https://file.aiquickdraw.com/mathbot/file/1725406412980h42c5gfi.jpg

The expression shown in the image is:

\[
\frac{\frac{1}{2} + \frac{b}{4}}{\frac{1}{2} + \frac{b}{3}}
\]

To simplify this expression, follow these steps:

### 1. Simplify the Numerator and Denominator:
- **Numerator**: \(\frac{1}{2} + \frac{b}{4}\)
  - Find a common denominator for the terms. The common denominator for \(2\) and \(4\) is \(4\).
  - Rewrite \(\frac{1}{2}\) as \(\frac{2}{4}\).
  - So, the numerator becomes \(\frac{2}{4} + \frac{b}{4} = \frac{2 + b}{4}\).

- **Denominator**: \(\frac{1}{2} + \frac{b}{3}\)
  - Find a common denominator for the terms. The common denominator for \(2\) and \(3\) is \(6\).
  - Rewrite \(\frac{1}{2}\) as \(\frac{3}{6}\) and \(\frac{b}{3}\) as \(\frac{2b}{6}\).
  - So, the denominator becomes \(\frac{3}{6} + \frac{2b}{6} = \frac{3 + 2b}{6}\).

### 2. Substitute Back:
Now the expression becomes:

\[
\frac{\frac{2 + b}{4}}{\frac{3 + 2b}{6}}
\]

### 3. Simplify the Fraction:
To simplify this complex fraction, multiply by the reciprocal of the denominator:

\[
\frac{2 + b}{4} \times \frac{6}{3 + 2b} = \frac{(2 + b) \times 6}{4 \times (3 + 2b)} = \frac{6(2 + b)}{4(3 + 2b)}
\]

### 4. Simplify Further:
Simplify the coefficients:

\[
\frac{6(2 + b)}{4(3 + 2b)} = \frac{3(2 + b)}{2(3 + 2b)}
\]

This is the simplified form of the expression.

Would you like any further details or have any questions?

Here are some related questions you might find helpful:
1. How do you simplify complex fractions in general?
2. What are the rules for finding a common denominator?
3. How do you multiply and divide fractions?
4. Can this expression be simplified further under certain conditions?
5. What happens if \(b = 0\) in this expression?

**Tip:** When simplifying fractions, always look for opportunities to factor and cancel common terms to achieve the simplest form.

Simplify the given algebraic expression involving fractions.

Learn how to simplify algebraic expressions involving complex fractions step-by-step. This problem covers key algebraic concepts, including finding common denominators and simplifying complex fractions. Suitable for students in Grades 9-12.

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