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
The one on the left is correct.
48 12 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
45÷3x2= 510÷5÷12= 2÷5x8=
144 6 2 3 4 24 2 3 4 48 3 4 16 4 64
8÷4x7= 8÷4x7= 2x5÷8= 6÷3x10÷5x9= 42x3÷9= 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= 3.4 - 1.2 - 0.65= 45-3+2= 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
4 is
correct.
9 12 3 9 36
1 4
2.25 is
15 3 5 17 is 15 8 7
Multiply before subtraction
correct.
correct.
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.
An exponent is a way to show repeated multiplication. 34 means 3x3x3x3=81 52 means 5x5=25 25 means 2x2x2x2x2=32
Practice: i) 52= j) 1.22= k) 23 + 52= l) 42 + 5 x 23=
23= 0.52=
32= 3.43=
33= 0.052=
25= 0.43=
92= 1.52=
43= 1.53=
53= 6.122= 23 x 34 =
25 - 42= 52 + 3 x 22 = 152 - 3 x 52 =
54 - 33= 23 - 6 ÷ 3 x 32=
50 ÷ 2 + 5 x 22=
m) 52(2) - 3 x 23=
The first thing always done is parenthesis or other grouping symbols. If there are nested grouping symbols, work from the inside out.
4 2 71 3 14
4 2 71 42 4 2 29 4 58 62
4 2 71 3 8 6
Work the inside parenthesis first. The next set of parenthesis has two operations inside. Always do multiplication before subtraction. Finish the inside of the second parenthesis. Multiply before subtracting.
Any symbol 65 8 7 23 5 65 56 85 9 3 3
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 34 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 ,
Aunt Sally
same division. Add and Subtract from left to right.
12 , and 4 12 are all the 4
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)= c) 7 – 3(2) + 5(14-3)= d) 2(3.14)(5)2 + 2(3.14)(5)(7) 81 ÷ 6+3(7 - 4)= 12 - 10 + 89 – 72= 0.2 +0.5(5 - 0.6)= 3(3) + 9(7-5)= 3.4 – 1.7 + 0.9 + 7.2 5 - 0.34(4 + 2)=
e)
7 4
3 = 50 6 3
3
16 12
5 = 25 6(3)
4
4 32 6 8(2 5) = 5
f)
30 60 56 8 = 14 9
6 2 52 = 7 18 3 5
9(28) 32 = 5
82 5
2
=
g)
5 82 32 = 9
5 38 32 = 9
h)
9(8) 32 = 5
0.3 + 0.81(8.1 - 2.435)=
7.1(0.5) - 3÷1.2=