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# Complex Fractions

## Solve fractions with fractional numerators and/or denominators.

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Complex Fractions
Gupta knows the area and width of a rectangle. He comes up with this equation for the length of the rectangle \begin{align*}\frac{\frac{2}{x^2-1}}{\frac{2x}{x+1}}\end{align*}. What is the length of the rectangle in simplified form?

### Complex Fractions

A complex fraction is a fraction that has fractions in the numerator and/or denominator. To simplify a complex fraction, you will need to combine all that you have learned about simplifying fractions in general.

Let's simplify the following complex fractions.

1. \begin{align*}\frac{\frac{9x}{x+2}}{\frac{3}{x^2-4}}\end{align*}

Rewrite the complex fraction as a division problem.

\begin{align*}\frac{\frac{9x}{x+2}}{\frac{3}{x^2-4}} = \frac{9x}{x+2} \div \frac{3}{x^2-4}\end{align*}

Flip the second fraction, change the problem to multiplication and simplify.

\begin{align*}\frac{9x}{x+2} \div \frac{3}{x^2-4} = \frac{9x}{x+2} \cdot \frac{x^2-4}{3} = \frac{\overset{3}{\bcancel{9}x}}{\cancel{x+2}} \cdot \frac{\cancel{(x+2)}(x-2)}{\bcancel{3}} = 3x(x-2)\end{align*}

1. \begin{align*}\frac{\frac{1}{x} + \frac{1}{x+1}}{4- \frac{1}{x}}\end{align*}

To simplify this complex fraction, we first need to add the fractions in the numerator and subtract the two in the denominator. The LCD of the numerator is \begin{align*}x(x+1)\end{align*} and the denominator is just \begin{align*}x\end{align*}.

\begin{align*}\frac{\frac{1}{x} + \frac{1}{x+1}}{4- \frac{1}{x}} = \frac{{\color{red}\frac{x+1}{x+1}} \cdot \frac{1}{x} + \frac{1}{x+1} \cdot {\color{blue}\frac{x}{x}}}{{\color{blue}\frac{x}{x}} \cdot 4- \frac{1}{x}} = \frac{\frac{x+1}{x(x+1)} + \frac{x}{x(x+1)}}{\frac{4x}{x} - \frac{1}{x}} = \frac{\frac{2x+1}{x(x+1)}}{\frac{4x-1}{x}}\end{align*}

Divide and simplify if possible.

\begin{align*}\frac{\frac{2x+1}{x(x+1)}}{\frac{4x-1}{x}} = \frac{2x+1}{x(x+1)} \div \frac{4x-1}{x} = \frac{2x+1}{\cancel{x}(x+1)} \cdot \frac{\cancel{x}}{4x-1} = \frac{2x+1}{(x+1)(4x-1)}\end{align*}

1.  \begin{align*}\frac{\frac{5-x}{x^2+6x+8} + \frac{x}{x+4}}{\frac{6}{x+2} - \frac{2x+3}{x^2-3x-10}}\end{align*}

First, add the fractions in the numerator and subtract the ones in the denominator.

\begin{align*}\frac{\frac{5-x}{x^2+6x+8} + \frac{x}{x+4}}{\frac{6}{x+2} - \frac{2x+3}{x^2-3x-10}} = \frac{\frac{5-x}{(x+4){\color{red}(x+2)}} + \frac{x}{x+4} \cdot {\color{red}\frac{x+2}{x+2}}}{{\color{blue}\frac{x-5}{x-5}} \cdot \frac{6}{x+2} - \frac{2x+3}{(x+2){\color{blue}(x-5)}}} = \frac{\frac{5-x+x(x+2)}{(x+4)(x+2)}}{\frac{6(x-5)-(2x+3)}{(x+2)(x-5)}} = \frac{\frac{x^2+x+5}{(x+4)(x+2)}}{\frac{4x-36}{(x+2)(x-5)}}\end{align*}

Now, rewrite as a division problem, flip, multiply, and simplify.

\begin{align*}\frac{\frac{x^2+x+5}{(x+4)(x+2)}}{\frac{4x-36}{(x+2)(x-5)}} = \frac{x^2+x+5}{(x+4)(x+2)} \div \frac{4x-36}{(x+2)(x-5)} = \frac{x^2+x+5}{(x+4)\cancel{(x+2)}} \cdot \frac{\cancel{(x+2)}(x-5)}{4(x-9)}\end{align*}

\begin{align*}= \frac{(x^2+x+5)(x-5)}{4(x+4)(x-9)}\end{align*}

### Examples

#### Example 1

Earlier, you were asked to determine the length of a given rectangle in simplified form.

Rewrite the complex fraction as a division problem.

\begin{align*}\frac{\frac{2}{x^2-1}}{\frac{2x}{x+1}} = \frac{2}{x^2-1} \div \frac{2x}{x+1}\end{align*}

Flip the second fraction, change the problem to multiplication and simplify.

\begin{align*}\frac{2}{x^2-1} \div \frac{2x}{x+1} = \frac{2}{x^2-1} \cdot \frac{x+1}{2x} = \frac{\bcancel{2}}{\bcancel{(x+1)}(x-1)} \cdot \frac{\bcancel{(x+1)}}{\bcancel{2}x} = \frac {1}{x^2-x}\end{align*}

Therefore, the length of the rectangle in simplified form is \begin{align*}\frac {1}{x^2-x}\end{align*}.

Simplify the complex fractions.

#### Example 2

\begin{align*}\frac{\frac{5x-20}{x^2}}{\frac{x-4}{x}}\end{align*}

Rewrite the fraction as a division problem and simplify.

\begin{align*}\frac{\frac{5x-20}{x^2}}{\frac{x-4}{x}} = \frac{5x-20}{x^2} \div \frac{x-4}{x} = \frac{5 \cancel{(x-4)}}{x^{\cancel{2}}} \cdot \frac{\cancel{x}}{\cancel{x-4}} = \frac{5}{x}\end{align*}

#### Example 3

\begin{align*}\frac{\frac{1-x}{x} - \frac{2}{x-1}}{1 + \frac{1}{x}}\end{align*}

Add the fractions in the numerator and denominator together.

\begin{align*}\frac{\frac{1-x}{x} - \frac{2}{x-1}}{1+\frac{1}{x}} = \frac{\frac{x-1}{x-1} \cdot \frac{1-x}{x} - \frac{2}{x-1} \cdot \frac{x}{x}}{\frac{x}{x} \cdot 1+ \frac{1}{x}} = \frac{\frac{(x-1)(1-x)-2x}{x(x-1)}}{\frac{x+1}{x}} = \frac{\frac{-x^2+1}{x(x-1)}}{\frac{x+1}{x}} \end{align*}

Now, rewrite the fraction as a division problem and simplify.

\begin{align*}\frac{-x^2+1}{x(x-1)} \div \frac{x+1}{x} &= \frac{-(x^2-1)}{x(x-1)} \cdot \frac{x}{x+1} \\ &= \frac{-\cancel{(x-1)} \cancel{(x+1)}}{\cancel{x} \cancel{(x-1)}} \cdot \frac{\cancel{x}} {\cancel{x+1}} \\ &= -1\end{align*}

#### Example 4

\begin{align*}\frac{\frac{\text{-}3}{\text{-}4x^2-5x+6} + \frac{\text{-}4}{\text{-}4x+3}}{\text{-}\frac{1}{\text{-}4x^2-5x+6} + \frac{2}{\text{-}4x+3}} \end{align*}

Add the numerator and the denominator of this complex fraction.

\begin{align*}\frac{\frac{\text{-}3}{\text{-}4x^2-5x+6} + \frac{\text{-}4}{\text{-}4x+3}}{\text{-}\frac{1}{\text{-}4x^2-5x+6} + \frac{2}{\text{-}4x+3}} &= \frac{\frac{\text{-}3}{( \text{-}4x+3)(x+2)} + \frac{( \text{-}4)}{(\text{-}4x+3)} \cdot \frac{(x+2)}{(x+2)}}{\frac{\text{-}1}{( \text{-}4x+3)(x+2)} + \frac{2}{\text{-}4x+3} \cdot \frac{(x+2)}{(x+2)}} \\ \\ &= \frac{\frac{\text{-}4x-11}{\text{-}4x^2-5x+6}}{\frac{2x+3}{\text{-}4x^2-5x+6}} \\ \\ &= \frac{\text{-}4x-11}{\text{-}4x^2-5x+6} \cdot \frac{\text{-}4x^2-5x+6}{2x+3}\\ \\ &= \frac{\text{-}4x-11}{2x+3}\end{align*}

### Review

Simplify the complex fractions.

1. \begin{align*}\frac{\frac{2x}{5}}{\frac{8}{7}}\end{align*}
2. \begin{align*}\frac{\frac{4}{x^2-9}}{\frac{6x}{x+3}}\end{align*}
3. \begin{align*}\frac{\frac{7x^3}{x^2+5x+6}}{\frac{35x^2}{x+2}}\end{align*}
4. \begin{align*}\frac{\frac{24x+3}{3x+1}}{\frac{16x+2}{6x^2-13x-5}}\end{align*}
5. \begin{align*}\frac{\frac{4}{x-1} + \frac{1}{x}}{\frac{1}{x} -5}\end{align*}
6. \begin{align*}\frac{\frac{3x}{x+4} - \frac{1}{x}}{\frac{3x-4}{x^2+6x+8}}\end{align*}
7. \begin{align*}\frac{8- \frac{3x}{x+5}}{\frac{10}{x+5} + \frac{5}{x+1}}\end{align*}
8. \begin{align*}\frac{\frac{x}{x+3} - \frac{4}{2x+1}}{\frac{3}{2x+1} + \frac{6}{x^2-9}}\end{align*}
9. \begin{align*}\frac{\frac{x+3}{x} + \frac{2x}{5-x}}{\frac{3}{2x} - \frac{4x}{x-5}}\end{align*}
10. \begin{align*}\frac{\frac{2x}{5x^2-13x-6} + \frac{1}{x-3}}{\frac{4}{5x+2} - \frac{5x}{5x^2-3x-2}}\end{align*}
11. \begin{align*}\frac{\frac{3x}{x^2-4} + \frac{x+4}{x^2+3x+2}}{\frac{x+1}{x^2-x-2} - \frac{2x}{x^2+2x+1}}\end{align*}

Use the following pattern to answer the next four questions.

\begin{align*}2+\frac{1}{1+\frac{1}{2}}, \ 2+\frac{1}{1+\frac{1}{2+\frac{2}{3}}}, \ 2 + \frac{1}{1+\frac{1}{2+\frac{2}{3+\frac{3}{4}}}}\end{align*}

1. Find the next two terms in the pattern.
2. Using your graphing calculator, simplify each term in the pattern to a decimal.
4. Find the sixth term in the pattern. Does it support your conjecture?

To see the Review answers, open this PDF file and look for section 9.13.

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### Vocabulary Language: English

Complex Fraction

A fraction with rational expression(s) in the numerator and denominator (a fraction composed of other fractions) is known as a complex fraction.