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Study Guides > Prealgebra

Subtracting Fractions With Common Denominators

Learning Outcomes

  • Use fraction circles to find the difference between two fractions with like denominators
  • Subtract fractions with a like denominator without fraction circles
  • Subtract fractions with like denominators that contain variables

Model Fraction Subtraction

Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into [latex]12[/latex] slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or [latex]\frac{7}{12}[/latex] of the pizza) left in the box. If Leonardo eats [latex]2[/latex] of these remaining pieces (or [latex]\frac{2}{12}[/latex] of the pizza), how much is left? There would be [latex]5[/latex] pieces left (or [latex]\frac{5}{12}[/latex] of the pizza). [latex-display]\frac{7}{12}-\frac{2}{12}=\frac{5}{12}[/latex-display] Let’s use fraction circles to model the same example, [latex]\frac{7}{12}-\frac{2}{12}[/latex]. Start with seven [latex]\frac{1}{12}[/latex] pieces. Take away two [latex]\frac{1}{12}[/latex] pieces. How many twelfths are left? The bottom reads 7 twelfths minus 2 twelfths equals 5 twelfths. Above 7 twelfths, there is a circle divided into 12 equal pieces, with 7 pieces shaded in orange. Above 2 twelfths, the same circle is shown, but 2 of the 7 pieces are shaded in grey. Above 5 twelfths, the 2 grey pieces are no longer shaded, so there is a circle divided into 12 pieces with 5 of the pieces shaded in orange. Again, we have five twelfths, [latex]\frac{5}{12}[/latex]. Doing the Manipulative Mathematics activity "Model Fraction Subtraction" will help you develop a better understanding of subtracting fractions.

Example

Use fraction circles to find the difference: [latex]\frac{4}{5}-\frac{1}{5}[/latex] Solution: Start with four [latex]\frac{1}{5}[/latex] pieces. Take away one [latex]\frac{1}{5}[/latex] piece. Count how many fifths are left. There are three [latex]\frac{1}{5}[/latex] pieces left. The bottom reads 4 fifths minus 1 fifth equals 3 fifths. Above 4 fifths, there is a circle divided into 5 equal pieces, with 4 pieces shaded in orange. Above 1 fifth, the same circle is shown, but 1 of the 4 shaded pieces is shaded in grey. Above 3 fifths, the 1 grey piece is no longer shaded, so there is a circle divided into 5 pieces with 3 of the pieces shaded in orange.

Try It

#146190 [ohm_question height="270"]146190[/ohm_question]

Subtract Fractions with a Common Denominator

We subtract fractions with a common denominator in much the same way as we add fractions with a common denominator.

Fraction Subtraction

If [latex]a,b,\text{ and }c[/latex] are numbers where [latex]c\ne 0[/latex], then [latex-display]\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}[/latex-display] To subtract fractions with a common denominators, we subtract the numerators and place the difference over the common denominator.

Example

Find the difference: [latex]\frac{23}{24}-\frac{14}{24}[/latex]

Answer: Solution:

[latex]\frac{23}{24}-\frac{14}{24}[/latex]
Subtract the numerators and place the difference over the common denominator. [latex]\frac{23 - 14}{24}[/latex]
Simplify the numerator. [latex]\frac{9}{24}[/latex]
Simplify the fraction by removing common factors. [latex]\frac{3}{8}[/latex]

Try It

#146191 [ohm_question height="270"]146191[/ohm_question]
Watch the following video for more examples of subtracting fractions with like denominators. https://youtu.be/7CeAQcpOJw0

Example

Find the difference: [latex]\frac{y}{6}-\frac{1}{6}[/latex]

Answer: Solution:

[latex]\frac{y}{6}-\frac{1}{6}[/latex]
Subtract the numerators and place the difference over the common denominator. [latex]\frac{y - 1}{6}[/latex]
The fraction is simplified because we cannot combine the terms in the numerator.

Try it

#146192 [ohm_question height="270"]146192[/ohm_question]

Example

Find the difference: [latex]-\frac{10}{x}-\frac{4}{x}[/latex]

Answer: Solution: Remember, the fraction [latex]-\frac{10}{x}[/latex] can be written as [latex]\frac{-10}{x}[/latex].

[latex]-\frac{10}{x}-\frac{4}{x}[/latex]
Subtract the numerators. [latex]\frac{-10 - 4}{x}[/latex]
Simplify. [latex]\frac{-14}{x}[/latex]
Rewrite with the negative sign in front of the fraction. [latex]-\frac{14}{x}[/latex]

Try It

#146249 [ohm_question height="270"]146249[/ohm_question]
Now lets do an example that involves both addition and subtraction.

Example

Simplify: [latex]\frac{3}{8}+\left(-\frac{5}{8}\right)-\frac{1}{8}[/latex]

Answer: Solution:

[latex]\frac{3}{8}+\left(-\frac{5}{8}\right)-\frac{1}{8}[/latex]
Combine the numerators over the common denominator. [latex]\frac{3+\left(-5\right)-1}{8}[/latex]
Simplify the numerator, working left to right. [latex]\frac{-2 - 1}{8}[/latex]
Subtract the terms in the numerator. [latex]\frac{-3}{8}[/latex]
Rewrite with the negative sign in front of the fraction. [latex]-\frac{3}{8}[/latex]

Try It

#146250 [ohm_question height="270"]146250[/ohm_question]
In the next video we show more examples of subtracting fractions with a common denominator.  Make note of the second example, it addresses a common mistake made by students when simplifying fractions with variables. https://youtu.be/-mLFZT2KgWI

Licenses & Attributions

CC licensed content, Original

  • Subtract Fractions with Variables and Common Denominators. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.

CC licensed content, Shared previously

  • Ex: Subtract Fractions with Like Denominators. Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.

CC licensed content, Specific attribution