Question

If \( \mathbf{x} \) and \( \mathbf{y} \) are positive integers then the following is always true?
{2x - 3y < 0
(1) x = (y - 1)
(2) x > y

• 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. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed

Hint:

In Data Sufficiency, proving a "No" is just as sufficient as proving a "Yes". However, if substituting values gives a "Yes" in one case and a "No" in another, the statement is insufficient.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This is a Data Sufficiency question. We need to determine if we can conclusively answer "Yes" or "No" to the question "Is \( 2x - 3y < 0 \) ALWAYS true?" given that \( x \) and \( y \) are positive integers (\( 1, 2, 3, \dots \)), using the provided statements.


Step 2: Detailed Explanation:
Let's evaluate Statement (1) alone:
Statement (1): \( x = y - 1 \)
Substitute this value of \( x \) into the expression \( 2x - 3y \):
\[ 2(y - 1) - 3y = 2y - 2 - 3y = -y - 2 \] Since \( x \) is a positive integer (\( x \ge 1 \)), we know that \( y - 1 \ge 1 \implies y \ge 2 \).
For any value of \( y \ge 2 \), the expression \( -y - 2 \) will always be a negative number.
Therefore, \( -y - 2 < 0 \) is ALWAYS true.
This means Statement (1) alone is sufficient to answer the question with a definitive "Yes".
Let's evaluate Statement (2) alone:
Statement (2): \( x > y \)
Let's test with different positive integer values.
Case A: Let \( y = 1 \) and \( x = 2 \). (Satisfies \( x > y \))
\[ 2x - 3y = 2(2) - 3(1) = 4 - 3 = 1 \] Here, \( 1 < 0 \) is False.
Case B: Let \( y = 3 \) and \( x = 4 \). (Satisfies \( x > y \))
\[ 2x - 3y = 2(4) - 3(3) = 8 - 9 = -1 \] Here, \( -1 < 0 \) is True.
Because we can get both a "False" and a "True" answer depending on the numbers chosen, Statement (2) alone is NOT sufficient.


Step 3: Final Answer:
Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.