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 \( y + \frac{y-3}{3} \), we first need to combine the terms. The term \( y \) can be written as \( \frac{3y}{3} \) for ease of addition: \[ y + \frac{y-3}{3} = \frac{3y}{3} + \frac{y-3}{3} = \frac{3y + (y - 3)}{3} = \frac{3y + y - 3}{3} = \frac{4y - 3}{3}. \] Now, we can express the entire complex fraction: \[ \frac{y+\frac{y-3}{3}}{\frac{5}{9}+\frac{2}{3y}} = \frac{\frac{4y - 3}{3}}{\frac{5}{9} + \frac{2}{3y}}. \] Next, we need to find the common denominator of the denominator \( \frac{5}{9} + \frac{2}{3y} \). The least common multiple (LCM) of the denominators 9 and \( 3y \) is \( 9y \). We rewrite each fraction with this common denominator: \[ \frac{5}{9} = \frac{5y}{9y}, \] \[ \frac{2}{3y} = \frac{6}{9y}. \] Thus, we combine these fractions: \[ \frac{5}{9} + \frac{2}{3y} = \frac{5y + 6}{9y}. \] Now, we have the complex fraction written as: \[ \frac{\frac{4y - 3}{3}}{\frac{5y + 6}{9y}}. \] To simplify this, we can multiply by the reciprocal: \[ \frac{4y - 3}{3} \cdot \frac{9y}{5y + 6} = \frac{9y(4y - 3)}{3(5y + 6)}. \] The common denominator for the initial expression \( y + \frac{y-3}{3} \) is therefore \( 3 \) since that was the denominator used to combine those terms. For the denominator \( \frac{5}{9} + \frac{2}{3y} \), the common denominator is \( 9y \). In conclusion, the common denominators of the components of the complex fraction are \( 3 \) and \( 9y \) respectively.