To determine the common denominator of the expression \( y + \frac{y-3}{3} \) in the complex fraction \( \frac{y + \frac{y-3}{3}}{\frac{5}{9} + \frac{2}{3y}} \), we can follow these steps:
### Numerator Analysis:
1. First, consider the numerator: \( y + \frac{y-3}{3} \).
2. To combine these terms, we need a common denominator.
3. The first term \( y \) can be rewritten as \( \frac{y}{1} \), and the second term is \( \frac{y-3}{3} \).
4. The common denominator of \( \frac{y}{1} \) and \( \frac{y-3}{3} \) is 3.
5. Convert the term \( y \) to have the same denominator 3:
[tex]\[
y = \frac{3y}{3}
\][/tex]
6. Now, our numerator \( y + \frac{y-3}{3} \) becomes: \( \frac{3y}{3} + \frac{y-3}{3} \).
7. Combine the fractions:
[tex]\[
\frac{3y + (y - 3)}{3} = \frac{3y + y - 3}{3} = \frac{4y - 3}{3}
\][/tex]
### Denominator Analysis:
Now let's focus on the denominator \( \frac{5}{9} + \frac{2}{3y} \).
8. The denominators in the terms are 9 and \( 3y \). To find the common denominator for this part, we'll consider:
9. Rewrite both fractions with a common denominator:
[tex]\[
\frac{5}{9} + \frac{2}{3y}
\][/tex]
The slowest common multiple (LCM) of 9 and \( 3y \) is \( 9y \).
With all of the above analysis, let's look at the provided options for the common denominator specifically asked in the problem:
- \( 3y(y-3) \)
- \( y(y-3) \)
- \( 3y \)
- 3
Given our calculations and the details specified in the problem, the correct answer for the common denominator of \( y + \frac{y-3}{3} \) term itself is:
[tex]\[
3
\][/tex]
