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Study Guides > Mathematics for the Liberal Arts Corequisite

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

Model Fraction Subtraction

Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into 1212 slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or 712{\Large\frac{7}{12}} of the pizza) left in the box. If Leonardo eats 22 of these remaining pieces (or 212{\Large\frac{2}{12}} of the pizza), how much is left? There would be 55 pieces left (or 512{\Large\frac{5}{12}} of the pizza).

712212=512{\Large\frac{7}{12}}-{\Large\frac{2}{12}}={\Large\frac{5}{12}}

Let’s use fraction circles to model the same example, 712212{\Large\frac{7}{12}}-{\Large\frac{2}{12}}. Start with seven 112{\Large\frac{1}{12}} pieces. Take away two 112{\Large\frac{1}{12}} 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, 512{\Large\frac{5}{12}}.

Example

Use fraction circles to find the difference: 4515{\Large\frac{4}{5}}-{\Large\frac{1}{5}} Solution: Start with four 15{\Large\frac{1}{5}} pieces. Take away one 15{\Large\frac{1}{5}} piece. Count how many fifths are left. There are three 15{\Large\frac{1}{5}} 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

[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 a,b, and ca,b,\text{ and }c are numbers where c0c\ne 0, then

acbc=abc{\Large\frac{a}{c}}-{\Large\frac{b}{c}}={\Large\frac{a-b}{c}}

To subtract fractions with a common denominators, we subtract the numerators and place the difference over the common denominator.

Example

Find the difference: 23241424{\Large\frac{23}{24}}-{\Large\frac{14}{24}}

Answer: Solution:

23241424{\Large\frac{23}{24}}-{\Large\frac{14}{24}}
Subtract the numerators and place the difference over the common denominator. 231424{\Large\frac{23 - 14}{24}}
Simplify the numerator. 924{\Large\frac{9}{24}}
Simplify the fraction by removing common factors. 38{\Large\frac{3}{8}}

Try It

[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: 10747{\Large-\frac{10}{7}-\frac{4}{7}}

Answer: Solution: Remember, the fraction 107{\Large-\frac{10}{7}} can be written as 107{\Large\frac{-10}{7}}

10747-\frac{10}{7}-\frac{4}{7}
Subtract the numerators. 1047{\Large\frac{-10 - 4}{7}}
Simplify. 147{\Large\frac{-14}{7}}
Rewrite with the negative sign in front of the fraction. 147{\Large-\frac{14}{7}}

Now lets do an example that involves both addition and subtraction.

Example

Simplify: 38+(58)18{\Large\frac{3}{8}}+\left(-{\Large\frac{5}{8}}\right)-{\Large\frac{1}{8}}

Answer: Solution:

38+(58)18\Large\frac{3}{8}+\left(-\frac{5}{8}\right)-\frac{1}{8}
Combine the numerators over the common denominator. 3+(5)18{\Large\frac{3+\left(-5\right)-1}{8}}
Simplify the numerator, working left to right. 218{\Large\frac{-2 - 1}{8}}
Subtract the terms in the numerator. 38{\Large\frac{-3}{8}}
Rewrite with the negative sign in front of the fraction. 38{\Large-\frac{3}{8}}

Try It

[ohm_question height="270"]146250[/ohm_question]

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