CR.17: Complex Fractions

Learning Outcomes

Simplify complex rational expressions

Fractions and rational expressions can be interpreted as quotients. When both the dividend (numerator) and divisor (denominator) include fractions or rational expressions, you have something more complex than usual. Do not fear—you have all the tools you need to simplify these quotients!

A complex fraction is the quotient of two fractions. These complex fractions are never considered to be in simplest form, but they can always be simplified using division of fractions. Remember, to divide fractions, you multiply by the reciprocal.

Before you multiply the numbers, it is often helpful to factor the fractions. You can then cancel factors.

Example

Simplify.

$\displaystyle\Large \dfrac{\,\frac{12}{35}\,}{\,\frac{6}{7}\,}$

Show Solution

Rewrite the complex fraction as a division problem.

$\displaystyle\large \frac{\,\frac{12}{35}\,}{\,\frac{6}{7}\,}=\normalsize\frac{12}{35}\div \frac{6}{7}$

Rewrite the division as multiplication and take the reciprocal of the divisor.

$=\frac{12}{35}\cdot \frac{7}{6}$

Factor the numerator and denominator looking for common factors before multiplying numbers together.

$\begin{array}{l}=\frac{2\cdot 6\cdot 7}{5\cdot 7\cdot 6}\\\\=\frac{2}{5}\cdot \frac{6\cdot 7}{6\cdot 7}\\\\=\frac{2}{5}\cdot 1\end{array}$

$\displaystyle\Large \frac{\,\frac{12}{35}\,}{\,\frac{6}{7}\,}=\normalsize\frac{2}{5}$

Try It

Simplify.

$\displaystyle\Large \dfrac{\,\frac{6}{28}\,}{\,\frac{2}{7}\,}$

Show Solution

\frac{\frac{6}{28}}{\frac{2}{7}} = \frac{3}{4}

$\frac{\frac{6}{28}}{\frac{2}{7}}=\frac{3}{4}$

If two fractions appear in the numerator or denominator (or both), first combine them. Then simplify the quotient as shown above.

Example

Simplify.

$\displaystyle\Large \frac{\,\frac{3}{4}+\frac{1}{2}\,}{\,\frac{4}{5}-\frac{1}{10}\,}$

Show Solution

First combine the numerator and denominator by adding or subtracting. You may need to find a common denominator first. Note that we do not show the steps for finding a common denominator, so please review that in the previous section if you are confused.

$\displaystyle\Large \frac{\,\frac{3}{4}+\frac{1}{2}\,}{\,\frac{4}{5}-\frac{1}{10}\,}=\frac{\,\frac{5}{4}\,}{\,\frac{7}{10}\,}$

Rewrite the complex fraction as a division problem.

$\displaystyle\Large \frac{\,\,\frac{5}{4}\,\,}{\,\,\frac{7}{10}\,\,}=\normalsize\frac{5}{4}\div \frac{7}{10}$

Rewrite the division as multiplication and take the reciprocal of the divisor.

$=\dfrac{5}{4}\cdot \frac{10}{7}$

Multiply and simplify as needed.

$\dfrac{5}{4}\cdot \frac{10}{7}=\frac{5\cdot5\cdot2}{2\cdot2\cdot7}=\frac{25}{14}$

$\displaystyle\Large \frac{\,\frac{3}{4}+\frac{1}{2}\,}{\,\frac{4}{5}-\frac{1}{10}\,}=\normalsize\frac{25}{14}$

Try It

Simplify.

$\displaystyle\Large \frac{\,\frac{1}{3}-\frac{1}{4}\,}{\,\frac{7}{12}-\frac{1}{48}\,}$

Show Solution

\frac{\frac{1}{3} - \frac{1}{4}}{\frac{7}{12} - \frac{1}{48}} = \frac{4}{27}

$\displaystyle\Large \frac{\,\frac{1}{3}-\frac{1}{4}\,}{\,\frac{7}{12}-\frac{1}{48}\,}=\normalsize\frac{4}{27}$

In the following video, we will show a couple more examples of how to simplify complex fractions.

Simplifying Complex Rational Expressions

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression $\dfrac{a}{\dfrac{1}{b}+c}$ can be simplified by rewriting the numerator as the fraction $\dfrac{a}{1}$ and combining the expressions in the denominator as $\dfrac{1+bc}{b}$ . We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get $\dfrac{a}{1}\cdot \dfrac{b}{1+bc}$ which is equal to $\dfrac{ab}{1+bc}$ .

How To: Given a complex rational expression, simplify it

Combine the expressions in the numerator into a single rational expression by adding or subtracting.
Combine the expressions in the denominator into a single rational expression by adding or subtracting.
Rewrite as the numerator divided by the denominator.
Rewrite as multiplication.
Multiply.
Simplify.

Example: Simplifying Complex Rational Expressions

Simplify: $\dfrac{y+\dfrac{1}{x}}{\dfrac{x}{y}}$ .

Show Solution

Begin by combining the expressions in the numerator into one expression.

\begin{array}{cc} y \cdot \frac{x}{x} + \frac{1}{x} & Multiply by \frac{x}{x} to get LCD as denominator . \\ \frac{x y}{x} + \frac{1}{x} \\ \frac{x y + 1}{x} & Add numerators . \end{array}

$\begin{array}{cc}y\cdot \dfrac{x}{x}+\dfrac{1}{x}\hfill & \text{Multiply by }\dfrac{x}{x}\text{to get LCD as denominator}.\hfill \\ \dfrac{xy}{x}+\dfrac{1}{x}\hfill & \\ \dfrac{xy+1}{x}\hfill & \text{Add numerators}.\hfill \end{array}$

Now the numerator is a single rational expression and the denominator is a single rational expression.

\frac{\frac{x y + 1}{x}}{\frac{x}{y}}

$\dfrac{\dfrac{xy+1}{x}}{\dfrac{x}{y}}$

We can rewrite this as division and then multiplication.

\begin{array}{cc} \frac{x y + 1}{x} \div \frac{x}{y} \\ \frac{x y + 1}{x} \cdot \frac{y}{x} & Rewrite as multiplication . \\ \frac{y (x y + 1)}{x^{2}} & Multiply . \end{array}

$\begin{array}{cc}\dfrac{xy+1}{x}\div \dfrac{x}{y}\hfill & \\ \dfrac{xy+1}{x}\cdot \dfrac{y}{x}\hfill & \text{Rewrite as multiplication}\text{.}\hfill \\ \dfrac{y\left(xy+1\right)}{{x}^{2}}\hfill & \text{Multiply}\text{.}\hfill \end{array}$

Try It

Simplify: $\dfrac{\dfrac{x}{y}-\dfrac{y}{x}}{y}$

Show Solution

$\dfrac{{x}^{2}-{y}^{2}}{x{y}^{2}}$

Q & A

Can a complex rational expression always be simplified?

Yes. We can always rewrite a complex rational expression as a simplified rational expression.

Example

Simplify.

$\dfrac{\dfrac{5x^2}{9}}{\dfrac{15x^3}{27}}$

Show Solution

State the quotient in simplest form. Rewrite division as multiplication by the reciprocal.

$\frac{5x^{2}}{9}\cdot\frac{27}{15x^{3}}$

Factor the numerators and denominators.

$\frac{5\cdot{x}\cdot{x}}{3\cdot3}\cdot\frac{3\cdot3\cdot3}{5\cdot3\cdot{x}\cdot{x}\cdot{x}}$

Cancel common factors.

$\large\begin{array}{c}\frac{\cancel{5}\cdot{\cancel{x}}\cdot{\cancel{x}}}{\cancel{3}\cdot\cancel{3}}\cdot\frac{\cancel{3}\cdot\cancel{3}\cdot\cancel{3}}{\cancel{5}\cdot\cancel{3}\cdot{\cancel{x}}\cdot{\cancel{x}}\cdot{x}}=\frac{1}{x}\end{array}$

$\displaystyle \frac{5{{x}^{2}}}{9}\div \frac{15{{x}^{3}}}{27}=\frac{1}{x}$

Example

Simplify.

$\dfrac{\dfrac{3x^2}{x+2}}{\dfrac{6x^4}{x^2+5x+6}}$

Show Solution

Rewrite division as multiplication by the reciprocal.

$\frac{3x^{2}}{x+2}\cdot\frac{\left(x^{2}+5x+6\right)}{6x^{4}}$

Factor the numerators and denominators.

$\frac{3\cdot{x}\cdot{x}}{x+2}\cdot\frac{\left(x+2\right)\left(x+3\right)}{2\cdot3\cdot{x}\cdot{x}\cdot{x}\cdot{x}}$

Cancel common factors.

$\dfrac{\cancel{3}\cdot{\cancel{x}}\cdot{\cancel{x}}}{\cancel{x+2}}\cdot\frac{\cancel{\left(x+2\right)}\left(x+3\right)}{2\cdot\cancel{3}\cdot{\cancel{x}}\cdot{\cancel{x}}\cdot{x}\cdot{x}}$

Simplify.

$\frac{(x+3)}{2{{x}^{2}}}$

$\displaystyle \frac{3{{x}^{2}}}{x+2}\div \frac{6{{x}^{4}}}{({{x}^{2}}+5x+6)}=\frac{x+3}{2{{x}^{2}}}$ .

Notice that once you rewrite the division as multiplication by a reciprocal, you follow the same process you used to multiply rational expressions.

In the video that follows, we present another example of dividing rational expressions.

Try It

Simplify.

$\dfrac{\dfrac{16c^4d^3}{5c}}{\dfrac{4cd^4}{d^2}}$

Show Solution

$\dfrac{\dfrac{16c^4d^3}{5c}}{\dfrac{4cd^4}{d^2}}=\dfrac{16c^4d^3}{5c} \cdot \dfrac{d^2}{4cd^4}=\dfrac{16c^4d^5}{20c^2d^4}=\dfrac{4c^2d}{5}$

Example

Simplify.

$\displaystyle\Large \frac{\,\,\frac{x+5}{{{x}^{2}}-16}\,}{\,\,\frac{{{x}^{2}}-\,\,25}{x-4}\,}$

Show Solution

Rewrite the complex rational expression as a division problem.

$=\frac{x+5}{{{x}^{2}}-16}\div \frac{{{x}^{2}}-25}{x-4}$

Rewrite the division as multiplication and take the reciprocal of the divisor. Note that the excluded values for this are $-4$ , $4$ , $-5$ , and $5$ , because those values make the denominators of one of the fractions zero.

$=\frac{x+5}{{{x}^{2}}-16}\cdot \frac{x-4}{{{x}^{2}}-25}$

Factor the numerator and denominator, looking for common factors. In this case, $x+5$ and $x–4$ are common factors of the numerator and denominator.

$\begin{array}{l}=\frac{(x+5)(x-4)}{(x+4)(x-4)(x+5)(x-5)}\\\\=\frac{1}{(x+4)(x-5)}\end{array}$

$\displaystyle\Large \frac{\,\,\frac{x+5}{{{x}^{2}}-16}\,}{\,\,\frac{{{x}^{2}}-25}{x-4}\,}\normalsize=\frac{1}{(x+4)(x-5)},x\ne -4,4,-5,5$

Try It

Simplify.

$\displaystyle\Large \frac{\,\,\frac{t}{t+9}\,}{\,\,\frac{4}{t^2-81}\,}$

Show Solution

$\displaystyle\Large \frac{\,\,\frac{t}{t+9}\,}{\,\,\frac{4}{t^2-81}\,}\normalsize=\dfrac{t}{t+9} \cdot \dfrac{t^2-81}{4}=\dfrac{t}{t+9} \cdot \dfrac{(t+9)(t-9)}{4}=\frac{t(t-9)}{4}$

In the next video example, we will show that simplifying a complex fraction may require factoring first.

The same ideas can be used when simplifying complex rational expressions that include more than one rational expression in the numerator or denominator. However, there is a shortcut that can be used. Compare these two examples of simplifying a complex fraction.

Example

Simplify.

$\displaystyle\dfrac{\,\,\normalsize1-\dfrac{9}{{{x}^{2}}}\,\,}{\,\,\normalsize1+\dfrac{5}{x}\normalsize+\dfrac{6}{{{x}^{2}}}\,\,}$

Show Solution

Combine the expressions in the numerator and denominator. To do this, rewrite the expressions using a common denominator. There is an excluded value of $0$ because this makes the denominators of the fractions zero.

$\begin{array}{l}=\frac{\frac{{{x}^{2}}}{{{x}^{2}}}-\frac{9}{{{x}^{2}}}}{\frac{{{x}^{2}}}{{{x}^{2}}}+\frac{5x}{{{x}^{2}}}+\frac{6}{{{x}^{2}}}}\\\\=\frac{\frac{{{x}^{2}}-9}{{{x}^{2}}}}{\frac{{{x}^{2}}+5x+6}{{{x}^{2}}}}\end{array}$

Rewrite the complex rational expression as a division problem. (When you are comfortable with the step of rewriting the complex rational fraction as a division problem, you might skip this step and go straight to rewriting it as multiplication.)

$=\frac{{{x}^{2}}-9}{{{x}^{2}}}\div \frac{{{x}^{2}}+5x+6}{{{x}^{2}}}$

Rewrite the division as multiplication and take the reciprocal of the divisor.

$=\frac{{{x}^{2}}-9}{{{x}^{2}}}\cdot \frac{{{x}^{2}}}{{{x}^{2}}+5x+6}$

Factor the numerator and denominator looking for common factors. In this case, $x+3$ and $x^{2}$ are common factors. We can now see there are two additional excluded values, $-2$ and $-3$ .

$\begin{array}{l}=\frac{(x+3)(x-3){{x}^{2}}}{{{x}^{2}}(x+3)(x+2)}\\\\=\frac{(x-3)}{(x+2)}\cdot \frac{{{x}^{2}}(x+3)}{{{x}^{2}}(x+3)}\end{array}$

$\frac{1-\frac{9}{{{x}^{2}}}}{1+\frac{5}{x}+\frac{6}{{{x}^{2}}}}=\frac{x-3}{x+2},x\ne -3,-2,0$

Example

Simplify.

$\frac{1-\frac{9}{{{x}^{2}}}}{1+\frac{5}{x}+\frac{6}{{{x}^{2}}}}$

Show Solution

Before combining the expressions, find a common denominator for all of the rational expressions. In this case, $x^{2}$ is a common denominator. Multiply by $1$ in the form of a fraction with the common denominator in both the numerator and denominator. In this case, multiply by $\frac{{{x}^{2}}}{{{x}^{2}}}$ . There is an excluded value of $0$ because this makes the denominators of the fractions zero.

$\begin{array}{l}=\frac{1-\frac{9}{{{x}^{2}}}}{1+\frac{5}{x}+\frac{6}{{{x}^{2}}}}\cdot \frac{{{x}^{2}}}{{{x}^{2}}}\\\\=\frac{\left( 1-\frac{9}{{{x}^{2}}} \right){{x}^{2}}}{\left( 1+\frac{5}{x}+\frac{6}{{{x}^{2}}} \right){{x}^{2}}}\\\\=\frac{{{x}^{2}}-9}{{{x}^{2}}+5x+6}\end{array}$

Notice that the expression is no longer complex! You can simplify by factoring and identifying common factors. We can now see there are two additional excluded values, $-2$ and $-3$ .

$\begin{array}{l}=\frac{(x+3)(x-3)}{(x+3)(x+2)}\\\\=\frac{x+3}{x+3}\cdot \frac{x-3}{x+2}\\\\=1\cdot \frac{x-3}{x+2}\end{array}$

$\frac{1-\frac{9}{{{x}^{2}}}}{1+\frac{5}{x}+\frac{6}{{{x}^{2}}}}=\frac{x-3}{x+2},x\ne -3,-2,0$

You may find the second method easier to use, but do try both ways to see what you prefer.

Try It

Simplify.

$\displaystyle\dfrac{\,\,\normalsize \dfrac{7}{{{y}^2}}-\dfrac{1}{{2y}}\,\,}{\,\,\normalsize \dfrac{2}{y}\normalsize+\dfrac{7}{{3y}}\,\,}$

Show Solution

$\displaystyle\dfrac{\,\,\normalsize \dfrac{7}{{{y}^2}}-\dfrac{1}{{2y}}\,\,}{\,\,\normalsize \dfrac{2}{y}\normalsize+\dfrac{7}{{3y}}\,\,}=\frac{3(14-y)}{26y}$

Watch the video example below for a similar problem.

Example

Simplify.

$\displaystyle\dfrac{\,\,\normalsize \dfrac{7}{y}+y\,\,}{\,\,\normalsize \dfrac{5}{y}\normalsize-y\,\,}$

Show Solution

First, find the LCD of the denominators that occur in the numerator of the complex fraction.

If necessary, factor the denominators before finding the LCD. In this problem, there is just one denominator, $y$ . The LCD of the numerator is $y$ .

Multiply the $y$ in the numerator by $\frac{y}{y}$ , so that the denominators match the LCD.

$\dfrac{\dfrac{7}{y}+y}{\dfrac{5}{y}-y}=\dfrac{\dfrac{7}{y}+y \cdot \dfrac{y}{y}}{\dfrac{5}{y}-y}=\dfrac{\dfrac{7}{y}+\dfrac{y^2}{y}}{\dfrac{5}{y}-y}$

Add the fractions in the numerator.

$\dfrac{\dfrac{7}{y}+\dfrac{y^2}{y}}{\dfrac{5}{y}-y}=\dfrac{\dfrac{7+y^2}{y}}{\dfrac{5}{y}-y}$

At this point, repeat the preceding steps for the denominator of the complex fraction.

$\dfrac{\dfrac{7+y^2}{y}}{\dfrac{5}{y}-y}=\dfrac{\dfrac{7+y^2}{y}}{\dfrac{5}{y}-y \cdot \dfrac{y}{y}}=\dfrac{\dfrac{7+y^2}{y}}{\dfrac{5}{y}-\dfrac{y^2}{y}}=\dfrac{\dfrac{7+y^2}{y}}{\dfrac{5-y^2}{y}}$

You are now ready to divide the numerator by the denominator. To accomplish this, multiply the numerator by the reciprocal of the denominator.

$\dfrac{\dfrac{7+y^2}{y}}{\dfrac{5-y^2}{y}}=\dfrac{7+y^2}{y} \cdot \dfrac{y}{5-y^2}$

The $y$ on the top and bottom cancel, therefore:
$\dfrac{\dfrac{7}{y}+y}{\dfrac{5}{y}-y}=\dfrac{7+y^2}{5-y^2}$

Try It

Simplify.

$\displaystyle\dfrac{\,\,\normalsize \dfrac{9}{x}-x\,\,}{\,\,\normalsize \dfrac{11}{x}\normalsize+x\,\,}$

Show Solution

$\dfrac{\dfrac{9}{x}-x}{\dfrac{11}{x}+x}=\dfrac{9-x^2}{x^2+11}$

Summary

Complex rational expressions are quotients with rational expressions in the divisor, dividend, or both. When written in fraction form, they appear to be fractions within a fraction. These can be simplified by first treating the quotient as a division problem. Then you can rewrite the division as multiplication and take the reciprocal of the divisor. Or you can simplify the complex rational expression by multiplying both the numerator and denominator by a denominator common to all rational expressions within the complex expression. This can help simplify the complex expression even faster.

Chapter CR: Corequisite Review

Learning Outcomes

Example

Try It

Example

Try It

Simplifying Complex Rational Expressions

How To: Given a complex rational expression, simplify it

Example: Simplifying Complex Rational Expressions

Try It

Q & A

Example

Example

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Example

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Example

Example

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Example

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Summary