequality sign: The question is equivalent to asking: Is \( w + x>y + z \)? We have no information about the values of w, x, y, and z. We can pick values to make this true or false.

If \(w=5, x=5, y=1, z=1\), then \(10>2\). The answer is "Yes".
If \(w=1, x=1, y=5, z=5\), then \(2>10\). The answer is "No".
Since we can get both "Yes" and "No", Statement (2) is not sufficient.
Analyze Both Statements Together:
From Statement (1), we know that \( w + x<y + z \). From Statement (2), we know that \( a<0 \). The question is: Is \( (w + x)a<(y + z)a \)? Since \(a\) is negative, this inequality is true if and only if \( w + x>y + z \). However, Statement (1) tells us definitively that \( w + x<y + z \). Therefore, the condition \( w + x>y + z \) is false. This means the answer to the question "Is \( (w + x)a<(y + z)a \)" is a definite "No". Since we have a definite "No" answer, the statements together are sufficient.