Let's simplify the given complex rational expression step by step.

The expression is:

$\frac{\frac{1}{5y} + \frac{1}{z-5}}{\frac{1}{2z} + \frac{5y}{2z^2 - 10z}}$

Step 1: Simplify the denominator

Start by simplifying the denominator $\frac{1}{2z} + \frac{5y}{2z^2 - 10z}$.

Notice that the term $2z^2 - 10z$ can be factored as:

$2z^2 - 10z = 2z(z - 5)$

So, the denominator becomes:

$\frac{1}{2z} + \frac{5y}{2z(z - 5)}$

Now, to add these fractions, we need a common denominator. The least common denominator (LCD) is $2z(z - 5)$. Rewrite both fractions with this common denominator:

$\frac{1}{2z} = \frac{z - 5}{2z(z - 5)}, \quad \frac{5y}{2z(z - 5)} = \frac{5y}{2z(z - 5)}$

Now, add these two fractions:

$\frac{z - 5}{2z(z - 5)} + \frac{5y}{2z(z - 5)} = \frac{z - 5 + 5y}{2z(z - 5)}$

Step 2: Simplify the numerator

Now simplify the numerator $\frac{1}{5y} + \frac{1}{z - 5}$.

The common denominator here is $5y(z - 5)$. Rewrite both fractions:

$\frac{1}{5y} = \frac{z - 5}{5y(z - 5)}, \quad \frac{1}{z - 5} = \frac{5y}{5y(z - 5)}$

Adding them gives:

$\frac{z - 5 + 5y}{5y(z - 5)}$

Step 3: Combine everything

Now, we have:

$\frac{\frac{z - 5 + 5y}{5y(z - 5)}}{\frac{z - 5 + 5y}{2z(z - 5)}}$

Cancel the common terms $(z - 5 + 5y)$ from the numerator and denominator:

$\frac{1}{5y} \div \frac{1}{2z} = \frac{2z}{5y}$

Final Answer:

The simplified expression is:

$\frac{2z}{5y}$

Would you like more details or have any questions about the steps? Here are some questions you might consider:

1. How do we factor more complicated quadratic expressions?
2. What happens if the numerator and denominator don't have common terms?
3. How do you find the least common denominator in different types of fractions?
4. Can the expression be simplified further with additional constraints?
5. What are common mistakes when simplifying complex rational expressions?

Tip: Always check for common factors between the numerator and the denominator to simplify rational expressions efficiently!