Step 1: Translate the Ratios into an Equation
The problem states:

The ratio of $x$ to $\(\frac{y}{3}\)$ is $\(\frac{1}{3}\)$ times the ratio of $y$ to $x$.
We can write the ratios as fractions:
1. Ratio of $x$ to $\(\frac{y}{3}\)$ :

$$
\(\frac{x}{\frac{y}{3}}\)
$$

2. Ratio of $y$ to $x$ :

$$
\(\frac{y}{x}\)
$$


Now, set up the full equation:

$$
\(\frac{x}{\frac{y}{3}}=\frac{1}{3} \cdot\left(\frac{y}{x}\right)\)
$$

Step 2: Simplify the Equation
First, simplify the complex fraction on the left side:

$$
\(\frac{x}{\frac{y}{3}}=x \cdot \frac{3}{y}=\frac{3 x}{y}\)
$$


Now substitute this back into the main equation:

$$
\(\frac{3 x}{y}=\frac{1}{3} \cdot \frac{y}{x}\)
$$


Step 3: Solve for the Expression $\(\frac{y}{x}\)$
The question asks for the value of $\(\frac{y}{x}\)$. We can rearrange the equation to isolate this expression.
1. Multiply both sides by 3 to clear the fraction on the right:

$$
\(\begin{gathered}
3 \cdot \frac{3 x}{y}=3 \cdot \frac{1}{3} \cdot \frac{y}{x} \\
\frac{9 x}{y}=\frac{y}{x}
\end{gathered}\)
$$

2. To relate $\(\frac{y}{x}\)$ to itself, we can use the identity $\(\frac{9 x}{y}=9 \cdot \frac{x}{y}\)$. Since $\(\frac{x}{y}\)$ is the reciprocal of $\(\frac{y}{x}\)$, let $\(k=\frac{y}{x}\)$. Then $\(\frac{x}{y}=\frac{1}{k}\)$.

Substitute $k$ into the equation:

$$
\(\begin{gathered}
9 \cdot \frac{1}{k}=k \\
\frac{9}{k}=k
\end{gathered}\)
$$

3. Solve for $k$ :

$$
\(\begin{gathered}
9=k^2 \\
k= \pm \sqrt{9} \\
k=3 \text { or } k=-3
\end{gathered}\)
$$


Since $\(k=\frac{y}{x}\)$, the possible values for $\(\frac{y}{x}\)$ are 3 and -3 .

Step 4: Check the Options
We found two possible values: 3 and -3 .
- (A) -3
- (F) 3

The values that could be the value of $\(\frac{y}{x}\)$ are $\(-\mathbf{3}\)$ and $\(\mathbf{3}\)$.