What is the common denominator of \( y+\frac{y-3}{3} \) in the complex fraction \( \frac{y+\frac{y-3}{3}}{\frac{5}{9}+\frac{2}{3 y}} ? \)

To find the common denominator of the complex fraction \( \frac{y+\frac{y-3}{3}}{\frac{5}{9}+\frac{2}{3 y}} \), we will first identify the denominators in both the numerator and the denominator of the complex fraction. 1. **Numerator**: The expression in the numerator is \( y + \frac{y-3}{3} \). - The term \( y \) can be considered as \( \frac{3y}{3} \) to have a common denominator with \( \frac{y-3}{3} \). - Thus, the numerator becomes: \[ \frac{3y}{3} + \frac{y-3}{3} = \frac{3y + (y - 3)}{3} = \frac{4y - 3}{3} \] 2. **Denominator**: The expression in the denominator is \( \frac{5}{9} + \frac{2}{3y} \). - The denominators here are \( 9 \) and \( 3y \). - The least common multiple (LCM) of \( 9 \) and \( 3y \) can be found by considering the factors: - \( 9 = 3^2 \) - \( 3y = 3 \cdot y \) - The LCM will be \( 9y \). Now we have: - The numerator as \( \frac{4y - 3}{3} \) - The denominator as \( \frac{5}{9} + \frac{2}{3y} \) with a common denominator of \( 9y \). Next, we can rewrite the denominator: \[ \frac{5}{9} = \frac{5y}{9y} \quad \text{and} \quad \frac{2}{3y} = \frac{6}{9y} \] Thus, the denominator becomes: \[ \frac{5y + 6}{9y} \] Now, we can express the entire complex fraction: \[ \frac{\frac{4y - 3}{3}}{\frac{5y + 6}{9y}} = \frac{4y - 3}{3} \cdot \frac{9y}{5y + 6} = \frac{(4y - 3) \cdot 9y}{3(5y + 6)} \] The common denominator of the entire complex fraction is \( 9y \). Thus, the common denominator of \( y + \frac{y-3}{3} \) in the complex fraction is: \[ \boxed{9y} \]