1.3.7: Order of Operations with Fractions

Learning Objectives

  • Use order of operations to simplify expressions involving fractions

Key words

  • Fraction Bar: the bar in a fraction that acts like as grouping symbol

Using the order of operations with fractions

Recall that the order of operations outlines the order in which terms in an expression must be simplified.

Order of Operations

When simplifying mathematical expressions perform the operations in the following order:
1. Grouping Symbols

  • Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.

2. Exponents

  • Simplify all expressions with exponents.

3. Multiplication and Division from left to right

  • Perform all multiplication and division in order from left to right. These operations have equal priority.

4. Addition and Subtraction

  • Perform all addition and subtraction in order from left to right. These operations have equal priority.

Performing the order of operations for fractional expressions is no different than order of operations for integers.

Example

Simplify: [latex]\frac{3}{5} \cdot \frac{2}{3} \div \frac{1}{6}[/latex]

Solution:

One term [latex]\frac{3}{5} \cdot \frac{2}{3} \div \frac{1}{6}[/latex]
Multiply or divide from left to right. [latex]\large\frac{\color{red}{\cancel{3}} \cdot 2}{5 \cdot \color{red}{\cancel{3}}} \div \frac{1}{6}[/latex]
Simplify by cancelling. [latex]\frac{2}{5}\div \frac{1}{6}[/latex]
Turn the division into multiplication of the reciprocal. [latex]\frac{2}{5}\cdot \frac{6}{1}[/latex]
Multiply. [latex]\frac{2\cdot 6}{5\cdot 1}[/latex]
Simplify. [latex]\frac{12}{5}[/latex]

Try It

The following video provides examples of how simplify fractional expression with order of operations.

Simplifying an Expression With a Fraction Bar

Fraction bars act as grouping symbols. The expressions above and below the fraction bar should be treated as if they were in parentheses. For example, [latex]\frac{4+8}{5 - 3}[/latex] means [latex]\left(4+8\right)\div\left(5 - 3\right)[/latex]. The order of operations tells us to simplify the numerator and the denominator first—as if there were parentheses—before we divide.

Simplify an expression with a fraction bar

  1. Simplify the numerator.
  2. Simplify the denominator.
  3. Simplify the fraction.

Example

Simplify: [latex]\frac{4+8}{5 - 3}[/latex]

Try It

the following video provides another example of how to simplify various expressions that contain a fraction bar.

Example

Simplify: [latex]\frac{4 - 2\left(3\right)}{{2}^{2}+2}[/latex]

Try It

Example

Simplify: [latex]\frac{{\left(8 - 4\right)}^{2}}{{8}^{2}-{4}^{2}}[/latex]

Try It

Example

Simplify: [latex]\frac{4\left(-3\right)+6\left(-2\right)}{-3\left(2\right)-2}[/latex]

Try It

Watch this video to see another example of how to simplify an expression with a fraction bar that contains several different operations.

When there are two or more fractions, we work each fraction separately, then add or subtract them once they are simplified to lowest terms.

Example

Simplify:  [latex]\frac{-3^2+4\cdot (-6)}{(-5)^2-(6-2)} - \left [\frac{6-2^2}{-4}\right ]^{2}[/latex]

Solution

Let’s work each fraction separately, then subtract them.

1.   [latex]\frac{-3^2+4\cdot (-6)}{(-5)^2-(6-2)}[/latex]           Exponent and parentheses

[latex]=\frac{-9+(-24)}{25-(4)}[/latex]        Addition and subtraction

[latex]=\frac{-33}{21}[/latex]        Simplify

[latex]=\frac{-11}{7}[/latex]

 

2.   [latex]\left [\frac{6-2^2}{-4}\right ]^{2}[/latex]          Exponent

[latex]=\left [\frac{6-4}{-4}\right ]^{2}[/latex]         Subtract

[latex]=\left [\frac{2}{-4}\right ]^{2}[/latex]         Simplify

[latex]=\left [\frac{1}{-2}\right ]^{2}[/latex]        Square the numerator and denominator

[latex]=\frac{1}{4}[/latex]

 

Finally, we subtract:

[latex]\frac{-3^2+4\cdot (-6)}{(-5)^2-(6-2)} - \left [\frac{6-2^2}{-4}\right ]^{2}[/latex]

[latex]=\frac{-11}{7}-\frac{1}{4}[/latex]       LCD = 28.  Build equivalent fraction with LCD

[latex]=\frac{-11\cdot 4}{7\cdot 4}-\frac{1\cdot 7}{4\cdot 7}[/latex]         Multiply numerators and denominators

[latex]=\frac{-44}{28}-\frac{7}{28}[/latex]          Subtract numerators and keep common denominator

[latex]=\frac{-44-7}{28}[/latex]         Subtract

[latex]=\frac{-51}{28}[/latex]

 

Try It

Simplify:  [latex]\frac{4\cdot 3 - 7}{5\cdot 2 - 4}-\frac{3-5^2}{(7-3)^2 - 5}[/latex]