Order of Operations
Look at both of the problems. Notice the difference in the way they are solved.
48  6  2
48  6  2 The only difference in the way these problems were
done is the order the operations were performed.
 8 2
 16
 48  12
The one on the left is correct.
4
Multiplication and division are always done left to right.
Remember multiplication is commutative and associative, but division is not. You can do
problems that contain only multiplication in any order, but if division is in the problem,
then the order is important.
81  3  5
 27  5
 135
Practice:
a) 5÷10x200=
b) 4.5x2÷5=
c) 4x3÷5x24=
4  21  2
 84  2
 42
144  6  2  3  4
 24  2  3  4
 48  3  4
 16  4  64
45÷3x2=
8÷4x7=
6÷3x10÷5x9=
510÷5÷12=
8÷4x7=
42x3÷9=
2÷5x8=
2x5÷8=
2x5x8=
Addition and subtraction work the same way. Subtraction isn’t commutative. Remember
to think “add the opposite” when subtracting, but do it left to right.
Practice:
d) 9+3-5=
e) 534-83+29=
9-3+5=
45-3+2=
3.4 - 1.2 - 0.65=
5.34 - 0.24 +
4.999=
12-6-3+7=
5.34 + 0.24 4.999=
Multiplication and division are always done before addition and subtraction. Write each
step out completely under the previous step.
We use three ways to indicate multiplication. 3x4, 3(4) and 3·4
5  5  2  8  Multiplication before 8  5  7 8 
subtraction.
25  16 
40  56 
Note: Write the new
problem after
9
96
multiplying.
Practice:
f) 45+3x2=
20-6÷3x9=
2.3÷4+5=
 
all mean multiplication.
8  4 6   3  5
2 6  3  5
12  3  5
9  5  14
5÷8 - 0.003=
g) 4·5+3·2
4÷5-3x2=
12÷4+5(2)=
8 - 2.3÷4 + 5=
There is a mistake in each of the following problems. Discover what was done
incorrectly.
h)
12  4  2 
8  2  16
9  12  3 
9  36 
4 is
correct.
1
15  3  5 
15  8 
7
2.25 is
4
correct.
Multiply before subtraction
Exponents are done before multiplication and division.
9  23  7 2 
43  2  32  2  5  Note in the
examples, the
9  8  49 
64  2  9  2  5  exponents are done
and the rest of the
72  49 
32  18  5
problem is written
23
14  5  19
down.
If you scratch to
the side and skip
writing all steps,
you will make
mistakes.
Practice:
i) 52=
j) 1.22=
23=
0.52=
k) 23 + 52=
17 is
correct.
An exponent is a way to
show repeated
multiplication.
34 means 3x3x3x3=81
52 means 5x5=25
25 means 2x2x2x2x2=32
32=
33=
25=
92=
3.43=
0.052=
0.43=
1.52=
25 - 42=
54 - 33=
l) 42 + 5 x 23=
52 + 3 x 22 =
23 - 6 ÷ 3 x 32=
m) 52(2) - 3 x 23=
152 - 3 x 52 =
The first thing always done is parenthesis or other grouping symbols.
If there are nested grouping symbols, work from the inside out.
43=
53=
1.53=
6.122=
23 x 34 =
50 ÷ 2 + 5 x 22=


4  2  71  3 14   
4  2 71  3  8  6  
Work the inside parenthesis first.
4  2  71  42  
The next set of parenthesis has two operations inside.
Always do multiplication before subtraction.
Finish the inside of the second parenthesis.
4  2  29  
4  58  62
Any symbol
65  8  7

23  5
65  56

85
9
3
3
Multiply before subtracting.
that separates the problem into parts acts like a parenthesis.
The division bar groups the operations.
In the numerator the multiplication is done before subtraction.
In the denominator the exponent is done first.
Finally the division is done.
To remember the order of operations use the mnemonic devise Please Excuse My
Dear Aunt Sally.
Please
Parenthesis and other grouping
34  5= The subtraction is done
first
symbols. ( ), { }, [ ], ,
3(-1)=-3
because it is in the
parenthesis.
Excuse
Exponents
144 - 43=
x2
The exponent is done before the
subtraction.
144 - 64=80
My Dear
Multiply and Divide from left to
4  6  8  24  8  3
right. 3x5, 3 5 and 3(5) all mean 72  3  4  24  4  96
multiply.
12  4 ,
12
, and 4 12 are all the
4
same division.
Add and Subtract from left to
right.
Aunt Sally
7 - 5+8=2+8=10
Practice: If necessary round each answer to the nearest thousandth.
a) 25 - 8(3+2)=
5 + 8( 3+2)=
30÷6x18=
b) 18 - 8(4 - 2)=
81 ÷ 6+3(7 - 4)=
3(3) + 9(7-5)=
c) 7 – 3(2) + 5(14-3)=
12 - 10 + 89 – 72=
3.4 – 1.7 + 0.9 + 7.2
d) 2(3.14)(5)2 +
2(3.14)(5)(7)
0.2 +0.5(5 - 0.6)=
5 - 0.34(4 + 2)=
7  4
3
=
50  6  3
3
e)
f)
g)
30   60  56 
=
8
14  9
5 82  32 
=
9
16  12 
5
=
25  6(3)
4
4  32  6
 8(2  5) =
5
82   5
2
62  52
=
7  18  3  5 
9(28)
 32 =
5
=
5  38  32 
=
9
h)
9(8)
 32 =
5
0.3 + 0.81(8.1 - 2.435)=
7.1(0.5) - 3÷1.2=