Introduction

Let's consider the following expression:

5 + 3 \times 7

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

To help us, we have the following rule:

When an expression contains only multiplication and addition (or subtraction), the multiplication should always be carried out first.

So, we must first carry out the multiplication (3\times 7). We can emphasize this using parentheses (\phantom{0}){:}

5 + \big({\color{blue}3 \times 7}\big)

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

Now, we evaluate our expression as follows: \begin{align*} 5 + \big({\color{blue}3 \times 7}\big) &=\\[5pt] 5 + {\color{blue}21} &=\\[5pt] 26 & \end{align*}

Watch out! If we carried out the operations in a different order, we'd obtain an incorrect answer!

For example, if we perform the addition first, we get

\begin{align*} \big({\color{red}{5 + 3}}\big) \times 7 &=\\[5pt] {\color{red}{8}}\times 7 &=\\[5pt] 56&\qquad{\color{red}{\times}} \end{align*}

which is incorrect!

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Example: Evaluating Expressions Where Multiplication Comes First

Evaluate 5 \times 4 -7.

EXPLANATION

Multiplication should be carried out before addition or subtraction. We can emphasize this by using parentheses:

5 \times 4 -7 = ({\color{blue}5 \times 4})-7

We carry out the operations inside the parentheses first. We have

{\color{blue}5 \times 4} = {\color{blue}20}.

Therefore,

\begin{align*} ({\color{blue}5 \times 4}) -7 &=\\[3pt] {\color{blue}20} -7 &=\\[3pt] 13 &. \end{align*}

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Practice: Evaluating Expressions Where Multiplication Comes First

Evaluate $4 \times 7 + 3.$

a
 $82$
b
 $31$
c
 $24$
d
 $34$
e
 $84$
Practice: Evaluating Expressions Where Multiplication Comes First

$4 \times 7 - 3 =$

a
 $25$
b
 $15$
c
 $16$
d
 $31$
e
 $23$
Example: Evaluating Expressions Where Multiplication Comes Second

Evaluate 4 + 10 \times 2.

EXPLANATION

Multiplication should be carried out before addition or subtraction. We can emphasize this by using parentheses:

4+ 10 \times 2 = 4 + ({\color{blue}10 \times 2})

We carry out the operations inside the parentheses first. We have

{\color{blue}10 \times 2} = {\color{blue}20}.

Therefore,

\begin{align*} 4 + ({\color{blue}10 \times 2}) &=\\[3pt] 4 + {\color{blue}20} &=\\[3pt] 24 &. \end{align*}

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Practice: Evaluating Expressions Where Multiplication Comes Second

Evaluate $3 + 4 \times 7.$

a
 $25$
b
 $84$
c
 $88$
d
 $23$
e
 $31$
Practice: Evaluating Expressions Where Multiplication Comes Second

$25 - 4 \times 3 =$

a
 $60$
b
 $15$
c
 $66$
d
 $63$
e
 $13$
Example: Evaluating Expressions Containing Two Products

Evaluate 3 \times 9 - 5 \times 5.

EXPLANATION

Multiplication should be carried out before addition or subtraction. We can emphasize this by using parentheses:

3 \times 9 - 5 \times 5 = ({\color{blue}3 \times 9}) - ({\color{red}5 \times 5})

Now, we evaluate the products:

\begin{align*} {\color{blue}3 \times 9} &= {\color{blue}27} \\[5pt] {\color{red}5\times 5} &= {\color{red}25} \end{align*}

Therefore,

\begin{align*} ({\color{blue}3 \times 9}) - ({\color{red}5 \times 5}) &=\\[3pt] {\color{blue}27} - {\color{red}25} &=\\[3pt] 2 &. \end{align*}

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Practice: Evaluating Expressions Containing Two Products

Evaluate $3 \times 4 + 2 \times 3.$

a
 $54$
b
 $22$
c
 $18$
d
 $12$
e
 $42$
Practice: Evaluating Expressions Containing Two Products

$3 \times 4 - 2 \times 3 =$

a
 $12$
b
 $18$
c
 $30$
d
 $6$
e
 $24$
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