https://r2.erweima.ai/mathbot/image/1725587385947ioc0ah61.jpg

The expression in the image is:

\[
\frac{\frac{uv + 5u + 3v + 15}{5v}}{1 + \frac{5}{v}}
\]

Let’s simplify this step by step.

### Step 1: Simplify the numerator \(\frac{uv + 5u + 3v + 15}{5v}\)
Each term in the numerator can be divided by \(5v\):

- \(\frac{uv}{5v} = \frac{u}{5}\)
- \(\frac{5u}{5v} = \frac{u}{v}\)
- \(\frac{3v}{5v} = \frac{3}{5}\)
- \(\frac{15}{5v} = \frac{3}{v}\)

Thus, the simplified numerator becomes:

\[
\frac{u}{5} + \frac{u}{v} + \frac{3}{5} + \frac{3}{v}
\]

### Step 2: Keep the denominator as it is
The denominator is:

\[
1 + \frac{5}{v}
\]

### Step 3: Final expression
The expression becomes:

\[
\frac{\frac{u}{5} + \frac{u}{v} + \frac{3}{5} + \frac{3}{v}}{1 + \frac{5}{v}}
\]

This is the simplified version. Do you want me to continue simplifying further or evaluate this for specific values of \(u\) and \(v\)? 

Here are some follow-up questions related to this:

1. Do you want to substitute specific values for \(u\) and \(v\)?
2. Are you interested in rewriting the expression with a common denominator for easier manipulation?
3. Should we check for any possible factorization in the numerator?
4. Do you need a graphical interpretation of this expression?
5. Would you like to explore limits as \(v \to 0\) or \(v \to \infty\)?

**Tip**: Simplifying fractions is easier when you look for common factors or attempt to combine terms into a single fraction early.

Simplify the given algebraic expression: ((uv + 5u + 3v + 15) / 5v) / (1 + 5/v).

Learn how to simplify complex rational expressions with variables u and v. This example covers dividing algebraic fractions, finding common denominators, and simplifying step-by-step. Suitable for students in Grades 10-12.

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