Question

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

\\(\circ\\) $3y(y - 3)$
\\(\circ\\) $y(y - 3)$
\\(\circ\\) $3y$
\\(\circ\\) $3$

Explanation:

Step1: Analyze the terms in \( y + \frac{y - 3}{3} \)

The first term is \( y \), which can be written as \( \frac{y}{1} \), and the second term is \( \frac{y - 3}{3} \). The denominators here are \( 1 \) and \( 3 \). But wait, maybe we misread? Wait, no—wait, actually, in the complex fraction's numerator, it's \( y + \frac{y - 3}{3} \). Wait, no, maybe we need to check again. Wait, no, the question is about the common denominator of \( y + \frac{y - 3}{3} \). Wait, \( y \) is a polynomial, and \( \frac{y - 3}{3} \) has denominator \( 3 \). But wait, maybe we made a mistake. Wait, no—wait, actually, when we have \( y \), we can think of it as \( \frac{y}{1} \), and the other term is \( \frac{y - 3}{3} \). But the common denominator for \( 1 \) and \( 3 \) is \( 3 \)? Wait, no, that can't be. Wait, maybe the question is about the numerator of the complex fraction, which is \( y + \frac{y - 3}{3} \), and we need to find the common denominator when combining these terms. Wait, \( y \) is a monomial, and \( \frac{y - 3}{3} \) has denominator \( 3 \). So to combine \( y \) (which is \( \frac{y}{1} \)) and \( \frac{y - 3}{3} \), the common denominator would be \( 3 \)? But that's not one of the options. Wait, no, maybe we misread the problem. Wait, the complex fraction is \( \frac{y + \frac{y - 3}{3}}{\frac{5}{9} + \frac{2}{3y}} \). Wait, no, the question is about the common denominator of \( y + \frac{y - 3}{3} \). Wait, no, maybe the first term is \( y \), but maybe it's a typo? Wait, no, let's re-express \( y + \frac{y - 3}{3} \). To combine these, we need a common denominator. The denominator of \( y \) (if we write it as a fraction) is \( 1 \), and the denominator of the second term is \( 3 \). The least common denominator (LCD) of \( 1 \) and \( 3 \) is \( 3 \). But that's not matching the options. Wait, maybe we made a mistake. Wait, no—wait, maybe the question is actually about the numerator and denominator of the complex fraction? Wait, no, the question says: "What is the common denominator of \( y + \frac{y - 3}{3} \) in the complex fraction...". Wait, maybe we misread the numerator. Wait, no, the numerator is \( y + \frac{y - 3}{3} \), and we need to find the common denominator when simplifying that numerator. Wait, \( y \) can be written as \( \frac{3y}{3} \) (multiplying numerator and denominator by \( 3 \)) to have the same denominator as \( \frac{y - 3}{3} \). So the common denominator for \( y \) (written as \( \frac{3y}{3} \)) and \( \frac{y - 3}{3} \) is \( 3 \). But that's option D? Wait, no, the options are \( 3y(y - 3) \), \( y(y - 3) \), \( 3y \), \( 3 \). So the answer should be \( 3 \)? Wait, but that seems too simple. Wait, maybe we made a mistake. Wait, no—wait, let's check again. The expression is \( y + \frac{y - 3}{3} \). To add these, we need a common denominator. The first term is \( y = \frac{y}{1} \), the second is \( \frac{y - 3}{3} \). The least common denominator of \( 1 \) and \( 3 \) is \( 3 \). So the common denominator is \( 3 \).

Step2: Verify the options

Looking at the options, the last option is \( 3 \), which matches our calculation.

Answer:

3 (the option with "3")