Question
Which expression is equivalent to the following complex fraction? \( \frac{2-\frac{1}{y}}{3+\frac{1}{y}} \)
Ask by Pearson Harrington. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression simplifies to \( \frac{2y - 1}{3y + 1} \).
Solution
1. Multiply the numerator and denominator by \( y \) to eliminate the fractions:
\[
\frac{2 - \frac{1}{y}}{3 + \frac{1}{y}} \cdot \frac{y}{y} = \frac{y \left( 2 - \frac{1}{y} \right)}{y \left( 3 + \frac{1}{y} \right)}
\]
2. Distribute \( y \) in both the numerator and the denominator:
\[
\text{Numerator: } y \cdot 2 - y \cdot \frac{1}{y} = 2y - 1
\]
\[
\text{Denominator: } y \cdot 3 + y \cdot \frac{1}{y} = 3y + 1
\]
3. Therefore, the simplified expression is:
\[
\frac{2y - 1}{3y + 1}
\]
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Simplify this solution Bonus Knowledge
To simplify the complex fraction \( \frac{2-\frac{1}{y}}{3+\frac{1}{y}} \), we start by finding a common denominator for both the numerator and the denominator. The common denominator is \( y \). Rewriting the numerator: \[ 2 - \frac{1}{y} = \frac{2y - 1}{y} \] Rewriting the denominator: \[ 3 + \frac{1}{y} = \frac{3y + 1}{y} \] Now, substituting these back into the complex fraction: \[ \frac{\frac{2y - 1}{y}}{\frac{3y + 1}{y}} \] Since we have the same denominators in both the numerator and denominator, we can simplify: \[ = \frac{2y - 1}{3y + 1} \] Thus, the expression equivalent to the given complex fraction is: \[ \frac{2y - 1}{3y + 1} \]
