Introduction

Some expressions contain multiple operations. To evaluate these correctly, we follow a sequence of steps called the order of operations.

According to the order of operations, if both addition ( + ) and subtraction ( - ) appear in an expression, then we complete each operation from left to right.

For example, let's look at the following expression:

12 - 5 + 4

To evaluate this expression, we add and subtract from left to right: \begin{align} {\color{blue}12 - 5} + 4 &=\\[5pt] {\color{blue}7} + 4&=\\[5pt] 11& \phantom{=} \color{green} \checkmark \end{align}

Watch out! You have to evaluate the expression from left to right. If you evaluate the expression in the opposite direction, from right to left, then it does not come out to the same answer.

\begin{align} 12 - {\color{red}5 + 4} &=\\[5pt] 12 - {\color{red}9} &=\\[5pt] 3& \phantom{=} \color{red} \times \end{align}

FLAG
Example: Evaluating an Expression Containing Addition and Subtraction

Evaluate 6+8-12.

EXPLANATION

To evaluate the given expression, we follow the order of operations.

In this case, we add and subtract from left to right:

\begin{align*} {\color{blue}{6+8}}-12 &=\\[5pt] {\color{blue}{14}}-12 &=\\[5pt] 2& \end{align*}

FLAG
Practice: Evaluating an Expression Containing Addition and Subtraction

$9 + 6 - 5 = $

a
 $8$
b
 $10$
c
 $14$
d
 $2$
e
 $20$
Practice: Evaluating an Expression Containing Addition and Subtraction

$12-6+4 =$

a
 $2$
b
 $6$
c
 $10$
d
 $4$
e
 $8$
The Order of Operations With Multiplication and Division

The order of operations also states that if both multiplication ( \times ) and division ( \div ) appear in an expression, we complete them from left to right.

For example, to evaluate 18 \div 2 \times 3, we multiply and divide from left to right:

\begin{align} {\color{blue}18 \div 2} \times 3 &=\\ {\color{blue}9} \times 3 &= \\ 27& \phantom{=} \color{green} \checkmark \end{align}

Watch out! You have to evaluate the expression from left to right. If you evaluate the expression in the opposite direction, from right to left, then it does not come out to the same answer.

\begin{align} 18 \div {\color{red}2 \times 3} &=\\[5pt] 18 \div {\color{red}6} &=\\[5pt] 3& \phantom{=} \color{red} \times \end{align}

FLAG
Example: Evaluating an Expression With Multiplication and Division

Calculate 12 \div 6 \times 4 .

EXPLANATION

To calculate the given expression, we follow the order of operations.

In this case, we multiply and divide from left to right:

\begin{align} 12 \div 6 \times 4&=\\[5pt] {\color{blue}12 \div 6} \times 4&=\\[5pt] {\color{blue}2} \times 4 &=\\[5pt] 8& \end{align}

FLAG
Practice: Evaluating an Expression With Multiplication and Division

$3 \times 6 \div 2 =$

a
 $18$
b
 $9$
c
 $36$
d
 $6$
e
 $12$
Practice: Evaluating an Expression With Multiplication and Division

$21 \div 7 \times 4 = $

a
 $15$
b
 $7$
c
 $3$
d
 $12$
e
 $4$
Evaluating Numerical Expressions With a Mix of Operations

When an expression contains a mix of addition, subtraction, multiplication, and division, the order of operations tells us:

  • First, we multiply and divide from left to right.

  • Then, we add and subtract from left to right.

Let's use the order of operations to evaluate the expression 18 - 9 \div 3.

First, we multiply and divide, from left to right: \begin{align} 18 - \underbrace{\color{blue}{9 \div 3}}_{1\textrm{st}} & = 18 - {\color{blue}3}& \end{align}

Now, we add and subtract from left to right:

\begin{align} 18 - 3 &= 15 \quad{\color{green}\checkmark} \end{align}

Watch out! In this question, it's really important that we use the order of operations. If we only solve from left to right, we get an incorrect result:

\begin{align} {\color{red} 18-9} \div 3 &= \\[5pt] {\color{red}9} \div 3 &=\\[5pt] 3& \phantom{=} {\color{red}\times} \end{align}

FLAG
Example: Evaluating an Expression With Two Mixed Operations

Evaluate 3 + 2 \times 4.

EXPLANATION

First, we multiply and divide, from left to right:

\begin{align} 3 + 2 \times 4 &=\\[5pt] 3 + {\color{blue}2 \times 4} &=\\[5pt] 3 + {\color{blue}8} & \end{align}

Now, we add:

\begin{align} 3 + 8 &= 11 \end{align}

FLAG
Practice: Evaluating an Expression With Two Mixed Operations

$5 + 10 \div 5 = $

a
 $7$
b
 $5$
c
 $3$
d
 $25$
e
 $15$
Practice: Evaluating an Expression With Two Mixed Operations

$9 - 4 \times 2 = $

a
 $1$
b
 $7$
c
 $11$
d
 $6$
e
 $10$
Example: Evaluating an Expression With Three Mixed Operations

Calculate 4 \times 5 + 6 \div 3.

EXPLANATION

To evaluate the given expression, we follow the order of operations.

First, we multiply and divide, from left to right:

\begin{align} 4 \times 5 + 6 \div 3 &=\\[5pt] {\color{blue}4 \times 5} + 6 \div 3 &= \\[5pt] {\color{blue}20} + 6 \div 3 &=\\[5pt] {\color{blue}20} + {\color{red} 6 \div 3}&=\\[5pt] {\color{blue}20} + {\color{red} 2}& \end{align}

Now, we add:

\begin{align} {\color{blue}20} + {\color{red} 2} &= 22 \end{align}

FLAG
Practice: Evaluating an Expression With Three Mixed Operations

$20\div 4 \times 1\times 5=$

a
 $10$
b
 $25$
c
 $30$
d
 $20$
e
 $15$
Practice: Evaluating an Expression With Three Mixed Operations

$3\times 4 - 6\div 2=$

a
 $8$
b
 $7$
c
 $10$
d
 $9$
e
 $12$
Practice: Evaluating an Expression With Three Mixed Operations

$20 - 10 \div 5 + 2 = $

a
 $8$
b
 $4$
c
 $20$
d
 $15$
e
 $19$
Flag Content
Did you notice an error, or do you simply believe that something could be improved? Please explain below.
SUBMIT
CANCEL