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Let's analyze the math problem you've uploaded.

### Given:
1. \(\frac{a + b}{a - b} = 3\)
2. \(\frac{a}{b} = 2\)

We need to simplify and solve the expression:

\[
\frac{3a + 3b}{a + 4b} + \frac{5b}{a}
\]

### Step 1: Use \(\frac{a}{b} = 2\)
From this equation:
\[
a = 2b
\]

### Step 2: Substitute \(a = 2b\) in the expression \(\frac{3a + 3b}{a + 4b} + \frac{5b}{a}\).

#### First part of the expression: \(\frac{3a + 3b}{a + 4b}\)
\[
\frac{3(2b) + 3b}{2b + 4b} = \frac{6b + 3b}{6b} = \frac{9b}{6b} = \frac{3}{2}
\]

#### Second part of the expression: \(\frac{5b}{a}\)
Since \(a = 2b\),
\[
\frac{5b}{a} = \frac{5b}{2b} = \frac{5}{2}
\]

### Step 3: Combine both parts
\[
\frac{3}{2} + \frac{5}{2} = \frac{3 + 5}{2} = \frac{8}{2} = 4
\]

### Final Answer:
The value of the given expression is \(4\).

Would you like further clarification on any steps?

Here are five related questions to expand this problem:
1. How do we solve similar problems involving systems of equations?
2. What are some common strategies for solving algebraic fractions?
3. How can we simplify complex fractions when there are multiple variables?
4. How does substitution help in solving multi-variable expressions?
5. What is the importance of finding common denominators in fraction problems?

**Tip:** Always double-check your substitutions when working with multiple equations to ensure consistency.

Simplify the expression \(\frac{3a + 3b}{a + 4b} + \frac{5b}{a}\) given that \(\frac{a + b}{a - b} = 3\) and \(\frac{a}{b} = 2\).

This page simplifies the expression \(\frac{3a + 3b}{a + 4b} + \frac{5b}{a}\) using the given conditions \(\frac{a + b}{a - b} = 3\) and \(\frac{a}{b} = 2\). Learn how to apply substitution and simplify complex fractions in algebra. Suitable for students in Grades 9-10.

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