1.04 Order of operations with whole numbers
The order of operations is the way we understand and remember how to evaluate expressions involving two or more operations (add, subtract, multiply, divide).
This convention is used so that everyone can agree about how to write and perform mathematics.
The order is as follows:
- Evaluate whatever is contained in brackets.
- Evaluate any multiplication or division, reading from left to right.
- Evaluate any addition or subtraction, reading from left to right.
There's no standout reason why this should be the order but it is the one everybody uses. Before it becomes natural, this is something you will have to learn by heart. Good quality calculators are programmed to this automatically!
Worked example
Find the value of $34-8\times7\div2$34−8×7÷2.
Think: What operations are used in the expression? Are there any brackets?
Do: Referring back to the order of operations, we see that brackets are performed first. There are no brackets so we can ignore this step.
Multiplication and division come after, both are in this expression so we evaluate each one from left to right
| $34-8\times7\div2$34−8×7÷2 | $=$= | $34-56\div2$34−56÷2 | (evaluate the product $8\times7$8×7) |
| $=$= | $34-28$34−28 | (evaluate the quotient $56\div2$56÷2) |
We are left with one final operation. Evaluating it we find $34-8\times7\div2=34-28$34−8×7÷2=34−28$=$=$6$6
Practice questions
Question 1
Consider the expression $6+16\div4$6+16÷4.
Which operation should we perform first?
Add $6$6 and $16$16.
ADivide $16$16 by $4$4.
BNow, evaluate $6+16\div4$6+16÷4 by following the order of operations.
Question 2
Consider the expression $4+7\times3$4+7×3.
Which two of the following expressions give the same value as the given expression?
$7\times3+4$7×3+4
A$3\times4+7$3×4+7
B$\left(4+7\right)\times3$(4+7)×3
C$4+\left(7\times3\right)$4+(7×3)
DHow do the brackets in $\left(4+7\right)\times3$(4+7)×3 change the order of operations from the original expression $4+7\times3$4+7×3?
They don't do anything.
AThey cause the addition to be evaluated before the multiplication.
BThey cause the multiplication to be evaluated before the addition.
C
Question 3
Consider the expression $4+18-5\times2$4+18−5×2.
Which of the following is the value of the given expression?
$12$12
A$30$30
B$34$34
CWithout changing the order of the numbers and operations, write the expression which would evaluate to $30$30.
Without changing the order of the numbers and operations, write the expression which would evaluate to $34$34.