Introduction

Let's consider the following subtraction problem:

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

When we subtract the fractional parts, we get

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

However, we cannot perform this subtraction because {\color{red}\dfrac{2}{3}} is larger than {\color{blue}\dfrac{1}{3}}.

To get around this problem, we write each mixed number as an improper fraction and then subtract:

  • Writing \color{blue}4\,\dfrac 1 3 as an improper fraction, we have \dfrac{(4\times 3)+1}{3} = {\color{blue}\dfrac{13}{3}}.

  • Writing \color{red}3\,\dfrac 2 3 as an improper fraction, we have \dfrac{(3\times 3) + 2}{3} = {\color{red}\dfrac {11} 3}.

We can now carry out the subtraction in the usual way.

\begin{align*} {\color{blue}4\,\dfrac{1}{3}} - {\color{red}3\,\dfrac{2}{3}} &= {\color{blue}\dfrac{13}{3}} - {\color{red}\dfrac{11}{3}} \\[5pt] &= \dfrac{13-11}{3} \\[5pt] &= \dfrac{2}{3} \end{align*}

Therefore, we conclude that

4 \,\dfrac{1}{3} - 3 \, \dfrac{2}{3} = \dfrac{2}{3}.

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Example: Subtracting Mixed Numbers

What is the value of 10 \,\dfrac{2}{9} - 9 \, \dfrac{4}{9} ?

EXPLANATION

When subtracting the fractional parts, we get

\dfrac{2}{9} - \dfrac{4}{9}.

However, we cannot perform this subtraction because \dfrac49 is larger than \dfrac29.

To get around this problem, we write each mixed number as an improper fraction.

  • Writing 10\,\dfrac 2 9 as an improper fraction, we have \dfrac{(10\times 9)+2}{9} = \dfrac{92}{9}.

  • Writing 9 \, \dfrac{4}{9} as an improper fraction, we have \dfrac{(9\times 9) + 4}{9} = \dfrac {85} 9.

We can now carry out the subtraction in the usual way: \dfrac{92}{9} - \dfrac {85} 9= \dfrac{92-85}{9} = \dfrac{7}{9}

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Practice: Subtracting Mixed Numbers

$6 \,\dfrac{1}{3} - 5 \, \dfrac{2}{3} \, = $

a
 $1 \, \dfrac{2}{3}$
b
 $\dfrac{1}{3}$
c
 $1 \, \dfrac{1}{3}$
d
 $1$
e
 $\dfrac{2}{3}$
Practice: Subtracting Mixed Numbers

$6 \,\dfrac{5}{9} - 5 \, \dfrac{7}{9} \, = $

a
 $\dfrac{7}{9}$
b
 $1 \, \dfrac{1}{9}$
c
 $1 \, \dfrac{2}{9}$
d
 $\dfrac{2}{9}$
e
 $\dfrac{5}{9}$
Practice: Subtracting Mixed Numbers

$3 \,\dfrac{3}{11} - 2 \, \dfrac{5}{11} \, =$

a
b
c
d
e
Example: Subtracting Mixed Numbers When Simplification Is Needed

Find the value of 7 \,\dfrac{2}{6} - 6 \, \dfrac{4}{6} .

EXPLANATION

When subtracting the fractional parts, we get

\dfrac{2}{6} - \dfrac{4}{6}.

However, we cannot perform this subtraction because \dfrac{4}{6} is larger than \dfrac{2}{6}.

To get around this problem, we write each mixed number as an improper fraction.

  • Writing 7\,\dfrac{2}{6} as an improper fraction, we have \dfrac{(7\times 6)+2}{6} = \dfrac{44}{6}.

  • Writing 6 \, \dfrac{4}{6} as an improper fraction, we have \dfrac{(6\times 6) + 4}{6} = \dfrac{40}{6}.

We can now carry out the subtraction: \dfrac{44}{6} - \dfrac {40}{6}= \dfrac{44-40}{6} = \dfrac{4}{6}

Finally, we can simplify \dfrac{4}{6} by dividing the numerator and denominator by 2{:} \dfrac{4}{6} = \dfrac{4\div 2}{6\div 2} = \dfrac{2}{3}

Therefore, we conclude that

7 \,\dfrac{2}{6} - 6 \, \dfrac{4}{6} = \dfrac23

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Practice: Subtracting Mixed Numbers When Simplification Is Needed

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

a
 $2$
b
 $\dfrac{3}{4}$
c
 $\dfrac{1}{2}$
d
 $\dfrac{1}{3}$
e
 $1$
Practice: Subtracting Mixed Numbers When Simplification Is Needed

Expressed as a fraction in its simplest form,

a
b
c
d
e
Practice: Subtracting Mixed Numbers When Simplification Is Needed

Expressed as a fraction in its simplest form,

a
b
c
d
e
Cases When the Difference of the Fractional Parts Gives a Whole

When subtracting mixed numbers, we sometimes get an answer that is an improper fraction. In such cases, it is best practice to leave the final answer as a mixed number.

For example, let's consider the following subtraction problem:

4 \,\dfrac{3}{5} - 1 \, \dfrac{4}{5}

When subtracting the fractional parts, we get

\dfrac{3}{5} - \dfrac{4}{5}.

We cannot perform this subtraction because \dfrac45 is larger than \dfrac35.

To get around this problem, we write each mixed number as an improper fraction.

  • Writing 4\,\dfrac 3 5 as an improper fraction, we have: \dfrac{(4\times 5)+3}{5} = \dfrac{23}{5}

  • Writing 1\,\dfrac 4 5 as an improper fraction, we have: \dfrac{(1\times 5) + 4}{5} = \dfrac 9 5

We can now carry out the subtraction:

\dfrac{23}{5} - \dfrac{9}{5} = \dfrac{23-9}{5} = \dfrac{14}{5}

Notice that our answer is an improper fraction because the numerator (14) is larger than the denominator (5). Therefore, we convert our answer to a mixed number:

\begin{align*} \dfrac{14}{5} &=14\div 5\\[5pt] &= 2\,\textrm{R}\,4\\[5pt] &= 2\,\dfrac45 \end{align*}

So, we conclude that

4 \,\dfrac{3}{5} - 1 \, \dfrac{4}{5} = 2\,\dfrac45.

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Example: Subtracting Mixed Numbers When the Difference of the Fractional Parts Gives a Whole

What is the value of 7 \, \dfrac{1}{4} - 2 \, \dfrac{3}{4} ?

EXPLANATION

When subtracting the fractional parts, we get

\dfrac{1}{4} - \dfrac{3}{4}.

However, we cannot perform this subtraction because \dfrac34 is larger than \dfrac14.

To get around this problem, we write each mixed number as an improper fraction.

  • Writing 7\,\dfrac 1 4 as an improper fraction, we have \dfrac{(4\times 7)+1}{4} = \dfrac{29}{4}.

  • Writing 2 \, \dfrac{3}{4} as an improper fraction, we have \dfrac{(2\times 4) + 3}{4} = \dfrac {11} 4.

We can now carry out the subtraction: \dfrac{29}{4} - \dfrac {11} 4 = \dfrac{29-11}{4} = \dfrac{18}{4}

We can simplify \dfrac{18}{4} by dividing the numerator and denominator by 2{:} \dfrac{18}{4} = \dfrac{18\div 2}{4\div 2} = \dfrac{9}{2}

Finally, we convert back to a mixed number: 9\div 2 = 4\,\textrm{R}1

So, our final answer is 4\,\dfrac{1}{2}.

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Practice: Subtracting Mixed Numbers When the Difference of the Fractional Parts Gives a Whole

Expressed as a mixed number,

a
b
c
d
e
Practice: Subtracting Mixed Numbers When the Difference of the Fractional Parts Gives a Whole

$10 \,\dfrac{3}{7} - 6 \, \dfrac{5}{7} \, = $

a
 $4$
b
 $4 \,\dfrac{2}{7}$
c
 $\dfrac{2}{7}$
d
 $3 \,\dfrac{5}{7}$
e
 $3$
Practice: Subtracting Mixed Numbers When the Difference of the Fractional Parts Gives a Whole

Expressed as a mixed number in its simplest form,

a
b
c
d
e
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