Question

which expression is equivalent to the following complex fraction?\\(\frac{1+\frac{1}{y}}{1-\frac{1}{y}}\\)\\(\bigcirc\\ \frac{(y + 1)(y - 1)}{y^2}\\)\\(\bigcirc\\ \frac{y + 1}{y - 1}\\)\\(\bigcirc\\ \frac{y - 1}{y + 1}\\)\\(\bigcirc\\ \frac{y^2}{(y + 1)(y - 1)}\\)

Explanation:

Step1: Simplify the numerator

The numerator of the complex fraction is \(1 + \frac{1}{y}\). To combine these terms, we find a common denominator, which is \(y\). So, \(1=\frac{y}{y}\), and then \(1+\frac{1}{y}=\frac{y}{y}+\frac{1}{y}=\frac{y + 1}{y}\).

Step2: Simplify the denominator

The denominator of the complex fraction is \(1-\frac{1}{y}\). Using the common denominator \(y\) again, \(1=\frac{y}{y}\), so \(1 - \frac{1}{y}=\frac{y}{y}-\frac{1}{y}=\frac{y - 1}{y}\).

Step3: Divide the simplified numerator by the simplified denominator

Now we have the complex fraction as \(\frac{\frac{y + 1}{y}}{\frac{y - 1}{y}}\). Dividing by a fraction is the same as multiplying by its reciprocal, so this becomes \(\frac{y + 1}{y}\times\frac{y}{y - 1}\). 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 \(\frac{y+1}{y-1}\))