Question

What is the value of \( w + q \)?
(1) \( 3w = 3q - 3 \)
(2) \( 5w + 5q = 5 \)

• 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 combining statements, always substitute known values from one statement into the other to simplify the equation.

Answer 1

The Correct Option is C

Step 1: Analyze statement (1).
Statement (1) tells us that \( 3w = 3q - 3 \). We can simplify this: \[ w = q - 1 \] This equation alone does not give us the value of \( w + q \), so statement (1) alone is not sufficient.
Step 2: Analyze statement (2).
Statement (2) tells us that \( 5w + 5q = 5 \), which can be simplified as: \[ w + q = 1 \] This gives us a direct relationship between \( w \) and \( q \), but we cannot use this information alone to find \( w + q \) because the individual values of \( w \) and \( q \) are unknown. Thus, statement (2) alone is also insufficient.
Step 3: Combine statements (1) and (2).
From statement (1), we know that \( w = q - 1 \), and from statement (2), we know that \( w + q = 1 \). Substituting \( w = q - 1 \) into \( w + q = 1 \), we get: \[ (q - 1) + q = 1 \quad \implies \quad 2q - 1 = 1 \quad \implies \quad 2q = 2 \quad \implies \quad q = 1 \] Substituting \( q = 1 \) back into \( w = q - 1 \): \[ w = 1 - 1 = 0 \] Thus, \( w + q = 0 + 1 = 1 \). \[ \boxed{1} \]