Question
What is the common denominator of \( y+\frac{y-3}{3} \) in the complex fraction \( \frac{y+\frac{y-3}{3}}{\frac{5}{9}+\frac{2}{3 y}} ? \)
Ask by Mills Fernandez. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The common denominator of the complex fraction is 9y.
Solution
To find the common denominator of the complex fraction \( \frac{y+\frac{y-3}{3}}{\frac{5}{9}+\frac{2}{3y}} \), we need to simplify the expression first.
The common denominator of the complex fraction is the least common multiple (LCM) of the denominators of the fractions in the numerator and the denominator.
Let's simplify the expression step by step:
1. Simplify the numerator:
\[ y + \frac{y-3}{3} = \frac{3y + y - 3}{3} = \frac{4y - 3}{3} \]
2. Simplify the denominator:
\[ \frac{5}{9} + \frac{2}{3y} = \frac{5y + 2}{9y} \]
3. Now, we can rewrite the complex fraction with the simplified numerator and denominator:
\[ \frac{\frac{4y - 3}{3}}{\frac{5y + 2}{9y}} \]
4. To find the common denominator, we need to find the LCM of the denominators in the numerator and the denominator. In this case, the LCM is 9y.
Therefore, the common denominator of the complex fraction is 9y.
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Simplify this solution Bonus Knowledge
To find the common denominator of the expression \( y + \frac{y-3}{3} \), we first need to analyze the components involved. Starting with the fraction \( \frac{y-3}{3} \), the denominator is \( 3 \). For the term \( y \), we can express it as \( \frac{3y}{3} \) in order to have a common denominator with the fraction. Therefore, we rewrite the expression: \[ y + \frac{y-3}{3} = \frac{3y}{3} + \frac{y-3}{3} \] Now we can combine the two fractions: \[ \frac{3y + (y-3)}{3} = \frac{3y + y - 3}{3} = \frac{4y - 3}{3} \] Thus, the common denominator of \( y + \frac{y-3}{3} \) is \( 3 \).
