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Section13.4Complex Fractions

In this section, we will learn how to simplify complex fractions, which have fractions in the numerator and/or denominator of another fraction.

Figure13.4.1Alternative Video Lesson

Subsection13.4.1Simplifying Complex Fractions

Consider the rational expression

\begin{equation*} \dfrac{\frac{6}{x-4}}{\dfrac{6}{x-4}+3}\text{.} \end{equation*}

It's difficult to quickly evaluate this expression, or determine the important information such as its domain. This type of rational expression, which contains a “fraction within a fraction,” is referred to as a complex fraction. Our goal is to simplify such a fraction so that it has a single numerator and a single denominator, neither of which contain any fractions themselves.

A complex fraction may have fractions in its numerator and/or denominator. Here is an example to show how we use division to simplify a complex fraction.

\begin{align*} \frac{\frac{1}{2}}{3}\amp=\frac{1}{2}\div3\\ \amp=\frac{1}{2}\div\frac{3}{1}\\ \amp=\frac{1}{2}\cdot\frac{1}{3}\\ \amp=\frac{1}{6} \end{align*}

What if the expression above was instead \(\frac{\frac{1}{2}}{\frac{1}{3}+\frac{1}{4}}\text{?}\) We would no longer be able to simply multiply by the reciprocal, as we don't immediately know the reciprocal of that denominator \(\frac{1}{3}+\frac{1}{4}\text{.}\) Instead, we could multiply the “main” numerator and denominator by something that eliminates all of the “internal” denominators. (We'll use the LCD to determine this). For example, we will multiply \(\frac{\frac{1}{2}}{3}\) by \(\frac{2}{2}\text{:}\)

\begin{align*} \frac{\frac{1}{2}}{3}\amp=\frac{\frac{1}{2}}{3}\multiplyright{\frac{2}{2}}\\ \amp=\frac{1}{6} \end{align*}
Remark13.4.2

In the last example, it's important to identify which fraction bar is the “main” fraction bar, and which fractions are “internal.” Comparing the two expressions below, we see that they are not equivalent.

\begin{align*} \frac{\frac{1}{2}}{3}\amp=\frac{\frac{1}{2}}{3}\multiplyright{\frac{2}{2}}\amp\amp\text{versus}\amp\frac{1}{\frac{2}{3}}\amp=\frac{1}{\frac{2}{3}}\multiplyright{\frac{3}{3}}\\ \amp=\frac{1}{6}\amp\amp\amp\amp=\frac{3}{2} \end{align*}

To remove multiple fractions in a complex fraction, we need to multiply the numerator and denominator by the LCD of all the internal fractions, as we will show in the next example.

Example13.4.3

Simplify the complex fraction \(\dfrac{\frac{1}{2}}{\frac{1}{3}+\frac{1}{4}}\text{.}\)

Solution

The internal denominators are \(2\text{,}\) \(3\text{,}\) and \(4\text{,}\) so the LCD is \(12\text{.}\) We will thus multiply the main numerator and denominator by \(12\) and simplify the result:

\begin{align*} \frac{\frac{1}{2}}{\frac{1}{3}+\frac{1}{4}}\amp=\frac{\frac{1}{2}}{\frac{1}{3}+\frac{1}{4}}\multiplyright{\frac{12}{12}}\\ \amp=\frac{\frac{1}{2}\multiplyright{12}}{\left(\frac{1}{3}+\frac{1}{4}\right)\multiplyright{12}}\\ \amp=\frac{6}{4+3}\\ \amp=\frac{6}{7} \end{align*}

Next we will evaluate a function that is a complex fraction and then simplify the result.

Example13.4.4

Find each function value for \(f(x)=\frac{\frac{x+2}{x+3}}{\frac{2}{x+3}-\frac{3}{x-1}}\text{.}\)

  1. \(f(4)\)

  2. \(f(0)\)

  3. \(f(-3)\)

  4. \(f(-11)\)

Solution

We will determine each function value by replacing \(x\) with the specified number and then simplify the complex fraction:

  1. \begin{align*} f(\substitute{4})\amp=\frac{\frac{\substitute{4}+2}{\substitute{4}+3}}{\frac{2}{\substitute{4}+3}-\frac{3}{\substitute{4}-1}}\\ \amp=\frac{\frac{6}{7}}{\frac{2}{7}-\frac{3}{3}}\\ \amp=\frac{\frac{6}{7}}{\frac{2}{7}-1}\multiplyright{\frac{7}{7}}\\ \amp=\frac{6}{2-7}\\ \amp=-\frac{6}{5} \end{align*}
  2. \begin{align*} f(\substitute{0})\amp=\frac{\frac{\substitute{0}+2}{\substitute{0}+3}}{\frac{2}{\substitute{0}+3}-\frac{3}{\substitute{0}-1}}\\ \amp=\frac{\frac{2}{3}}{\frac{2}{3}-\frac{3}{-1}}\\ \amp=\frac{\frac{2}{3}}{\frac{2}{3}+3}\multiplyright{\frac{3}{3}}\\ \amp=\frac{2}{2+9}\\ \amp=-\frac{2}{11} \end{align*}

  3. When evaluating \(f\) at \(-3\text{,}\) we can quickly see that this results in division by zero:

    \begin{align*} f(\substitute{-3})\amp=\frac{\frac{\substitute{-3}+2}{\substitute{-3}+3}}{\frac{2}{\substitute{-3}+3}-\frac{3}{\substitute{-3}-1}}\\ \amp=\frac{\frac{2}{0}}{\frac{2}{0}-\frac{3}{-4}} \end{align*}

    Thus \(f(-3)\) is undefined.

  4. \begin{align*} f(\substitute{-11})\amp=\frac{\frac{\substitute{-11}+2}{\substitute{-11}+3}}{\frac{2}{\substitute{-11}+3}-\frac{3}{\substitute{-11}-1}}\\ \amp=\frac{\frac{-8}{-9}}{\frac{2}{-8}-\frac{3}{-12}}\\ \amp=\frac{\frac{8}{9}}{-\frac{1}{4}+\frac{1}{4}}\\ \amp=\frac{\frac{8}{9}}{0} \end{align*}

    Therefore \(f(-11)\) is undefined.

We have simplified complex fractions involving numbers and now we will apply the same concept to complex fractions with variables.

Example13.4.5

Simplify the complex fraction \(\dfrac{3}{\frac{1}{y}+\frac{5}{y^2}}\text{.}\)

Solution

To start, we will look at the internal denominators and identify the LCD of \(y^2\text{.}\) We'll start by multiplying the main numerator and denominator by the LCD, and then simplify. Since we are multiplying by \(\frac{y^2}{y^2}\text{,}\) it is important to note that \(y\) cannot be \(0\text{,}\) since \(\frac{0}{0}\) is undefined.

\begin{align*} \frac{3}{\frac{1}{y}+\frac{5}{y^2}}\amp=\frac{3}{\frac{1}{y}+\frac{5}{y^2}}\multiplyright{\frac{y^2}{y^2}}\\ \amp=\frac{3\multiplyright{y^2}}{\frac{1}{y}\multiplyright{y^2}+\frac{5}{y^2}\multiplyright{y^2}}\\ \amp=\frac{3y^2}{y+5}, \text{ for }y\neq 0 \end{align*}
Example13.4.6

Simplify the complex fraction \(\dfrac{\frac{5x-6}{2x+1}}{\frac{3x+2}{2x+1}}\text{.}\)

Solution

The internal denominators are both \(2x+1\text{,}\) so this is the LCD and we will multiply the main numerator and denominator by this expression. Since we are multiplying by \(\frac{2x+1}{2x+1}\text{,}\) what \(x\)-value would cause \(2x+1\) to equal \(0\text{?}\) Solving \(2x+1=0\) leads to \(x=-\frac{1}{2}\text{.}\) So \(x\) cannot be \(-\frac{1}{2}\text{,}\) since \(\frac{0}{0}\) is undefined.

\begin{align*} \frac{\frac{5x-6}{2x+1}}{\frac{3x+2}{2x+1}}\amp=\frac{\frac{5x-6}{2x+1}}{\frac{3x+2}{2x+1}}\multiplyright{\frac{2x+1}{2x+1}}\\ \amp=\frac{5x-6}{3x+2}, \text{ for }x\neq -\frac{1}{2} \end{align*}
Example13.4.7

Completely simplify the function defined by \(f(x)=\frac{\frac{x+2}{x+3}}{\frac{2}{x+3}-\frac{3}{x-1}}\text{.}\) Then determine the domain of this function.

Solution

The LCD of the internal denominators is \((x+3)(x-1)\text{.}\) We will thus multiply the main numerator and denominator by the expression \((x+3)(x-1)\) and then simplify the resulting expression.

\begin{align*} f(x)\amp=\frac{\frac{x+2}{x+3}}{\frac{2}{x+3}-\frac{3}{x-1}}\\ f(x)\amp=\frac{\frac{x+2}{x+3}}{\frac{2}{x+3}-\frac{3}{x-1}}\multiplyright{\frac{(x+3)(x-1)}{(x+3)(x-1)}}\\ f(x)\amp=\frac{\frac{x+2}{x+3}\multiplyright{(x+3)(x-1)}}{\left(\frac{2}{x+3}-\frac{3}{x-1}\right)\multiplyright{(x+3)(x-1)}}\\ f(x)\amp=\frac{\frac{x+2}{x+3}\multiplyright{(x+3)(x-1)}}{\frac{2}{x+3}\multiplyright{(x+3)(x-1)}-\frac{3}{x-1}\multiplyright{(x+3)(x-1)}}\\ f(x)\amp=\frac{(x+2)(x-1)}{2(x-1)-3(x+3)}, \text{ for }x\neq -3, x\neq 1\\ f(x)\amp=\frac{(x+2)(x-1)}{2x-2-3x-9}, \text{ for }x\neq -3, x\neq 1\\ f(x)\amp=\frac{(x+2)(x-1)}{-x-11}, \text{ for }x\neq -3, x\neq 1 \end{align*}

In the original (unsimplified) function, we could see that \(x\neq -3\) and \(x\neq 1\text{.}\) In the simplified function, we need \(-x-11\neq 0\text{,}\) so we can also see that \(x\neq -11\text{.}\) Therefore the domain of the function \(f\) is \(\{x\mid x\neq -11, -3, 1\}.\)

Remark13.4.8

Note that in Example 13.4.7, the denominator of \(-x-11\) could have the factor \(-1\) factored out. So an equivalent way to write this function would be:

\begin{align*} f(x)\amp=\frac{(x+2)(x-1)}{-x-11}, \text{ for }x\neq -3, x\neq 1\\ f(x)\amp=\frac{(x+2)(x-1)}{-(x+11)}, \text{ for }x\neq -3, x\neq 1\\ f(x)\amp=-\frac{(x+2)(x-1)}{x+11}, \text{ for }x\neq -3, x\neq 1 \end{align*}
Example13.4.9

Simplify the complex fraction \(\dfrac{2\left(\frac{-4x+3}{x-2}\right)+3}{\frac{-4x+3}{x-2}+4}\text{.}\)

Solution

The only internal denominator is \(x-2\text{,}\) so we will begin by multiplying the main numerator and denominator by this. Afterwards, we'll simplify the resulting expression.

\begin{align*} \frac{2\left(\frac{-4x+3}{x-2}\right)+3}{\frac{-4x+3}{x-2}+4}\amp=\frac{2\left(\frac{-4x+3}{x-2}\right)+3}{\frac{-4x+3}{x-2}+4}\multiplyright{\frac{x-2}{x-2}}\\ \amp=\frac{2\left(\frac{-4x+3}{x-2}\right)\multiplyright{(x-2)}+3\multiplyright{(x-2)}}{\left(\frac{-4x+3}{x-2}\right)\multiplyright{(x-2)}+4\multiplyright{(x-2)}}\\ \amp=\frac{2(-4x+3)+3(x-2)}{(-4x+3)+4(x-2)}, \text{ for }x\neq 2\\ \amp=\frac{-8x+6+3x-6}{-4x+3+4x-8}\\ \amp=\frac{-5x}{-5}\\ \amp=x, \text{ for } x\neq 2 \end{align*}
Example13.4.10

Simplify the complex fraction \(\dfrac{\frac{5}{x}+\frac{4}{y}}{\frac{3}{x}-\frac{2}{y}}\text{.}\)

Solution

We multiply the numerator and denominator by the common denominator of \(x\) and \(y\text{,}\) which is \(xy\text{:}\)

\begin{align*} \frac{\frac{5}{x}+\frac{4}{y}}{\frac{3}{x}-\frac{2}{y}}\amp=\frac{\frac{5}{x}+\frac{4}{y}}{\frac{3}{x}-\frac{2}{y}}\multiplyright{\frac{xy}{xy}}\\ \amp=\frac{\left(\frac{5}{x}+\frac{4}{y}\right)\multiplyright{xy}}{\left(\frac{3}{x}-\frac{2}{y}\right)\multiplyright{xy}}\\ \amp=\frac{\frac{5}{x}\multiplyright{xy}+\frac{4}{y}\multiplyright{xy}}{\frac{3}{x}\multiplyright{xy}-\frac{2}{y}\multiplyright{xy}}\\ \amp=\frac{5y+4x}{3y-2x} \end{align*}
Example13.4.11

Simplify the complex fraction \(\dfrac{\frac{t}{t+3}+\frac{2}{t-3}}{1-\frac{t}{t^2-9}}\text{.}\)

Solution

First, we check all quadratic polynomials to see if they can be factored and factor them:

\begin{equation*} \frac{\frac{t}{t+3}+\frac{2}{t-3}}{1-\frac{t}{t^2-9}}=\frac{\frac{t}{t+3}+\frac{2}{t-3}}{1-\frac{t}{(t-3)(t+3)}} \end{equation*}

Next, we identify the common denominator of the three fractions, which is \((t+3)(t-3)\text{.}\) We then multiply the main numerator and denominator by that expression:

\begin{align*} \frac{\frac{t}{t+3}+\frac{2}{t-3}}{1-\frac{t}{t^2-9}}\amp=\frac{\frac{t}{t+3}+\frac{2}{t-3}}{1-\frac{t}{(t-3)(t+3)}}\multiplyright{\frac{(t+3)(t-3)}{(t+3)(t-3)}}\\ \amp=\frac{\frac{t}{t+3}\multiplyright{(t+3)(t-3)}+\frac{2}{t-3}\multiplyright{(t+3)(t-3)}}{1\multiplyright{(t+3)(t-3)}-\frac{t}{(t-3)(t+3)}\multiplyright{(t+3)(t-3)}}\\ \amp=\frac{t(t-3)+2(t+3)}{(t+3)(t-3)-t}\\ \amp=\frac{t^2-3t+2t+6}{t^2-9-t}\\ \amp=\frac{t^2-t+6}{t^2-t-9} \end{align*}

Note that since both the numerator and denominator are prime trinomials, this expression can neither factor nor simplify any further.

SubsectionExercises

Simplifying Complex Numerical Fractions

1

Calculate the following. Use an improper fraction in your answer.

  1. \(\displaystyle{\frac{ 4 }{ \frac{9}{8} } =}\)

  2. \(\displaystyle{\frac{ \frac{4}{9} }{8} =}\)

2

Calculate the following. Use an improper fraction in your answer.

  1. \(\displaystyle{\frac{ 5 }{ \frac{3}{7} } =}\)

  2. \(\displaystyle{\frac{ \frac{5}{3} }{7} =}\)

3

Calculate the following. Use an improper fraction in your answer.

\(\displaystyle{\frac{ \frac{1}{6} + \frac{5}{3} }{ \frac{1}{5} } =}\)

4

Calculate the following. Use an improper fraction in your answer.

\(\displaystyle{\frac{ \frac{1}{6} - \frac{5}{2} }{ \frac{1}{4} } =}\)

5

Calculate the following. Use an improper fraction in your answer.

\(\displaystyle{\frac{1}{ \frac{4}{5} - \frac{3}{2} } =}\)

6

Calculate the following. Use an improper fraction in your answer.

\(\displaystyle{\frac{2}{ \frac{2}{5} + \frac{6}{5} } =}\)

7

Calculate the following. Use an improper fraction in your answer.

  1. \(\displaystyle{\frac{ \frac{2}{3} }{ \frac{2}{7} } =}\)

  2. \(\displaystyle{\frac{ \frac{y}{t} }{ \frac{x}{r} } =}\)

8

Calculate the following. Use an improper fraction in your answer.

  1. \(\displaystyle{\frac{ \frac{9}{7} }{ \frac{3}{2} } =}\)

  2. \(\displaystyle{\frac{ \frac{y}{r} }{ \frac{x}{t} } =}\)

Simplifying Complex Fractions with One Variable

9

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{4s-7}{s}}}{{\frac{s-6}{s}}}}\) =

10

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{9p+1}{p}}}{{\frac{p-8}{p}}}}\) =

11

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{r}{\left(r-5\right)^{2}}}}{{\frac{10r}{r^{2}-25}}}=}\)

12

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{k}{\left(k-2\right)^{2}}}}{{\frac{5k}{k^{2}-4}}}=}\)

13

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{9+\frac{1}{m}}}{{m+9}}=}\)

14

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{5+\frac{1}{x}}}{{x+4}}=}\)

15

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{3}{\frac{5}{x}-\frac{1}{x+2}}}=}\)

16

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{2}{\frac{5}{y}+\frac{6}{y+4}}}=}\)

17

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{5+\frac{1}{t-5}}}{{\frac{1}{t-5}-\frac{1}{6}}}=}\)

18

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{2+\frac{1}{t-10}}}{{\frac{1}{t-10}-\frac{1}{3}}}=}\)

19

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{1}{r+8}+\frac{4}{r-8}}}{{1-\frac{1}{r-8}}}=}\)

20

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{1}{r+4}+\frac{8}{r-4}}}{{8-\frac{1}{r-4}}}=}\)

21

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{1}{m-1}+\frac{3}{m-1}}}{{6-\frac{1}{m+1}}}=}\)

22

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{1}{m-7}+\frac{7}{m-7}}}{{3-\frac{1}{m+7}}}=}\)

23

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{4}{y-1}-10}}{{\frac{1}{y-1}+\frac{1}{y-10}}}=}\)

24

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{10}{y-1}-5}}{{\frac{1}{y-1}+\frac{1}{y-7}}}=}\)

25

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{\frac{6y}{y^{2}-25}+1}{\frac{3}{y+5}+\frac{4}{y-5}}}=}\)

26

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{\frac{2y}{y^{2}-9}+3}{\frac{6}{y+3}+\frac{1}{y-3}}}=}\)

27

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{u}{u^{2}-100}-\frac{1}{u^{2}-100}}}{{\frac{1}{u+100}}}=}\)

28

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{r}{r^{2}-36}-\frac{1}{r^{2}-36}}}{{\frac{1}{r+36}}}=}\)

Simplifying Complex Fractions with More Than One Variable

29

Simplify this expression.

\(\displaystyle{\frac{{\frac{p}{q}}}{{\frac{3p}{2q^{2}}}}=}\)

30

Simplify this expression.

\(\displaystyle{\frac{{\frac{m}{n}}}{{\frac{6m}{5n^{2}}}}=}\)

31

Simplify this expression.

\(\displaystyle{\frac{{\frac{mn^{2}}{7k}}}{{\frac{m}{4nk}}}=}\)

32

Simplify this expression.

\(\displaystyle{\frac{{\frac{xy^{2}}{4z}}}{{\frac{x}{8yz}}}=}\)

33

Simplify this expression.

  1. \(\displaystyle{\frac{ \frac{x}{t} }{ r } =}\)

  2. \(\displaystyle{\frac{ x }{ \frac{t}{r} } =}\)

34

Simplify this expression.

  1. \(\displaystyle{\frac{ \frac{y}{r} }{ t } =}\)

  2. \(\displaystyle{\frac{ y }{ \frac{r}{t} } =}\)

35

Simplify this expression.

\(\displaystyle{{\frac{\frac{5}{y}}{20+5x}} =}\)

36

Simplify this expression.

\(\displaystyle{{\frac{\frac{3}{r}}{6+3t}} =}\)

37

Simplify this expression.

\(\displaystyle{{\frac{\frac{2}{r}+\frac{6}{y}}{\frac{2}{r}-\frac{4}{y}}} =}\)

38

Simplify this expression.

\(\displaystyle{{\frac{\frac{2}{r}+\frac{6}{x}}{\frac{8}{r}-\frac{2}{x}}} =}\)