Order of operations II
In life, the order in which we do things is important. For example, we put on socks then shoes, rather than shoes and then socks.
The same goes for solving maths problems with more than one operation. An operation is a mathematical process, such as addition, subtraction, multiplication or division. Other operations include raising a number to a power and taking a root of a number. An ‘operator’ is a symbol that indicates the type of operation, such as +, –, × or ÷.
There are a number of conventions (rules) which need to be followed in order to solve these problems correctly. The order goes:
Step 1: Do operations inside grouping symbols such as parentheses (...), brackets [...] and braces {...}.
Step 2: Do multiplication (including powers) and division (including roots) going from left to right.
Step 3: Do addition and subtraction going from left to right.
Examples
question 1
Evaluate: $5\times\left(6+6\right)$5×(6+6)
Think: Remember the order of operations.
Firstly, we'll solve what's inside the parentheses.
Secondly, we solve the multiplication.
There's no addition or subtraction in this problem.
Do:
| $5\times\left(6+6\right)$5×(6+6) | $=$= | $5\times12$5×12 |
| $=$= | $60$60 |
The same or applies no matter how many numbers there are.
question 2
Evaluate: $100-9\times6+18\div6$100−9×6+18÷6
Think: There are no brackets in this question, so firstly we'll solve the multiplication and division (going from left to right), then we will solve the addition and subtraction (going from left to right)
Do:
| $100-9\times6+18\div6$100−9×6+18÷6 | $=$= | $100-54+3$100−54+3 |
| $=$= | $49$49 |
question 3
Evaluate: $\left(6\times8+\left(30-22\right)\right)\div8$(6×8+(30−22))÷8
Think: Firstly, we'll solve what's inside the parentheses.
Secondly, we solve the multiplication and division going from left to right.
Then we'll do the addition and subtraction.
Do:
| $\left(6\times8+\left(30-22\right)\right)\div8$(6×8+(30−22))÷8 | $=$= | $\left(6\times8+8\right)\div8$(6×8+8)÷8 |
| $=$= | $\left(48+8\right)\div8$(48+8)÷8 | |
| $=$= | $56\div8$56÷8 | |
| $=$= | $7$7 |
Question 4
Evaluate: $\left(48\div12+35\div5\right)\times3^2$(48÷12+35÷5)×32
Think: We need to simplify the problem by using our order of operation rules.
Firstly we solve what's inside the parentheses- division first and then addition.
Then we solve any other multiplication or division, working from left to right.
Then we solve any addition or subtraction.
Do:
| $\left(48\div12+35\div5\right)\times3^2$(48÷12+35÷5)×32 | $=$= | $\left(4+7\right)\times9$(4+7)×9 |
| $=$= | $11\times9$11×9 | |
| $=$= | $99$99 |
You can add in other steps of working if you need. That's just the way I did it.
Here's another example.
question 5
Evaluate: $48-6\times\left(8-4\right)+12$48−6×(8−4)+12
Think: $6\times\left(8-4\right)=6\times4$6×(8−4)=6×4 $=$= $24$24
Do:
| $48-6\times\left(8-4\right)+12$48−6×(8−4)+12 | $=$= | $48-24+12$48−24+12 |
| $=$= | $36$36 |
Question 6
Evaluate $2\times12+30-28$2×12+30−28
Question 7
Evaluate $-18+21\div7$−18+21÷7