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?

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

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

Order of Operations

Do things in Brackets First

yes   6 × (5 + 3) = 6 × 8 =
48
 
not   6 × (5 + 3) = 30 + 3 =
33
(wrong)

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

yes   5 × 22 = 5 × 4 =
20
 
not   5 × 22 = 102 =
100
(wrong)

Multiply or Divide before you Add or Subtract

yes   2 + 5 × 3 = 2 + 15 =
17
 
not   2 + 5 × 3 = 7 × 3 =
21
(wrong)

Otherwise just go left to right

yes   30 ÷ 5 × 3 = 6 × 3 =
18
 
not   30 ÷ 5 × 3 = 30 ÷ 15 =
2
(wrong)

How Do I Remember It All ... ? BODMAS !


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

Divide and Multiply rank equally (and go left to right).

Add and Subtract rank equally (and go left to right)

So do it this way:

bodmas

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.


Note: the only strange name is "Orders". "Exponents" is used in Canada, and so you might prefer "BEDMAS". There is also "Indices" which makes it "BIDMAS". In the US they say "Parentheses" instead of Brackets, so it is "PEMDAS"

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 / 2 ?

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

First 12 / 6 = 2, then 2 × 3 = 6, then 6 / 2 = 3

A practical example:

ball throw

Example: Sam threw a ball straight up at 20 meters per second, how far did it go in 2 seconds?

Sam uses this special formula that includes gravity:

height = velocity × time − (1/2) × 9.8 × time2

Sam puts in the velocity of 20 meters per second and time of 2 seconds:

height = 20 × 2 − (1/2) × 9.8 × 22

Now for the calculations!

20 × 2 − (1/2) × 9.8 × 22    
20 × 2 − 0.5 × 9.8 × 22   Start inside Brackets
20 × 2 − 0.5 × 9.8 × 4   Then Orders (22=4)
40 − 19.6   Then the Multiplies
20.4   Subtract and DONE !

The ball reaches 20.4 meters after 2 seconds

Exponents of Exponents ...

What about this example?

432

Exponents are special: they go top-down (do the exponent at the top first). So we calculate this way:

Start with:   432
32 = 3×3:   49
49 = 4×4×4×4×4×4×4×4×4:   262144

 

 

And finally, what about the example from the beginning?

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 !
 
Order of Operations Worksheets