Question

which expression is equivalent to the following complex fraction?

\\\frac{1+\frac{1}{y}}{1-\frac{1}{y}}\\

\\\frac{(y+1)(y-1)}{y^2}\\

\\\frac{y+1}{y-1}\\

\\\frac{y-1}{y+1}\\

\\\frac{y^2}{(y+1)(y-1)}\\

Explanation:

<pre_analysis>
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<reasoning>

Simplify the numerator and denominator

\[

$$\begin{aligned} 1 + \frac{1}{y} &= \frac{y + 1}{y} \\ 1 - \frac{1}{y} &= \frac{y - 1}{y} \end{aligned}$$

\]

Divide the simplified expressions

\[
\frac{\frac{y + 1}{y}}{\frac{y - 1}{y}} = \frac{y + 1}{y} \cdot \frac{y}{y - 1} = \frac{y + 1}{y - 1} \quad (y
eq 0, y
eq 1)
\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{(y+1)(y-1)}{y^2}\)</mcq-option>
<mcq-correct>(B) \(\frac{y+1}{y-1}\)</mcq-correct>
<mcq-option>(C) \(\frac{y-1}{y+1}\)</mcq-option>
<mcq-option>(D) \(\frac{y^2}{(y+1)(y-1)}\)</mcq-option>
</answer>

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"Complex Fractions"
]
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</post_analysis>

Answer:

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<reasoning>

Simplify the numerator and denominator

\[

$$\begin{aligned} 1 + \frac{1}{y} &= \frac{y + 1}{y} \\ 1 - \frac{1}{y} &= \frac{y - 1}{y} \end{aligned}$$

\]

Divide the simplified expressions

\[
\frac{\frac{y + 1}{y}}{\frac{y - 1}{y}} = \frac{y + 1}{y} \cdot \frac{y}{y - 1} = \frac{y + 1}{y - 1} \quad (y
eq 0, y
eq 1)
\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{(y+1)(y-1)}{y^2}\)</mcq-option>
<mcq-correct>(B) \(\frac{y+1}{y-1}\)</mcq-correct>
<mcq-option>(C) \(\frac{y-1}{y+1}\)</mcq-option>
<mcq-option>(D) \(\frac{y^2}{(y+1)(y-1)}\)</mcq-option>
</answer>

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