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Answer

$$\frac{\frac{m}{n}+1}{\frac{m}{n}-1}=\frac{m+n}{m-n}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}\frac{\frac{m}{n}+1}{\frac{m}{n}-1}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\frac{m+n}{n}}{\frac{m-n}{n}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{m+n}{m-n}\end{aligned} $$
①

Add $ \dfrac{m}{n} $ and $ 1 $ to get $ \dfrac{ \color{purple}{ m+n } }{ n }$.

Step 1: Write $ 1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{n}$.

$$ \begin{aligned} \frac{m}{n} +1 & \xlongequal{\text{Step 1}} \frac{m}{n} + \frac{1}{\color{red}{1}} = \frac{ m }{ n } + \frac{ 1 \cdot \color{blue}{ n }}{ 1 \cdot \color{blue}{ n }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ m } }{ n } + \frac{ \color{purple}{ n } }{ n }=\frac{ \color{purple}{ m+n } }{ n } \end{aligned} $$
②

Subtract $1$ from $ \dfrac{m}{n} $ to get $ \dfrac{ \color{purple}{ m-n } }{ n }$.

Step 1: Write $ 1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{n}$.

$$ \begin{aligned} \frac{m}{n} -1 & \xlongequal{\text{Step 1}} \frac{m}{n} - \frac{1}{\color{red}{1}} = \frac{ m }{ n } - \frac{ 1 \cdot \color{blue}{ n }}{ 1 \cdot \color{blue}{ n }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ m } }{ n } - \frac{ \color{purple}{ n } }{ n }=\frac{ \color{purple}{ m-n } }{ n } \end{aligned} $$
③

Divide $ \dfrac{m+n}{n} $ by $ \dfrac{m-n}{n} $ to get $ \dfrac{ m+n }{ m-n } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Cancel $ \color{red}{ n } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{m+n}{n} }{ \frac{\color{blue}{m-n}}{\color{blue}{n}} } & \xlongequal{\text{Step 1}} \frac{m+n}{n} \cdot \frac{\color{blue}{n}}{\color{blue}{m-n}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{m+n}{\color{red}{1}} \cdot \frac{\color{red}{1}}{m-n} \xlongequal{\text{Step 3}} \frac{ \left( m+n \right) \cdot 1 }{ 1 \cdot \left( m-n \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ m+n }{ m-n } \end{aligned} $$
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