Rewrite \( \quad 2+\frac{1}{b-2}=\frac{3 b}{b+2} \) as a proportion. Which of these proportions is equivalent to the original equation? \( \frac{3}{b-2}=\frac{3 b}{b+2} \) \( \frac{2 b+3}{b-2}=\frac{3 b}{b+2} \) \( \frac{2 b-3}{b-2}=\frac{3 b}{b+2} \)

To rewrite the equation \( 2+\frac{1}{b-2}=\frac{3 b}{b+2} \) as a proportion, we can express \( 2 \) in terms of the common denominator. Multiply both sides by \( (b-2)(b+2) \) to eliminate the denominators: \[ 2(b-2)(b+2) + (b+2) = 3b(b-2). \] Now reorganizing this, we end up with a single fraction on each side, leading us to the form of a proportion. When we analyze the proposed proportions, let's check which one matches: 1. **Option 1:** \( \frac{3}{b-2}=\frac{3 b}{b+2} \) is not equivalent because the left side does not represent \( 2 + \frac{1}{b-2} \). 2. **Option 2:** \( \frac{2b + 3}{b-2}=\frac{3b}{b+2} \) can be derived by substituting \( 2 \) as \( \frac{2b-4 + 4}{b-2} \) which simplifies correctly when set to this form. 3. **Option 3:** \( \frac{2b-3}{b-2}=\frac{3b}{b+2} \) is also not equivalent since it miscalculates the adjustment of the constant term. Therefore, the option that represents a proportion equivalent to the original equation is: \[ \frac{2b + 3}{b-2}=\frac{3b}{b+2}. \]