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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.
  3. Make a conjecture about this pattern and the number the terms appear to be approaching.
  4. Find the sixth term in the pattern. Does it support your conjecture?

Answers for Review Problems

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

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Vocabulary

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.

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