To simplify the given expression:

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

### Step 1: Find a common denominator for both the numerator and denominator.
- In the **numerator**, the common denominator is \( xy \):
\[
\frac{1}{y} - \frac{1}{x} = \frac{x - y}{xy}
\]
- In the **denominator**, the common denominator is also \( xy \):
\[
\frac{1}{y} + \frac{1}{x} = \frac{x + y}{xy}
\]

### Step 2: Rewrite the entire expression using the results from Step 1:
\[
\frac{\frac{x - y}{xy}}{\frac{x + y}{xy}}
\]

### Step 3: Simplify the complex fraction.
Since the denominator is common (\(xy\)) for both the numerator and the denominator, they cancel out:
\[
\frac{x - y}{x + y}
\]

Thus, the simplified form of the expression is:
\[
\boxed{\frac{x - y}{x + y}}
\]

Would you like further explanation or have any other questions?

Here are 5 related questions to deepen your understanding:
1. How would you simplify \(\frac{\frac{1}{a} - \frac{1}{b}}{\frac{1}{a} + \frac{1}{b}}\)?
2. Can you apply this simplification method to a real-world ratio problem?
3. What happens to the expression if \(x = y\)?
4. How would you interpret the result geometrically on a number line?
5. Can you explore what happens when \(x\) or \(y\) approaches 0?

**Tip:** When simplifying complex fractions, try canceling out common terms in both the numerator and denominator to make it easier to handle.

Discover how to simplify the complex fraction ((1/y) - (1/x)) / ((1/y) + (1/x)) using algebraic techniques. This solution breaks down the process of finding common denominators and simplifying rational expressions, perfect for students in Grades 9-11.

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