
Why do we need it?  How
to use brackets  The basic rules  The
complete rules  Using calculators  Quick
quiz 
Why
do we need an order of operations?
Example:
In a room there are 2 teacher's chairs and 3 tables each with 4 chairs
for the students. How many chairs are in the room?
We
know there are 14, but how do we write this calculation? If we just
write
2 + 3 x 4
how does a reader know whether the answer is
2
+ 3 = 5, then multiply by 4 to get 20 or
3
x 4 = 12, then 2 + 12 to get 14?
There are two steps needed to find the answer; addition and multiplication.
Without an agreed upon order of when we perform each of these operations
to calculate a written expression, we could get two different answers.
If we want to all get the same "correct" answer when we
only have the written expression to guide us, it is important that
we all interpret the expression the same way.
One
way of explaining the order is to use brackets. This always works.
To say that the 3 x 4 is done before the adding, we would use brackets
like this:
2
+ (3 x 4)
The
brackets show us that 3 x 4 needs to be worked out first and then
added to 2. However, we can also agree on an order of operations,
which is explained below.
Another
example: Calculate 15 10 ÷ 5
If
you do the subtraction first, you will get 1. If you do the division
first, which is actually correct according to the rules explained
below, you will get 13. We need an agreed order.
division
first (correct) 
subtraction
first 
blue
indicates the operations being worked on first 
15
 10 ÷ 5 
15
 10 ÷ 5 
=
15  10 ÷ 5 
=
15  10 ÷ 5 
=
15  2 = 13 
=
5 ÷ 5 = 1 
How
to use brackets
Brackets
are marks of inclusion which tell us which parts of an expression
go together. We use brackets in an expression to indicate which part
to calculate first. It can be useful to think of brackets as a circle
with the top and bottom deleted to remind you that brackets indicate
that everything inside the 'circle' is selfcontained and must be
worked out first. Although brackets usually look like ( ), brackets
can also look like { } or [ ] and need to be treated in the same way.
Brackets are sometimes referred to as "parentheses".
want
division first 
want
subtraction first 
blue
indicates the operations being worked on first 
15
 (10 ÷ 5) 
(15
 10) ÷ 5 
=
15  (10 ÷ 5) 
=
(15  10) ÷ 5 
=
15  2 = 13 
=
5 ÷ 5 = 1 
There
are more examples on how to use brackets in complicated examples below.
If
we used brackets consistently we would not have to be concerned with
the order of operations. We could just work from innermost brackets
outwards to eventually get our answer. However using lots of brackets
can become tedious and confusing, as in the following example, so
we need some agreed rules.
3
+ ((4÷2)x7)(6÷3)((4x2)+((8÷2)+(3x3)))

You
can check how to work out this monster by clicking here,
but the next section tells you how to avoid the worst monsters.
YOU
CAN ALWAYS USE BRACKETS TO SHOW HOW
A
CALCULATION SHOULD BE DONE
The basic rules
Many
years ago mathematicians decided on an 'order of operations' that
everyone should use when performing mathematical computations from
written instructions. This means that when presented by the same problem
everyone using this agreed convention of order of operations would
obtain the same answer. You could think of the order of operations
as a sort of 'maths grammar' which enables mathematicians to communicate
with each other and with machines all over the world.
It
is important to realise that the order of operations has nothing to
do with underlying mathematical principles: it is just convention.
Other rules could have been invented. However the convention needs
to be understood before it can be successfully applied to every problem.
The
four rules below are enough for most purposes:
• 
RULE
1: Calculate anything in brackets first, then apply the other
rules. (For further discussion about expressions with more than
one set of brackets, see the next section.)

• 
RULE
2: If a calculation involves only addition and subtraction, work
from left to right. 
• 
RULE
3: If a calculation involves only multiplication and division,
work from left to right.

• 
RULE
4: Do multiplication and division before addition and subtraction. 
Example
of Rule 2: 10  3 + 2
This involves only addition and subtraction, so we work from left
to right. 10  3 + 2 is equal to 9 because we calculate 10  3 first,
then add 2. We do NOT do 3+2 first, then subtract from 10.
Example
of Rule 3: 48 ÷ 2 x 3
This involves only multiplication and division, so we work from left
to right. 48÷ 2 x 3 is equal to 72, because 48 ÷
2 = 24 and 24 x 3 = 72. We
do NOT work out 2 x 3 = 6 and then do 48 ÷ 6 = 8.
Example
of Rule 4: 4 + 5 x 3
Multiplication has precedence over addition. 4 + 5
x 3 is equal to 19 because 5 x 3 = 15 and 4 + 15 is 19. We do NOT
work out 4 + 5 first to get 9 and then multiply by 3.
Example
of Rule 1: (4 + 5) x 3
(4
+ 5) x 3 is equal to 27, because we calculate the brackets first to
4 + 5 = 9 and then multiply by 3. We do NOT work out 5 x 3 and then
add 4.
Examples
using all of the rules together:
Example:
72 + 4 x 6 ÷ 2  8
Working
out 
Thinking
text 
72
+ 4 x 6 ÷ 2  8

We
have addition, multiplication, division and subtraction
in this expression. 
=

72
+ 
4

x
6

÷
2


8


Using
RULE 4 
=

72
+ 
4

x
6

÷
2


8


Using
RULE 3 
=

72
+ 
24

÷
2


8


Using
RULE 3 
=

72
+ 
12


8


Using
RULE 2 
=


84


8


Using
RULE 2 
=

76




Example:
15  12 ÷ (6 ÷ 2) x 4
Working
out 
Thinking
text 
15 

12
÷ (6 ÷ 2) 
x
4 
+ 3 
We
can see that we have brackets, multiplication and division
and addition and subtraction in this expression. 
=
15  
12
÷ (6 ÷ 2)

x
4 
+
3 
First,
the brackets. (Rule 1) 
=
15  
12
÷ 3

x
4 
+
3 
Then
division and multiplication, working from left to right.
(Rules 4 and 3) 
=
15  
4 
x
4 
+
3 
=
15  
16 

+
3 
Then
addition and subtraction, working from left to right.
(Rule 2) 
=
 1 


+
3 
=
2 





The complete rules
BODMAS,
BOMDAS, BEMDAS, BIDMAS etc..
Many
of us were taught to use the BODMAS or BOMDAS mnemonics or other variations
to determine the order of operations. They summarise the rules above:
brackets
first,
then multiplication or division (left to right)
then addition or subtraction(left to right).

Brackets 
Either BODMAS or BOMDAS must be interpreted as implying
the same order of operations. 
then 
Of 
or 
Multiplication 
or 
Division 
then 
Addition 
or

Subtraction 

However
blind adherence to these mnemonics (memory aides) without understanding
of the mathematical ideas they represent will
lead to misunderstandings and incorrect usage, particularly when they
are applied to more complicated expressions.
B O D M
A S  the "B"
"B"
comes first, so in evaluating an expression, do the brackets first.
We have talked about how and why brackets are used in the section
above, How to use brackets. But what if there
are several brackets?
Rule
for multiple brackets: If there are brackets in the expression,
calculate them first. If there is more than one set of brackets
then begin with the innermost brackets and work outwards. If there
is more than one set of brackets but they are isolated from each
other, then do them independently. 
Example:
Expressions with multiple brackets.
3 x ((2+(3x4)) + (5(8÷4)  9))
Working
Out 
Thinking 

We
work on the innermost brackets first. Here there are 2 isolated
sets of inner brackets.
Then we move to the next level of brackets and so on. 
B O M D
A S  the "O"
The
'O' in BODMAS stands for 'of', which is a verbal indication of multiplication.
It is really included as a convenient vowel for the mnemonic to work
as a word.
Did
you know that in other versions of the memory aide, such as
BIDMAS and BEDMAS, the 'O' has been replaced by 'I' for indicies
or 'E' for exponents respectively. This is useful as it extends
the mnemonic to expressions which involve squares etc. See below.

B O M D A S or B O D M A S  the "M"
and the "D"
Common
misconception 1: BOMDAS tells me to do multiplication before division.
Common
misconception 2: BODMAS tells me to do division before multiplication.
Actual
rule: Multiplication and division are inverse operations and as
such need to be treated equally. When confronted with multiplication
and division, always work from left to right. 
Example:
105 ÷ 3 x 5
105 ÷ 3 x 5
is equal to 175 because we work out 105 ÷ 3 = 35 first and
then multiply by 5. We do NOT work out 3 x 5 = 15 first and then divide
105 by 15. This would give us an incorrect answer of 7. BOMDAS
and BODMAS give the same answer, correctly interpreted.
B O M D A S or BODMAS  the "A" and
the "S"
Common
misconception 3: BOMDAS or BODMAS tells me to do addition before subtraction.
Common
misconception 4: It doesn't matter what order you do addition or subtraction.
Actual
rule: Addition and subtraction are inverse operations and as such
need to be treated equally. When confronted with addition and
subtraction, always work from left to right. 
Example:
3 + 7  4  9
3 + 7  4  9 is equal to  3 because we work out the addition first,
3 + 7 = 10, and 10  4 = 6, then 6  9 =  3.
We
can see that if we did NOT work from left to right and worked out
4  9 =  5 first, and then subtracted this from 3 + 7 then,
3 + 7  ( 5) = 10 + 5 = 15
B I D M
A S OR B E D M A S  the "I" or "E"
Powers, fractions
and roots
Powers
(also known as exponents or indicies), fractions and
roots are not covered by BODMAS or BOMDAS but we still need to know
how to handle them. Fractions, powers, roots and other self contained
parts of expressions should be treated as if they are in brackets,
i.e. work them out first.
Example:
4 + 2 ^{3 }x 6
We treat 2
^{3 } as if it is in brackets and work this out first. 2
^{3} is equal to
8. Then we continue with, 4 + 8 x 6 which is equal to 52 because we
do the multiplication first and then the addition.
Example:
(2 + 3) ^{2}
In
this expression the brackets around the addition of 2 + 3 indicate
that it is 2 + 3 that is raised to the power of 2, NOT just 3. We
must work out the brackets first and then square the answer. 5 ^{2}
= 10
Example:
Actual
rule: Fractions should be treated as if the numerator is in brackets
and the denominator is in brackets and the fraction bar (the "vinculum")
is division. 
Example:
We
work out the square root first to get 3 and then do the division and
multiplication working from left to right. 12 ÷ 3 x 2 is equal
to 8.
We cannot do anything with until
it has been simplified. Also, you will notice once again that if we
do the multiplication before the division then we will get a different
answer.
What
to remember:
Work
on one level at a time, starting at the top and going down.
Within
each level, work from left to right.
Brackets





powers,
roots and fractions





multiplication
and division





addition
and subtraction

Activities
Four
fours
Using
four fours and any mathematical operations and signs you wish, can
you make every number from 1 to 20. Can you make every number up to
100?
For
example, (4 +4) x 4  4 = 28 and 4 + (4 x 4)  4 = 16.
This
is an excellent activity for a class to do over a week. Make a large
chart with a space for one or more expressions for each number. Students
can enter their expressions on the class chart after they have been
checked. The teacher can decide what signs are allowed.
Manipulating
expressions
6
+ 17  15 x 4 ÷ 3
By
inserting brackets into this expression (as many as you like, wherever
you like) make expressions with as many answers as you can.
The
correct answer when there are no brackets is
6 + 17  15 x 4 ÷ 3 = 3.
This
set of inserted brackets changes the answer to 8.66:
6
+ (17  15) x 4 ÷ 3 = 6 + 2 x 4 ÷ 3 = 6 + 8 ÷
3 = 6 + 2.66 = 8.66
Using
calculators
Not
all calculators have correct order of operations built in. More sophisticated
calculators have programmed logic which enables them to use the standard
mathematical conventions. Others just process the information/keystrokes
exactly as they are entered.
Example:
If you need to calculate 1 + 5 x 7 and enter these 6 key presses:
some
calculators give 42 (1 + 5 gives 6, multiply by 7 gives 42) and others
give 36 (multiply first so 5 x 7 = 35, add 1 + 35 giving 36).
The second is the correct answer for the expression.
Find
out how your calculator works and check to see if it has brackets
to help be precise. Learn how to use the memory to keep intermediate
answers.
Quick quiz
1. 
Using
the example 10  1  2 , show why you need to follow the correct
order of operations. 


2. 
Calculate
the following expressions:

(a) 
11
x (3 + 2) x 4 ÷ 2 
(b) 
7
 18 ÷ 2 x 3 + 5 
(c) 
42
÷ 3 x 7 


3.

Calculate
9 + 4 ÷ 2 x 7  6 ÷ 3  4 x 2 + 8 ÷ 2 + 3
x 3 


4. 
Using
the expression in question 3, make 3 alternative expressions and
answers by inserting brackets. 


5. 
Find
the answer to,
showing the method you have used to ensure you follow the correct
order of operations, e.g. checklist, colour scheme, arrows etc.



6. 
Calculate
the following expressions:

(a) 
32
÷ 4^{2 }x (3  8) 
(b) 
81
÷ (4  7)^{3} 
(c) 

(d) 

(e) 



7. 
Find
out how to use your calculator to evaluate the expressions in
question 6. 


8. 
Bernie
is in the process of landscaping the gardens of two new townhouses.
If he buys 30 bundles of 12 wooden planks for the fence for each
house and 15 bundles of 10 hardwood planks for the decking for
each house, write an expression for the total number of planks
bought and then work it out. If Bernie then returned 2 bundles
of the wooden fence planks but bought 5 extra bundles of the hardwood
planks, write a new expression and then work out the answer. 


9. 
Two
thirds of all Year 8 students, one quarter of all Year 9 students,
only 30 Year 10 students and two fifths of Year 11 and 12 students
combined ride their bike to school. If there are 99 Year 8 students,
124 Year 9 students, 111 Year 10 students, 65 Year 11 students
and 50
Year 12 students attending the school, how many students ride
their bike to school.
Write a mathematical expression for the number of students who
ride to school and then find the answer.



10. 
(300
÷ (10 x 2)) x 4. Create an appropriate worded problem
from this mathematical expression.

To view
the quiz answers, click here.
Monster multiple brackets example!
3
+ ((4÷2)x7)  (6÷3)  ((4x2) + ((8÷2) + (3x3)))


=

3 + ((4÷2)x7)  (6÷3)  ((4x2) + ((8÷2) +
(3x3))) 

=

3
+ ((2)x7))  (2)  ((8) + ((4) + (9)) 

=

3
+ (14)  (2)  (8 + (13)) 

=

3
+ (14)  (2)  (21) 

=

3
+ 14  2  21 

=

17
 2  21 

=

15
 21 

=

6 

