The given expression is:

\[
\frac{5m}{10m^2 + 5mn}
\]

Let's simplify it step-by-step:

1. **Factor the denominator:**

The denominator \( 10m^2 + 5mn \) has a common factor of \( 5m \):

\[
10m^2 + 5mn = 5m(2m + n)
\]

2. **Rewrite the expression:**

\[
\frac{5m}{5m(2m + n)}
\]

3. **Cancel out the common factor \( 5m \) from the numerator and denominator:**

\[
\frac{5m}{5m(2m + n)} = \frac{1}{2m + n}
\]

So, the simplified expression is:

\[
\boxed{\frac{1}{2m + n}}
\]

Would you like more details or have any questions?

Here are 5 related questions you might explore:

1. How would you simplify \(\frac{7x}{14x^2 + 21xy}\)?
2. What happens when the numerator is a polynomial, like \( m^2 + 5m \)?
3. How do you factor expressions involving three variables, like \( 15a^2b + 10ab^2 \)?
4. How do you approach expressions with a binomial numerator, such as \( \frac{3m + n}{6m^2 + 9mn} \)?
5. What’s the difference between simplifying and factoring in algebra?

**Tip:** When simplifying fractions, always look for common factors in the numerator and denominator to cancel out.

Learn how to simplify the fraction 5m / (10m^2 + 5mn) with a detailed step-by-step solution. This problem involves algebraic concepts such as fraction simplification and factoring. Suitable for students in Grades 7-9.

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