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Order of Operations - BODMAS

Operations

"Operations" mean things like add, subtract, multiply, divide, squaring, etc. If it isn't a number it is probably an operation.

But, when you see something like...

7 + (6 × 52 + 3)

... what part should you calculate first?

Start at the left and go to the right?
Or go from right to left?

Calculate them in the wrong order, and you will get a wrong answer !

So, long ago people agreed to always follow certain rules when doing calculations, and they are:

Order of Operations

Do things in Brackets First. Example:

yes   6 × (5 + 3) = 6 × 8 =
48
 
no   6 × (5 + 3) = 30 + 3 =
33
(wrong)

Exponents (Powers, Roots) before Multiply, Divide, Add or Subtract. Example:

yes   5 × 22 = 5 × 4 =
20
 
no   5 × 22 = 102 =
100
(wrong)

Multiply or Divide before you Add or Subtract. Example:

yes   2 + 5 × 3 = 2 + 15 =
17
 
no   2 + 5 × 3 = 7 × 3 =
21
(wrong)

Otherwise just go left to right. Example:

yes   30 ÷ 5 × 3 = 6 × 3 =
18
 
no   30 ÷ 5 × 3 = 30 ÷ 15 =
2
(wrong)

How Do I Remember ? BODMAS !

 
B
Brackets first
O
Orders (ie Powers and Square Roots, etc.)
DM
Division and Multiplication (left-to-right)
AS
Addition and Subtraction (left-to-right)

The only strange name is "Orders". "Powers" would have been a better word, but who could remember "BPDMAS" ... ? "Exponents" is used in Canada, and so you might prefer "BEDMAS", there is also "Indices" so that makes it "BIDMAS". In the US they say "Parenthesis" instead of Brackets, so they say "PEMDAS"

Note: Divide and Multiply rank equally. Add and Subtract rank equally.

 

After you have done "B" and "O", just go from left to right doing any "D" or "M" as you find them.

Then go from left to right doing any "A" or "S" as you find them.

Examples

Example: How do you work out 3 + 6 × 2 ?

Multiplication before Addition:

First 6 × 2 = 12, then 3 + 12 = 15


Example: How do you work out (3 + 6) × 2 ?

Brackets first:

First (3 + 6) = 9, then 9 × 2 = 18


Example: How do you work out 12 / 6 × 3 ?

Division and Multiplication rank equally, so just go left to right:

First 12 / 6 = 2, then 2 × 3 = 6


Oh, yes, and what about 7 + (6 × 52 + 3) ?

7 + (6 × 52 + 3)  
7 + (6 × 25 + 3) Start inside Brackets, and then use "Orders" First
7 + (150 + 3) Then Multiply
7 + (153) Then Add
7 + 153 Brackets completed, last operation is add
160 DONE !