Introduction

Let's consider the following expression:

12\div 4 \times 7

Notice that this expression contains multiplication and division. So how do we compute its value?

To help us, we have the following rule:

When an expression contains only multiplication and division, we carry out the operations from left to right.

So, we must first carry out the division (12\div 4). We can emphasize this using parentheses (\phantom{0}){:}

\big(12\div 4\big) \times 7

Here, the parentheses tell us which part of the expression to do first.

Now, we evaluate our expression as follows:

\begin{align*} \big({\color{blue}{12\div 4}}\big) \times 7 &= \\[5pt] {\color{blue}{3}} \times 7 &= \\[5pt] 21& \end{align*}

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Example: Evaluating Expressions With Two Operations

What is 36\div 6 \times 10?

EXPLANATION

When the only operations are multiplication and division, we carry out the operations from left to right, using parentheses (\cdots) to make our working clearer:

\begin{align*} 36\div 6 \times 10 &= \\[5pt] ({\color{blue}{36\div 6}}) \times 10 &= \\[5pt] {\color{blue}{6}} \times 10 &= \\[5pt] 60& \end{align*}

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Practice: Evaluating Expressions With Two Operations

$54\div 9 \times 8 = $

a
 $32$
b
 $48$
c
 $40$
d
 $42$
e
 $64$
Practice: Evaluating Expressions With Two Operations

$14\times 2 \div 4 = $

a
 $7$
b
 $6$
c
 $8$
d
 $5$
e
 $9$
Swapping Multiplication and Division

Let's now consider the following expression:

10{\color{red}{\,\div\, 6\,}} {\color{blue}{\,\times\, 3}}

We carry out the operations from left to right. However, the division problem is difficult because 10 cannot be divided evenly into groups of 6. So, what do we do?

To help us, we have the following rule:

When the only operations are multiplication and division, we can swap the order in which the operations are carried out.

Therefore, we can swap the order of the multiplication and division as follows:

10{\color{blue}{\,\times\, 3}}{\color{red}{\,\div\, 6\,}}

Now, we evaluate our expression:

\begin{align*} 10\times 3 \div 6 &= \\[5pt] \big(10\times 3\big) \div 6 &= \\[5pt] 30 \div 6 &= \\[5pt] 5& \end{align*}

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Example: Swapping Multiplication and Division: Division First

Compute 2\div 8 \times 12.

EXPLANATION

When the only operations are multiplication and division,

  • we carry out the operations from left to right, using parentheses (\cdots) to make our working clearer, and

  • we can swap the order in which the operations are carried out.

Therefore,

\begin{align*} 2\div 8 \times 12 &= \\[5pt] 2\times 12 \div 8 &= \\[5pt] ({\color{blue}{2\times 12}}) \div 8 &= \\[5pt] {\color{blue}{24}} \div 8 &= \\[5pt] 3&. \end{align*}

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Practice: Swapping Multiplication and Division: Division First

$5\div 3 \times 6 = $

a
 $16$
b
 $12$
c
 $10$
d
 $15$
e
 $9$
Practice: Swapping Multiplication and Division: Division First

$4\div 3 \times 6 = $

a
 $8$
b
 $9$
c
 $2$
d
 $10$
e
 $6$
Example: Swapping Multiplication and Division: Multiplication First

Find 24\times 5 \div 3.

EXPLANATION

When the only operations are multiplication and division,

  • we carry out the operations from left to right, using parentheses (\cdots) to make our working clearer, and

  • we can swap the order in which the operations are carried out.

Therefore,

\begin{align*} 24\times 5 \div 3 &= \\[5pt] 24 \div 3 \times 5 &= \\[5pt] ({\color{blue}{24 \div 3}}) \times 5 &= \\[5pt] {\color{blue}{8}} \times 5 &= \\[5pt] 40&. \end{align*}

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Practice: Swapping Multiplication and Division: Multiplication First

$14\times 8 \div 7 = $

a
 $16$
b
 $12$
c
 $18$
d
 $14$
e
 $17$
Practice: Swapping Multiplication and Division: Multiplication First

$12\times 21 \div 4 = $

a
 $53$
b
 $68$
c
 $84$
d
 $64$
e
 $63$
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