Introduction

Remember that a unit fraction contains \color{blue}1 in the numerator.

For example, the following fractions are unit fractions since they each contain {\color{blue}{1}} in the numerator:

\dfrac{\color{blue}{1}}{2}, \qquad \dfrac{\color{blue}{1}}{5}, \qquad \dfrac{\color{blue}{1}}{8}

However, the following fractions are not-unit fractions since they do not contain {\color{blue}{1}} in the numerator:

\dfrac{\color{red}{2}}{3}, \qquad \dfrac{\color{red}{5}}{8}, \qquad \dfrac{\color{red}{7}}{12}

In this lesson, we will use models to divide whole numbers by non-unit fractions.

Note: Throughout this lesson, we'll be drawing numerous fraction models. It is highly recommended that you draw these models on a piece of paper to better understand their workings.

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A Worked Example

Let's use a model to find the result of the following division problem:

{\color{blue}{1}} \div \dfrac26

We start with a model that represents {\color{blue}{1}} whole:

We need to determine how many times \dfrac26 fits into {\color{blue}{1}} whole. So, we proceed as follows:

  • First, we split our whole into sixths (the denominator of our fraction):
  • Next, we create groups, where each group contains \dfrac26{:}
  • Finally, we count the groups. We see that there are \color{purple} 3 groups in total.

So, if we divide \color{blue}1 whole into groups of {\color{black}{\dfrac26}}, we get {\color{purple}{3}} groups in total.

Therefore, we conclude that

{\color{blue}{1}} \div {\color{black}\dfrac26} = {\color{purple}3}.

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Example: Finding a Missing Number

Use the model above to determine the missing number in the following division problem. 4 \div \fbox{[math]\,\phantom{\dfrac00}\,[/math]} = 10

EXPLANATION

Let's start by interpreting our model:

  • The model shows \color{blue}4 wholes. Each whole is split into fifths.
  • The fifths are grouped, and each group contains {\color{red}{\dfrac{2}{5}}}.
  • There are \color{purple} 10 groups in total.

So, if we divide \color{blue}4 wholes into groups of {\color{red}{\dfrac{2}{5}}}, we get {\color{purple}{10}} groups in total.

Therefore, the model represents the following division problem: {\color{blue}{4}} \div \fbox{[math]\color{red}\dfrac{2}{5}[/math]} = {\color{purple}{10}}

So, the missing number is \bbox[3pt, Gainsboro]{\color{red}\dfrac{2}{5}}.

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Practice: Finding a Missing Number

Use the model above to determine the missing number in the following division problem. \[ \fbox{$\,\phantom{0}\,$} \div \dfrac{2}{7} = 7 \]

a
 $7$
b
 $1$
c
 $3$
d
 $14$
e
 $2$
Practice: Finding a Missing Number

Use the model above to determine the missing number in the following division problem. \[ 3 \div \dfrac{3}{4} = \fbox{$\,\phantom{0}\,$} \]

a
 $4$
b
 $3$
c
 $12$
d
 $8$
e
 $2$
Example: Finding Two Missing Numbers

Use the model above to determine the missing numbers in the following division problem. \fbox{[math]\,\phantom{0^2}[/math]}\: \div \:\! \dfrac{2}{3} = \,\fbox{[math]\,\phantom{0^2}\,[/math]}

EXPLANATION

Let's start by interpreting our model:

  • The model shows \color{blue}2 wholes. Each whole is split into thirds.
  • The thirds are grouped, and each group contains {\color{red}{\dfrac23}}.
  • There are \color{purple} 3 groups in total.

So, if we divide \color{blue}2 wholes into groups of {\color{red}{\dfrac23}}, we get {\color{purple}{3}} groups in total.

Therefore, the model represents the following division problem: \fbox{[math]\,{\color{blue}2}\,[/math]} \div {\color{red}{\dfrac23}} = \fbox{[math]\color{purple}\,3\,[/math]}

Therefore, the missing numbers are \bbox[3pt, Gainsboro]{\color{blue}2} and \bbox[3pt, Gainsboro]{\color{purple}3}.

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Practice: Finding Two Missing Numbers

Use the model above to determine the missing numbers in the following division problem.

a
b
c
d
e
Practice: Finding Two Missing Numbers

Use the model above to determine the missing numbers in the following division problem.

a
b
c
d
e
Practice: Finding Two Missing Numbers

Use the model above to determine the missing numbers in the following division problem.

a
b
c
d
e
Example: Dividing a Whole Number by a Non-Unit Fraction

The model above represents 3 wholes. Use the model to determine the missing number in the following division problem. 3 \div \dfrac{3}{5} = \fbox{[math]\,\phantom{0}\,[/math]}

EXPLANATION

We proceed as follows:

  • First, we split each whole into fifths.
  • Next, we create groups of {\color{red}{\dfrac3{5}}}.
  • Then, we count the groups.

So, if we divide \color{blue}3 wholes into groups of {\color{red}{\dfrac3{5}}}, we get {\color{purple}{5}} groups in total.

Therefore, we conclude that

3 \div \dfrac{3}{{5}} = \fbox{5}.

So, the missing number is \bbox[3pt, Gainsboro]{\color{purple}5}.

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Practice: Dividing a Whole Number by a Non-Unit Fraction

The model above represents $1$ whole. Use the model to determine the missing number in the following division problem.

\[ 1 \div \dfrac{2}{8} = \fbox{$\,\phantom{0^i}\,$} \]

a
 $16$
b
 $8$
c
 $4$
d
 $2$
e
 $6$
Practice: Dividing a Whole Number by a Non-Unit Fraction

If the model above represents $3$ wholes, use the model to solve the following division problem. Express your answer as a whole number.

a
b
c
d
e
Practice: Dividing a Whole Number by a Non-Unit Fraction

The model above represents $4$ wholes. Use the model to determine the missing number in the following division problem. \[ 4 \div \dfrac{4}{5} = \fbox{$\,\phantom{0}\,$} \]

a
 $5$
b
 $\dfrac{4}{5}$
c
 $2$
d
 $\dfrac{1}{5}$
e
 $4$
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