Introduction

Subtracting fractions with unlike denominators is just like adding them. Before subtracting two fractions, we must express them as equivalent fractions with a common denominator.

For example, suppose that we want to find the value of

\dfrac 3 {\color{blue}{4}} - \dfrac 1 {\color{red}{2}}.

The denominators are {\color{blue}{4}} and {\color{red}{2}}. Since {\color{blue}{4}} is a multiple of {\color{red}{2}}, we can make a common denominator of {\color{blue}{4}}.

We can put \dfrac{1}{\color{red}2} over a denominator of 4 by multiplying the numerator and denominator by 2\mathbin{:} \dfrac{1}{{\color{red}{2}}} = \dfrac{1\times 2}{{\color{red}{2}}\times 2} = \dfrac{2}{\color{blue}4}

Now, let's subtract the fractions. We keep the denominator the same, and we subtract the numerators. \begin{align*} \dfrac{3}{\color{blue}4} - \dfrac 2{\color{blue}4} = \dfrac{1}{\color{blue}4} \end{align*}

Therefore, the answer is \dfrac{1}{4}.

If we use a fraction model instead, we get the same answer:


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Example: Subtracting Fractions Where One Denominator Is a Multiple of the Other

Find the value of \dfrac{2}{3} - \dfrac 1 6 .

EXPLANATION

To subtract two fractions with unlike denominators, we must express each fraction as an equivalent fraction with a common denominator.

Since \color{blue}6 is a multiple of {\color{red}{3}}, we can make a common denominator of {\color{blue}{6}}.

To put \dfrac 2 {\color{red}{3}} over a denominator of {\color{blue}{6}}, we multiply the numerator and denominator by 2\mathbin{:}

\dfrac{2}{\color{red}3} = \dfrac{2\times 2}{{\color{red}{3}}\times 2} = \dfrac{4}{\color{blue}6}

We can now subtract the fractions. We keep the denominator the same, and we subtract the numerators.

\begin{align*} \dfrac{4}{\color{blue}6} - \dfrac 1 {\color{blue}6} = \dfrac{3}{\color{blue}6} \end{align*}

We can simplify this fraction by dividing the numerator and denominator by 3\mathbin{:}

\dfrac{3\div 3}{6\div 3} = \dfrac{1}{2}

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Practice: Subtracting Fractions Where One Denominator Is a Multiple of the Other

$\dfrac{5}{8} - \dfrac 1 4 = $

a
 $\dfrac 4 4$
b
 $\dfrac 1 8$
c
 $\dfrac 3 8$
d
 $\dfrac 1 2$
e
 $\dfrac 1 4$
Practice: Subtracting Fractions Where One Denominator Is a Multiple of the Other

$\dfrac{7}{12} - \dfrac 1 4 = $

a
 $\dfrac{1}{3}$
b
 $\dfrac{1}{12}$
c
 $\dfrac{1}{6}$
d
 $\dfrac{2}{3}$
e
 $\dfrac{5}{6}$
Subtracting Fractions With Unlike Denominators and No Common Factors

When one denominator is not a multiple of the other, a common denominator can always be found by taking the product of the denominators.

For example, let's compute

\dfrac{1}{\color{red}2} - \dfrac{1}{\color{blue}3}.

The number {\color{red}2} is not a multiple of {\color{blue}3} and vice-versa. But we can make a common denominator of {\color{red}2} \times {\color{blue}3} = 6.

To put \dfrac 1 {\color{red}2} over a denominator of 6, we multiply the numerator and denominator by {\color{blue}3}\mathbin{:}

\dfrac{1}{{\color{red}2} } = \dfrac{1\times {\color{blue}3} }{{\color{red}2} \times {\color{blue}3} } = \dfrac{3}{6}

To put \dfrac 1 {\color{blue}3} over a denominator of 6, we multiply the numerator and denominator by {\color{red}2}\mathbin{:}

\dfrac{1}{\color{blue}3 } = \dfrac{1\times {\color{red}2} }{{\color{blue}3} \times {\color{red}2} } = \dfrac{2}{6}

We can now subtract the fractions. We keep the denominator the same and subtract the numerators:

\begin{align*} \dfrac{3}{6}- \dfrac{2}{6} = \dfrac{1}{6} \end{align*}

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Example: Making a Common Denominator by Multiplying the Denominators

Find the value of \dfrac{3}{4} - \dfrac 1 5 .

EXPLANATION

To subtract two fractions with unlike denominators, we must express each fraction as an equivalent fraction with a common denominator.

Let's look at the denominators:

\dfrac{3}{\color{red}4} - \dfrac{1}{\color{blue}5}

The number {\color{red}4} is not a multiple of {\color{blue}5} and vice-versa. But we can make a common denominator of {\color{red}4} \times {\color{blue}5} = 20.

To put \dfrac 3 {\color{red}4} over a denominator of 20, we multiply the numerator and denominator by {\color{blue}5}\mathbin{:}

\dfrac{3}{{\color{red}4} } = \dfrac{3\times {\color{blue}5} }{{\color{red}4} \times {\color{blue}5} } = \dfrac{15}{20}

To put \dfrac 1 {\color{blue}5} over a denominator of 20, we multiply the numerator and denominator by {\color{red}4}\mathbin{:}

\dfrac{1}{{\color{blue}5} } = \dfrac{1\times {\color{red}4} }{{\color{blue}5} \times {\color{red}4} } = \dfrac{4}{20}

We can now subtract the fractions. We keep the denominator the same, and we subtract the numerators:

\begin{align*} \dfrac{15}{20}- \dfrac{4}{20} = \dfrac{11}{20} \end{align*}

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Practice: Making a Common Denominator by Multiplying the Denominators

$\dfrac{2}{3} - \dfrac 1 4 = $

a
 $\dfrac{1}{4}$
b
 $\dfrac{1}{2}$
c
 $\dfrac{1}{3}$
d
 $\dfrac{5}{12}$
e
 $\dfrac{3}{12}$
Practice: Making a Common Denominator by Multiplying the Denominators

$\dfrac{2}{3} - \dfrac 2 5 = $

a
 $ \dfrac{4}{15}$
b
 $ \dfrac{7}{15}$
c
 $ \dfrac{1}{5}$
d
 $ \dfrac{2}{15}$
e
 $ \dfrac{1}{3}$
Subtracting Fractions Using the Least Common Multiple

When two denominators have a common factor, finding the lowest common multiple is sometimes easier.

For example, let's compute

\dfrac{5}{6} - \dfrac{1}{8}.

Notice that the denominators {6} and {8} have a common factor of 2. So let's look at their multiples:

  • Multiples of 6: \quad 6,12,18,{\color{blue}{24}},30,\ldots

  • Multiples of 8: \quad 8,16,{\color{blue}{24}},32,40,\ldots

The lowest common multiple is {\color{blue}{24}}. This is the lowest common denominator, too.

To put \dfrac 5 6 over a denominator of 24, we multiply the numerator and denominator by 4\mathbin{:} \dfrac{5}{6} = \dfrac{5\times 4}{6 \times 4 } = \dfrac{20}{24}

To put \dfrac 1 8 over a denominator of 24, we multiply the numerator and denominator by 3\mathbin{:} \dfrac{1}{8} = \dfrac{1\times 3 }{8 \times 3 } = \dfrac{3}{24}

We can now subtract the fractions. We keep the denominator the same, and we subtract the numerators:

\begin{align*} \dfrac{20}{24} - \dfrac{3}{24} = \dfrac{17}{24} \end{align*}

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Example: Subtracting Fractions Using the Least Common Multiple

What is the value of \dfrac{7}{15} - \dfrac 2 9 ?

EXPLANATION

To subtract two fractions with unlike denominators, we must express each fraction as an equivalent fraction with a common denominator.

Let's look at the denominators:

\dfrac{7}{15} - \dfrac{2}{9}

The denominators 15 and 9 share a common factor of 3. So let's look at their multiples:

  • Multiples of 15: \quad 15,30,{\color{blue}{45}},60,\ldots

  • Multiples of 9: \quad 9,18,27,36,{\color{blue}{45}},\ldots

To put \dfrac 7 {15} over a denominator of 45, we multiply the numerator and denominator by 3\mathbin{:}

\dfrac{7}{{15} } = \dfrac{7\times 3 }{{15} \times 3 } = \dfrac{21}{45}

To put \dfrac 2 9 over a denominator of 45, we multiply the numerator and denominator by 5\mathbin{:}

\dfrac{2}{{9} } = \dfrac{2\times 5 }{9 \times 5 } = \dfrac{10}{45}

We can now subtract the fractions. We keep the denominator the same, and we subtract the numerators:

\begin{align*} \dfrac{21}{45}- \dfrac{10}{45} = \dfrac{11}{45} \end{align*}

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Practice: Subtracting Fractions Using the Least Common Multiple

$\dfrac{10}{9} - \dfrac 1 6 = $

a
 $\dfrac{17}{18}$
b
 $\dfrac{53}{54}$
c
 $\dfrac{26}{27}$
d
 $\dfrac{8}{9}$
e
 $\dfrac{5}{6}$
Practice: Subtracting Fractions Using the Least Common Multiple

$\dfrac{5}{6} - \dfrac 7 {10} = $

a
 $\dfrac{7}{30}$
b
 $\dfrac{7}{15}$
c
 $\dfrac{4}{15}$
d
 $\dfrac{1}{30}$
e
 $\dfrac{2}{15}$
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