We've updated our
Privacy Policy effective December 15. Please read our updated Privacy Policy and tap

Study Guides > Prealgebra

Simplifying an Expression With a Fraction Bar

Learning Outcomes

  • Identify negative fractions that are equivalent given that their negative sign is in a different location
  • Simplify expressions that contain fraction bars using the order of operations

Where does the negative sign go in a fraction? Usually, the negative sign is placed in front of the fraction, but you will sometimes see a fraction with a negative numerator or denominator. Remember that fractions represent division. The fraction [latex]-\frac{1}{3}[/latex] could be the result of dividing [latex]\frac{-1}{3}[/latex], a negative by a positive, or of dividing [latex]\frac{1}{-3}[/latex], a positive by a negative. When the numerator and denominator have different signs, the quotient is negative.

Negative 1 over positive 3 is equal to negative one third. Negative over positive equals negative. Positive 1 over negative 3 is equal to negative one third. Positive over negative equals negative. If both the numerator and denominator are negative, then the fraction itself is positive because we are dividing a negative by a negative. [latex-display]\frac{-1}{-3}=\frac{1}{3}\frac{\text{negative}}{\text{negative}}=\text{positive}[/latex-display]

Placement of Negative Sign in a Fraction

For any positive numbers [latex]a\text{ and }b[/latex], [latex-display]\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}[/latex-display]

Example

Which of the following fractions are equivalent to [latex]\frac{7}{-8}?[/latex] [latex-display]\frac{-7}{-8},\frac{-7}{8},\frac{7}{8},-\frac{7}{8}[/latex-display] Solution: The quotient of a positive and a negative is a negative, so [latex]\frac{7}{-8}[/latex] is negative. Of the fractions listed, [latex]\frac{-7}{8}\text{and}-\frac{7}{8}[/latex] are also negative.

try it

#146162 [ohm_question height="270"]146162[/ohm_question]

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. We’ll add fraction bars to our set of grouping symbols from Use the Language of Algebra to have a more complete set here.

Grouping Symbols

Parentheses, brackets, braces, an absolute value sign, and a fraction bar are shown.

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]

Answer: Solution:

[latex]\frac{4+8}{5 - 3}[/latex]
Simplify the expression in the numerator. [latex]\frac{12}{5 - 3}[/latex]
Simplify the expression in the denominator. [latex]\frac{12}{2}[/latex]
Simplify the fraction. [latex]6[/latex]

Try It

#146163 [ohm_question height="270"]146163[/ohm_question]
the following video provides another example of how to simplify various expressions that contain a fraction bar. https://youtu.be/eF_jiZvb79w

Example

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

Answer: Solution:

[latex]\frac{4 - 2\left(3\right)}{{2}^{2}+2}[/latex]
Use the order of operations. Multiply in the numerator and use the exponent in the denominator. [latex]\frac{4 - 6}{4+2}[/latex]
Simplify the numerator and the denominator. [latex]\frac{-2}{6}[/latex]
Simplify the fraction. [latex]-\frac{1}{3}[/latex]

Try It

#146164 [ohm_question height="270"]146164[/ohm_question]

Example

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

Answer: Solution:

[latex]\frac{{\left(8 - 4\right)}^{2}}{{8}^{2}-{4}^{2}}[/latex]
Use the order of operations (parentheses first, then exponents). [latex]\frac{{\left(4\right)}^{2}}{64 - 16}[/latex]
Simplify the numerator and denominator. [latex]\frac{16}{48}[/latex]
Simplify the fraction. [latex]\frac{1}{3}[/latex]

Try It

#146165 [ohm_question height="270"]146165[/ohm_question]

Example

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

Answer: Solution:

[latex]\frac{4\left(-3\right)+6\left(-2\right)}{-3\left(2\right)-2}[/latex]
Multiply. [latex]\frac{-12+\left(-12\right)}{-6 - 2}[/latex]
Simplify. [latex]\frac{-24}{-8}[/latex]
Divide. [latex]3[/latex]

Try It

#146167 [ohm_question height="270"]146167[/ohm_question]
Watch this video to see another example of how to simplify an expression with a fraction bar that contains several different operations. https://youtu.be/fK7-w77cgVQ

Licenses & Attributions

CC licensed content, Shared previously

  • Simplify Basic Expressions in Fraction Form. Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
  • Ex 1: Simplify an Expression in Fraction form (Order of Operations). Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.

CC licensed content, Specific attribution