`(x)/(y)lt0`
`y gt0`
`{:("Quantity A","Quantity B"),(y-x,xy):}`

Options

A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the infromation given.

Answer

To solve the problem, we need to analyze the given conditions and compare the two quantities.

**Given:**
1. \(\frac{x}{y} < 0\)
2. \(y > 0\)

From the first condition, since \(\frac{x}{y} < 0\) and \(y\) is positive, it implies that \(x\) must be negative. This is because a positive \(y\) divided by a negative \(x\) results in a negative value.

### Step 1: Determine the signs of \(x\) and \(y\)
- We have \(x < 0\) (negative) and \(y > 0\) (positive).

### Step 2: Analyze Quantity A: \(y - x\)
- Since \(x\) is negative, \(-x\) is positive. Therefore, we can rewrite \(y - x\) as:
\[ y - x = y + (-x) \]
- Both \(y\) and \(-x\) are positive. The sum of two positive numbers is always positive. Thus:
\[ y - x > 0 \]
- Therefore, Quantity A is positive.

### Step 3: Analyze Quantity B: \(xy\)
- Here, we have \(x < 0\) and \(y > 0\). The product of a negative number and a positive number is negative. Thus:
\[ xy < 0 \]
- Therefore, Quantity B is negative.

### Step 4: Compare Quantity A and Quantity B
- From our analysis:
- Quantity A is positive.
- Quantity B is negative.

Since a positive quantity is always greater than a negative quantity, we conclude that:
\[ \text{Quantity A} > \text{Quantity B} \]

### Final Conclusion
The answer is that Quantity A is greater than Quantity B.

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