Simplifying and Evaluating Complex Fractions

Learning Outcomes

Simplify complex fractions that contain several different mathematical operations
Evaluate variable expressions with fractions

In Multiply and Divide Mixed Numbers and Complex Fractions, we saw that a complex fraction is a fraction in which the numerator or denominator contains a fraction. We simplified complex fractions by rewriting them as division problems. For example,

$\Large\frac{\LARGE{\frac{3}{4}}}{\LARGE{\frac{5}{8}}}=\Large\frac{3}{4}\div\Large\frac{5}{8}$

Now we will look at complex fractions in which the numerator or denominator can be simplified. To follow the order of operations, we simplify the numerator and denominator separately first. Then we divide the numerator by the denominator.

Simplify complex fractions

Simplify the numerator.
Simplify the denominator.
Divide the numerator by the denominator.
Simplify if possible.

Example

Simplify: $\Large\frac{{\Large{(\LARGE\frac{1}{2}})}^{2}}{4+{3}^{2}}$

Solution:

	$\Large\frac{{\left(\LARGE\frac{1}{2}\right)}^{2}}{4+{3}^{2}}$
Simplify the numerator.	$\Large\frac{\LARGE\frac{1}{4}}{4+{3}^{2}}$
Simplify the term with the exponent in the denominator.	$\Large\frac{\LARGE\frac{1}{4}}{4+9}$
Add the terms in the denominator.	$\Large\frac{\LARGE\frac{1}{4}}{13}$
Divide the numerator by the denominator.	$\Large\frac{1}{4}\div \normalsize13$
Rewrite as multiplication by the reciprocal.	$\Large\frac{1}{4}\cdot\Large\frac{1}{13}$
Multiply.	$\Large\frac{1}{52}$

Try It

Example

Simplify: $\Large\frac{\LARGE\frac{1}{2}+\LARGE\frac{2}{3}}{\LARGE\frac{3}{4}-\LARGE\frac{1}{6}}$

Show Solution

Solution:

	$\Large\frac{\LARGE\frac{1}{2}+\LARGE\frac{2}{3}}{\LARGE\frac{3}{4}-\LARGE\frac{1}{6}}$
Rewrite numerator with the LCD of $6$ and denominator with LCD of $12$ .	$\Large\frac{\LARGE\frac{3}{6}+\LARGE\frac{4}{6}}{\LARGE\frac{9}{12}-\LARGE\frac{2}{12}}$
Add in the numerator. Subtract in the denominator.	$\Large\frac{\LARGE\frac{7}{6}}{\LARGE\frac{7}{12}}$
Divide the numerator by the denominator.	$\Large\frac{7}{6}\div\Large\frac{7}{12}$
Rewrite as multiplication by the reciprocal.	$\Large\frac{7}{6}\cdot\Large\frac{12}{7}$
Rewrite, showing common factors.	$\Large\frac{\overline{)7}\cdot \overline{)6}\cdot 2}{\overline{)6}\cdot \overline{)7}\cdot 1}$
Simplify.	$2$

Try It

In the following video we sow more examples of simplifying complex expressions.

Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can also evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

Example

Evaluate $x+\Large\frac{1}{3}$ when

$x=-\Large\frac{1}{3}$
$x=-\Large\frac{3}{4}$

Solution
1. To evaluate $x+\Large\frac{1}{3}$ when $x=-\Large\frac{1}{3}$ , substitute $-\Large\frac{1}{3}$ for $x$ in the expression.

	$x+\Large\frac{1}{3}$
Substitute $\color{red}{--\Large\frac{1}{3}}$ for x.	$\color{red}{--\Large\frac{1}{3}} +\Large\frac{1}{3}$
Simplify.	$0$

2. To evaluate $x+\Large\frac{1}{3}$ when $x=-\Large\frac{3}{4}$ , we substitute $-\Large\frac{3}{4}$ for $x$ in the expression.

	$x+\Large\frac{1}{3}$
Substitute $\color{red}{--\Large\frac{3}{4}}$ for x.	$\color{red}{--\Large\frac{3}{4}} +\Large\frac{1}{3}$
Rewrite as equivalent fractions with the LCD, $12$ .	$-\Large\frac{3\cdot 3}{4\cdot 3}+\Large\frac{1\cdot 4}{3\cdot 4}$
Simplify the numerators and denominators.	$-\Large\frac{9}{12}+\Large\frac{4}{12}$
Add.	$-\Large\frac{5}{12}$

Try It

Example

Evaluate $y-\Large\frac{5}{6}$ when $y=-\Large\frac{2}{3}$

Show Solution

Solution:
We substitute $-\Large\frac{2}{3}$ for $y$ in the expression.

	$y-\Large\frac{5}{6}$
Substitute $\color{red}{--\Large\frac{2}{3}}$ for y.	$\color{red}{--\Large\frac{2}{3}} -\Large\frac{5}{6}$
Rewrite as equivalent fractions with the LCD, $6$ .	$-\Large\frac{4}{6}-\Large\frac{5}{6}$
Subtract.	$-\Large\frac{9}{6}$
Simplify.	$-\Large\frac{3}{2}$

Try It

Example

Evaluate $2{x}^{2}y$ when $x=\Large\frac{1}{4}$ and $y=-\Large\frac{2}{3}$

Show Solution

Solution:
Substitute the values into the expression. In $2{x}^{2}y$ , the exponent applies only to $x$ .

	$2{x}^{2}y$
Substitute $\color{red}{\Large\frac{1}{4}}$ for x and $\color{blue}{--\Large\frac{2}{3}}$ for y.	$2(\color{red}{\Large\frac{1}{4}})^{2}(\color{blue}{--\Large\frac{2}{3}})$
Simplify exponents first.	$2(\Large\frac{1}{16})(--\Large\frac{2}{3})$
Multiply. The product will be negative.	$--\Large\frac{2}{1}\cdot\Large\frac{1}{16}\cdot\Large\frac{2}{3}$
Simplify.	$--\Large\frac{4}{48}$
Remove the common factors.	$--\Large\frac{1\cdot\color{red}{4}}{\color{red}{4} \cdot\ 12}$
Simplify.	$--\Large\frac{1}{12}$

try It

Example

Evaluate $\Large\frac{p+q}{r}$ when $p=-4,q=-2$ , and $r=8$

Show Solution

Solution:
We substitute the values into the expression and simplify.

	$\Large\frac{p+q}{r}$
Substitute $\color{red}{--4}$ for p, $\color{blue}{--2}$ for q, and $\color{red}{8}$ for r.	$\Large\frac{\color{red}{--4} + \color{blue}{(--2)}}{\color{red}{8}}$
Add in the numerator first.	$-\Large\frac{6}{8}$
Simplify.	$-\Large\frac{3}{4}$

Module 4: Fractions