We are given the inequality:

$$
\(x<0<y\)
$$


This tells us two things:
1. $x$ is negative $(x<0)$.
2. $y$ is positive $(y>0)$.

We are then presented with two quantities:
- Quantity A: $\(\frac{x}{y}\)$
- Quantity B: $\(\frac{y}{x}\)$

Our goal is to compare these two quantities and determine which one is larger, or if they are equal, or if the relationship cannot be determined from the given information.

Analyzing the Given Information
First, let's recall that:
- $x$ is negative.
- $y$ is positive.

This means:
- $\(\frac{x}{y}\)$ : A negative number divided by a positive number is negative.
- $\(\frac{y}{x}\)$ : A positive number divided by a negative number is negative.

So both quantities are negative. To compare them, we can consider their absolute values or find a relationship between them.

Comparing the Two Quantities
Let's denote:
- Quantity A: $\(\frac{x}{y}\)$
- Quantity B: $\(\frac{y}{x}\)$

One way to compare them is to divide Quantity A by Quantity B:

$$
\(\frac{\text { Quantity A }}{\text { Quantity B }}=\frac{\frac{x}{y}}{\frac{y}{x}}=\frac{x}{y} \times \frac{x}{y}=\left(\frac{x}{y}\right)^2\)
$$


Since $\(\frac{x}{y}\)$ is a non-zero real number (because $x \neq 0$ and $y \neq 0$ ), its square is always positive:

$$
\(\left(\frac{x}{y}\right)^2>0\)
$$


This tells us that $\(\frac{\text { Quantity A }}{\text { Quantity B }}>0\)$, which implies that Quantity A and Quantity B have the same sign. We already knew both are negative, but this confirms that their ratio is positive.

However, this doesn't directly tell us which one is larger. Let's consider specific values to understand the relationship better.

Testing with Specific Numbers
Case 1: Let $x=-1$ and $y=1$
- Quantity A: $\(\frac{-1}{1}=-1\)$
- Quantity B: $\(\frac{1}{-1}=-1\)$

Here, Quantity A = Quantity B.
Case 2: Let $x=-2$ and $y=1$
- Quantity A: $\(\frac{-2}{1}=-2\)$
- Quantity B: $\(\frac{1}{-2}=-0.5\)$

Now, $\(-2<-0.5\)$, so Quantity A < Quantity B.


Case 3: Let $x=-1$ and $y=2$
- Quantity A: $\(\frac{-1}{2}=-0.5\)$
- Quantity B: $\(\frac{2}{-1}=-2\)$

Here, $\(-0.5>-2\)$, so Quantity $A>$ Quantity $B$.
From these examples, we see that:
- In some cases, Quantity A equals Quantity B.
- In others, Quantity A is less than Quantity B.
- And in yet others, Quantity A is greater than Quantity B.

This suggests that the relationship between Quantity A and Quantity B depends on the specific values of $x$ and $y$, not just their signs.

General Approach Without Specific Numbers
Let's consider the general case where $x<0$ and $y>0$. We want to compare $\(\frac{x}{y}$ and $\frac{y}{x}\)$.
One way is to subtract one from the other:

$$
\(\frac{x}{y}-\frac{y}{x}=\frac{x^2-y^2}{x y}=\frac{(x-y)(x+y)}{x y}\)
$$


Now, let's analyze the sign of this expression:
1. $x y: x<0$ and $y>0$, so $x y<0$.
2. $x-y: x$ is negative, $y$ is positive, so $x-y$ is negative (since subtracting a positive from a negative makes it more negative).
3. $x+y$ : This can be positive, negative, or zero depending on whether $\(|x|<y,|x|>y\)$, or $\(|x|=\)$ $\(y\)$


So, the sign of $\(\frac{(x-y)(x+y)}{x y}\)$ depends on $x+y$ :
- If $\(x+y>0\)$ (i.e., $\(|x|<y\)$ ):
$\circ x-y$ is negative, $x+y$ is positive, $x y$ is negative.
- Negative $\(\times\)$ Positive $=$ Negative, divided by Negative $=$ Positive .
- So, $\(\frac{x}{y}-\frac{y}{x}>0 \Rightarrow \frac{x}{y}>\frac{y}{x}\)$.
- If $\(x+y<0\)$ (i.e., $\(|x|>y\)$ ):
$\(\circ x-y\)$ is negative, $x+y$ is negative, $x y$ is negative.
- Negative $\(\times\)$ Negative $=$ Positive, divided by Negative $=$ Negative .
- So, $\(\frac{x}{y}-\frac{y}{x}<0 \Rightarrow \frac{x}{y}<\frac{y}{x}\)$.
- If $x+y=0$ (i.e., $|x|=y$ ):
- The numerator is zero, so $\(\frac{x}{y}-\frac{y}{x}=0 \Rightarrow \frac{x}{y}=\frac{y}{x}\)$.

This confirms our earlier observations with specific numbers:
- When $|x|=y$, the quantities are equal.
- When $|x|<y$, Quantity A is greater.
- When $|x|>y$, Quantity B is greater.

Conclusion
Given that $x<0<y$, the relationship between Quantity A and Quantity B depends on the relativ magnitudes of $|x|$ and $y$ :
1. If $|x|=y$, then Quantity $\(\mathrm{A}=\)$ Quantity B .
2. If $|x|<y$, then Quantity $\(\mathrm{A}>\)$ Quantity B .
3. If $|x|>y$, then Quantity $\(\mathrm{A}<\)$ Quantity B .

Since the given information does not specify the relationship between $|x|$ and $y$, we cannot determine a fixed relationship between Quantity A and Quantity B based solely on $\(x<0<y$\).