Question

If xy $\neq$ 0, is $\frac{1}{x} + \frac{1}{y} = 16$?
1. x + y = 16xy 
2. x = y 
 

• A. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed
• B. EACH statement ALONE is 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. Statement (2) ALONE is sufficient, but statement (1) alone is not 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:

In algebraic data sufficiency questions, always try to manipulate the expression in the question stem first. Here, changing $\frac{1}{x} + \frac{1}{y}$ to $\frac{x+y}{xy}$ makes it immediately obvious how Statement (1) relates to the question.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question asks whether the sum of the reciprocals of \(x\) and \(y\) is equal to 16. This is a "Yes/No" data sufficiency question. A statement is sufficient if it always leads to the answer "Yes" or always leads to the answer "No".

Step 2: Key Formula or Approach:
The expression in the question is \(\frac{1}{x} + \frac{1}{y}\). We can combine this into a single fraction: \[ \frac{1}{x} + \frac{1}{y} = \frac{y}{xy} + \frac{x}{xy} = \frac{x+y}{xy} \] So, the question is equivalent to asking: Is \(\frac{x+y}{xy} = 16\)?

Step 3: Detailed Explanation:
Analyzing Statement (1): x + y = 16xy
We are given the equation \(x+y = 16xy\).
The question is whether \(\frac{1}{x} + \frac{1}{y} = 16\), which we found is equivalent to asking if \(\frac{x+y}{xy} = 16\).
From the information in the problem stem, we know that \(xy \neq 0\). Therefore, we can divide the equation from Statement (1) by \(xy\): \[ \frac{x+y}{xy} = \frac{16xy}{xy} \] \[ \frac{x+y}{xy} = 16 \] This directly answers the question with a "Yes".
Therefore, Statement (1) ALONE is sufficient.
Analyzing Statement (2): x = y
This statement tells us that \(x\) and \(y\) are equal.
Let's substitute \(y=x\) into the expression from the question: \[ \frac{1}{x} + \frac{1}{y} = \frac{1}{x} + \frac{1}{x} = \frac{2}{x} \] The question becomes: Is \(\frac{2}{x} = 16\)?
This simplifies to: Is \(x = \frac{2}{16} = \frac{1}{8}\)?
We do not know the value of \(x\).
- If \(x = \frac{1}{8}\), then the answer is "Yes". - If \(x=1\), then \(\frac{2}{1} = 2 \neq 16\), so the answer is "No". Since we can get both a "Yes" and a "No" answer, this statement is not sufficient.

Step 4: Final Answer:
Statement (1) alone provides a definitive "Yes" answer, while Statement (2) does not. Therefore, Statement (1) alone is sufficient. The correct option is (C).