Question

Is \( x>y \)? (1) \( x = y + 2 \) (2) \( \frac{x}{2} = y - 1 \)

• 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:

For simple algebraic comparisons, directly compare the expressions or manipulate the equations to find the relationship between the variables.

Answer 1

The Correct Option is D

Step 1: Analyze statement (1).
Statement (1) tells us that \( x = y + 2 \), which is a direct comparison between \( x \) and \( y \). From this, it is clear that \( x>y \) because \( x \) is 2 units greater than \( y \). Thus, statement (1) alone is sufficient.
Step 2: Analyze statement (2).
Statement (2) tells us that \( \frac{x}{2} = y - 1 \). We can rearrange this equation as: \[ x = 2(y - 1) = 2y - 2 \] Comparing this with \( x = y + 2 \) from statement (1), we see that \( 2y - 2 = y + 2 \), which simplifies to: \[ y = 4 \] Substituting \( y = 4 \) back into \( x = y + 2 \), we get \( x = 6 \), so \( x>y \). Thus, statement (2) alone is also sufficient.
\[ \boxed{D} \]