To solve the expression \(\frac{\frac{5}{x}}{\frac{5}{x} + 2}\), follow these steps:

### Step 1: Simplify the denominator
The denominator is \(\frac{5}{x} + 2\). To combine these terms, express the number 2 with a common denominator:
\[
\frac{5}{x} + 2 = \frac{5}{x} + \frac{2x}{x} = \frac{5 + 2x}{x}
\]
So the denominator becomes \(\frac{5 + 2x}{x}\).

### Step 2: Substitute into the original expression
The expression now becomes:
\[
\frac{\frac{5}{x}}{\frac{5 + 2x}{x}}
\]

### Step 3: Simplify the division of fractions
When dividing fractions, multiply by the reciprocal of the denominator:
\[
\frac{\frac{5}{x}}{\frac{5 + 2x}{x}} = \frac{5}{x} \times \frac{x}{5 + 2x}
\]

### Step 4: Cancel out the common factor
The \(x\)'s in the numerator and denominator cancel out:
\[
\frac{5}{5 + 2x}
\]

Thus, the simplified expression is:
\[
\frac{5}{5 + 2x}
\]

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would you solve \(\frac{\frac{3}{x}}{\frac{3}{x} + 1}\)?
2. What happens if you set \(x = 0\) in the original expression?
3. How can you solve \(\frac{\frac{a}{b}}{\frac{a}{b} + c}\) in general?
4. What is the effect of increasing \(x\) on the simplified expression?
5. How would this change if the numerator was \(\frac{10}{x}\) instead of \(\frac{5}{x}\)?

**Tip**: When dividing fractions, always remember to multiply by the reciprocal!

Learn how to simplify the rational expression (5/x) / (5/x + 2) with this step-by-step guide. This solution involves key algebraic concepts such as simplifying fractions and dividing rational expressions. Ideal for students in Grades 8-10.

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