Question

If xy $\neq$ 0, is $a > \frac{y}{x}$?
1. $a = \frac{1}{x} + \frac{1}{y}$
2. x and y are positive integers 
 

• A. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
• C. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
• D. EACH statement ALONE is sufficient to answer the question asked
• E. Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient

Hint:

For "Yes/No" Data Sufficiency questions, your goal is to find a counterexample. If you can find one case that gives a "Yes" and another that gives a "No" (while satisfying the given statements), the information is not sufficient. Start with small, simple numbers like 1, 2, and -1 to test the statements.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept: 
This is a "Yes/No" data sufficiency question comparing the value of \(a\) to the ratio \(\frac{y}{x}\). We need to determine if the given statements provide enough information to always give a "Yes" or always give a "No" answer. (Note: The OCR for statement 1 was ambiguous; this solution assumes the most likely interpretation, $a = \frac{1}{x} + \frac{1}{y}$). 
 

Step 2: Key Formula or Approach: 
We will substitute the expression for \(a\) from Statement (1) into the inequality in the question and then use the constraints from Statement (2) to test if the inequality holds true for all possible values of \(x\) and \(y\). 
 

Step 3: Detailed Explanation: 
Analyzing Statement (1): $a = \frac{1}{x} + \frac{1}{y}$
Substituting this into the question, we get: \[ \text{Is } \frac{1}{x} + \frac{1}{y} > \frac{y}{x}? \] The variables \(x\) and \(y\) can be any non-zero real numbers. Let's test some cases. - Case 1: Let \(x=1, y=1\). The inequality becomes: Is \(\frac{1}{1} + \frac{1}{1} > \frac{1}{1}\)? Is \(2 > 1\)? Yes. - Case 2: Let \(x=1, y=2\). The inequality becomes: Is \(\frac{1}{1} + \frac{1}{2} > \frac{2}{1}\)? Is \(1.5 > 2\)? No. Since we can get both "Yes" and "No" answers, Statement (1) ALONE is not sufficient. 
Analyzing Statement (2): x and y are positive integers 
This statement gives us information about \(x\) and \(y\), but tells us nothing about \(a\). The value of \(a\) could be anything. - If \(a=100\), \(x=1\), \(y=1\), then \(a > y/x\) because \(100 > 1\). (Yes) - If \(a=0\), \(x=1\), \(y=1\), then \(a\) is not greater than \(y/x\) because \(0\) is not greater than \(1\). (No) Therefore, Statement (2) ALONE is not sufficient. 
Analyzing Statements (1) and (2) Together: 
We now have both pieces of information: - \(a = \frac{1}{x} + \frac{1}{y}\) - \(x\) and \(y\) are positive integers. The question is: Is \(\frac{1}{x} + \frac{1}{y} > \frac{y}{x}\)? 
Since \(x\) and \(y\) are positive, we can multiply the inequality by \(xy\) without changing the direction of the inequality sign. \[ \text{Is } (\frac{1}{x} + \frac{1}{y})xy > (\frac{y}{x})xy? \] \[ \text{Is } y + x > y^2? \] Let's test this with allowed values for \(x\) and \(y\). - Case 1: Let \(y=1\). The inequality becomes: Is \(1+x > 1^2\)? Is \(1+x > 1\)? Is \(x > 0\)? Since \(x\) is a positive integer, \(x \ge 1\), so \(x > 0\) is always true. In this case, the answer to the question is "Yes". For example, if \(x=5, y=1\), we check if \(5+1 > 1^2\), which is \(6>1\), a "Yes". - Case 2: Let \(y=2\). The inequality becomes: Is \(2+x > 2^2\)? Is \(2+x > 4\)? Is \(x > 2\)? This is not always true. - If we pick \(x=3, y=2\), then \(x > 2\) is true. The answer is "Yes". - If we pick \(x=1, y=2\), then \(x > 2\) is false. The answer is "No". - If we pick \(x=2, y=2\), then \(x > 2\) is false. The answer is "No". Since we can still get both "Yes" and "No" answers even with both statements, the information is not sufficient. 
 

Step 4: Final Answer: 
Even when combined, the statements are not sufficient to determine a definitive answer. The correct option is (A).