1.2 Order of operations for whole numbers

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In a computation in which more than one operation is involved, grouping symbols indicate which operation to perform first.

Grouping symbols

Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations are to be grouped together and considered as one number. The grouping symbols commonly used in mathematics are the following:

Parentheses:  (   )

Brackets:     [   ]

Braces:       {   }

Bar:           —

In a computation in which more than one operation is involved, grouping symbols indicate which operation to perform first. If possible, we perform operations inside grouping symbols first.

For example:

(5 · 5) + 20 = 45

whereas:

5 · (5 + 20) = 125

If there are no parentheses, you should always do multiplications and divisions first followed by additions and subtractions. You can always put your own parentheses into equations using this rule to make things easier for yourself, for example:

$\begin{array}{}a×b+c÷d=\left(a×b\right)+\left(c÷d\right)\\ 5×5+\text{20}÷4=\left(5×5\right)+\left(\text{20}÷4\right)\end{array}$

Grouping symbols examples

If possible, determine the value of each of the following.

Example 1

9 + (3 · 8)

Since 3 and 8 are within parentheses, they are to be combined first:

= $9+\text{24}$

= $\text{33}$

Thus, 9 + (3 · 8) = 33.

Example 2

(10 ÷ 0) · 6

Since (10 ÷ 0) is undefined, this operation is meaningless, and we attach no value to it. We write, “meaningless.”

Grouping symbols exercises

If possible, determine the value of each of the following.

16 – (3 · 2)

10

5 + (7 · 9)

61

(4 + 8) · 2

24

28 ÷ (18 – 11)

4

(33 ÷ 3) – 11

0

4 + (0 ÷ 0)

meaningless

Multiple grouping symbols

When a set of grouping symbols occurs inside another set of grouping symbols, we perform the operations within the innermost set first.

Multiple grouping symbol examples

Determine the value of each of the following.

2 + (8 · 3) – (5 + 6)

Combine 8 and 3 first, then combine 5 and 6.

= 2 + 24 – 11

Now combine left to right.

= 26 –11

= 15

$\text{10}+\left[\text{30}-\left(2\cdot 9\right)\right]$

Combine 2 and 9 since they occur in the innermost set of parentheses.

= $\text{10}+\left[\text{30}-\text{18}\right]$

Now combine 30 and 18.

= 10 + 12

= 22

Distributivity

If you see a multiplication outside parentheses like this:

$\begin{array}{}a\left(b+c\right)\\ 3\left(4-3\right)\end{array}$

then it means you have to multiply each part inside the parentheses by the number outside:

$\begin{array}{}a\left(b+c\right)=\text{ab}+\text{ac}\\ 3\left(4-3\right)=3×4-3×3=\text{12}-9=3\end{array}$

Sometimes you can simplify everything inside the parentheses into a single term. In fact, in the above example, it would have been smarter to have done this:

$3\left(4-3\right)=3×\left(1\right)=3$

This can happen with letters too:

$3\left(4a-3a\right)=3×\left(a\right)=3a$

The fact that $a\left(b+c\right)=\text{ab}+\text{ac}$ is know as the distributive property.

If there are two sets of parentheses multiplied by each other, then you can do it one step at a time:

$\begin{array}{}\left(a+b\right)\left(c+d\right)=a\left(c+d\right)+b\left(c+d\right)\\ =\text{ac}+\text{ad}+\text{bc}+\text{bd}\\ \left(a+3\right)\left(4+d\right)=a\left(4+d\right)+3\left(4+d\right)\\ =4a+\text{ad}+\text{12}+3d\end{array}$

Multiple grouping symbol exercises

Determine the value of each of the following:

$\left(\text{17}+8\right)+\left(9+\text{20}\right)$

54

$\left(\text{55}-6\right)+\left(\text{13}\cdot 2\right)$

23

$\text{23}+\left(\text{12}÷4\right)+\left(\text{11}\cdot 2\right)$

48

$\text{86}+\left[\text{14}+\left(\text{10}-8\right)\right]$

102

$\text{31}+\left(9+\left[1+\left(\text{35}-2\right)\right]\right)$

74

{6 – [24 ÷ (4 · 2)]} 3

9

Order of operations

Sometimes there are no grouping symbols indicating which operations to perform first. For example, suppose we wish to find the value of $\text{3}+\text{5}\cdot \text{2}$ . We could do either of two things:

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