Math Calculators, Lessons and Formulas

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Question

Answer

$$\dfrac{\dfrac{p}{p+4}}{1+\dfrac{1}{p}-4}=\dfrac{p^2}{-3p^2-11p+4}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}\frac{\frac{p}{p+4}}{1+\frac{1}{p}-4}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\frac{p}{p+4}}{\frac{p+1}{p}-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\frac{p}{p+4}}{\frac{-3p+1}{p}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{p^2}{-3p^2-11p+4}\end{aligned} $$
①

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

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 first fraction by $\color{blue}{ p }$.

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

Subtract $4$ from $ \dfrac{p+1}{p} $ to get $ \dfrac{ \color{purple}{ -3p+1 } }{ p }$.

Step 1: Write $ 4 $ 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 $\color{blue}{p}$.

$$ \begin{aligned} \frac{p+1}{p} -4 & \xlongequal{\text{Step 1}} \frac{p+1}{p} - \frac{4}{\color{red}{1}} = \frac{ p+1 }{ p } - \frac{ 4 \cdot \color{blue}{ p }}{ 1 \cdot \color{blue}{ p }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ p+1 } }{ p } - \frac{ \color{purple}{ 4p } }{ p }=\frac{ \color{purple}{ -3p+1 } }{ p } \end{aligned} $$
③

Divide $ \dfrac{p}{p+4} $ by $ \dfrac{-3p+1}{p} $ to get $ \dfrac{p^2}{-3p^2-11p+4} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{p}{p+4} }{ \frac{\color{blue}{-3p+1}}{\color{blue}{p}} } & \xlongequal{\text{Step 1}} \frac{p}{p+4} \cdot \frac{\color{blue}{p}}{\color{blue}{-3p+1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ p \cdot p }{ \left( p+4 \right) \cdot \left( -3p+1 \right) } \xlongequal{\text{Step 3}} \frac{ p^2 }{ -3p^2+p-12p+4 } = \\[1ex] &= \frac{p^2}{-3p^2-11p+4} \end{aligned} $$
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