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Let's solve the expression step by step:

You are asked to simplify the expression:

\[
\frac{\left( \frac{1}{y} + \frac{1}{x} \right)}{\frac{3}{xy}}
\]

### Step 1: Simplify the numerator
The numerator is \( \frac{1}{y} + \frac{1}{x} \). We can add these two fractions by getting a common denominator:

\[
\frac{1}{y} + \frac{1}{x} = \frac{x + y}{xy}
\]

### Step 2: Rewrite the entire expression
Now we have:

\[
\frac{\frac{x + y}{xy}}{\frac{3}{xy}}
\]

### Step 3: Divide the two fractions
When dividing two fractions, we multiply by the reciprocal of the denominator:

\[
\frac{\frac{x + y}{xy}}{\frac{3}{xy}} = \frac{x + y}{xy} \times \frac{xy}{3}
\]

### Step 4: Simplify the expression
The \( xy \) terms cancel out, leaving:

\[
\frac{x + y}{3}
\]

### Conclusion:
The expression simplifies to:

\[
\frac{x + y}{3}
\]

Thus, the correct answer is **D**.

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Would you like more details on any of these steps? Here are five related questions to explore:

1. How do you find a common denominator for fractions with different denominators?
2. What is the rule for dividing fractions?
3. How would the expression change if there were a constant in the numerator of the original equation?
4. How can you verify the final simplified answer?
5. What other algebraic expressions can be simplified using similar methods?

**Tip:** When simplifying complex fractions, always simplify the numerator and denominator separately before dividing.

Which of the following is equivalent to \( \frac{ \left( \frac{1}{y} + \frac{1}{x} \right) }{ \frac{3}{xy} } \)?

Learn how to simplify the expression \( \frac{ \frac{1}{y} + \frac{1}{x} }{ \frac{3}{xy} } \) step by step. This problem covers key algebra concepts, such as adding fractions with different denominators and dividing rational expressions. Suitable for students in Grades 8-10.

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