Question

which expression is equivalent to the following complex fraction?
\\(\frac{1 + \frac{1}{y}}{1 - \frac{1}{y}}\\)
options:
\\(\circ \frac{(y + 1)(y - 1)}{y^2}\\)
\\(\circ \frac{y + 1}{y - 1}\\) (marked with red dot)
\\(\circ \frac{y - 1}{y + 1}\\)
\\(\circ \frac{y^2}{(y + 1)(y - 1)}\\)

Explanation:

Step1: Simplify numerator and denominator

Simplify \(1 + \frac{1}{y}\) and \(1 - \frac{1}{y}\) separately. For \(1 + \frac{1}{y}\), find a common denominator \(y\): \(1=\frac{y}{y}\), so \(1 + \frac{1}{y}=\frac{y}{y}+\frac{1}{y}=\frac{y + 1}{y}\). Similarly, for \(1-\frac{1}{y}\), \(1=\frac{y}{y}\), so \(1-\frac{1}{y}=\frac{y}{y}-\frac{1}{y}=\frac{y - 1}{y}\).

Step2: Divide the two simplified fractions

The complex fraction \(\frac{1+\frac{1}{y}}{1 - \frac{1}{y}}\) becomes \(\frac{\frac{y + 1}{y}}{\frac{y - 1}{y}}\). Dividing by a fraction is multiplying by its reciprocal, so \(\frac{y + 1}{y}\times\frac{y}{y - 1}\).

Step3: Cancel common terms

The \(y\) in the numerator and denominator cancels out, leaving \(\frac{y + 1}{y - 1}\).

Answer:

\(\boldsymbol{\frac{y + 1}{y - 1}}\) (corresponding to the option with the red dot, which is the second option: \(\frac{y + 1}{y - 1}\))