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Order of Operations (Mathematics Lesson)

What Is The Order of Operations?

The order of operations gives the order in which operations (such as add, subtract, multiply, divide) should be performed in.

For example:



Here there is a subtraction and a multiplication. There will be a different answer to the expression depending on what order the operations are performed.

To decide what order, we use the order of operations. The order is given by BODMAS.

What Is BODMAS?

BODMAS is an acronym, giving the order that operations should be performed in. It stands for:



The order of operations is read from top to bottom.

Note: the operations with a curly bracket '{' are on the same level, and should be performed left to right.

Step 1
Brackets ( ).
Evaluate expressions within brackets.

Step 2
Orders.
Evaluate exponents and roots. For example, 22, 53, or √4.

Step 3
Division ÷, Multiplication ×.
Perform divisions and multiplications, reading left to right.

Step 4
Addtion +, Subtraction -.
Perform additions and subtractions, reading left to right.

A Real Example of Using Order of Operations

Question: Use the order of operations to evaluate the expression below.


Step 1
Brackets ( ).
Evaluate expressions within brackets.
8 × 2 = 16.



Step 2
Orders.
Evaluate exponents and roots.
32 = 9.



Step 3
Division ÷, Multiplication ×.
Perform divisions and multiplications, reading left to right.
16 ÷ 2 = 8.



Step 4
Addtion +, Subtraction -.
Perform additions and subtractions, reading left to right.

From left...
2 + 9 = 11.



...to right.
11 - 8 = 3.



Evaluating the expression using the order of operations gives 3.

Another Real Example of Using Order of Operations

the slider below shows an example where there are several operations within a bracket.
Interactive Test
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Top Tip
BODMAS, BIDMAS, BEDMAS, PEMDAS

In the UK and Australia, BODMAS or BIDMAS is used, where O is for Order and I is for Indices.

In Canada, BEDMAS is used, where E is for Exponent.

The three words (Order, Indices, Exponents) are all different words for the same thing.

In the US, the acronym PEMDAS used: Parentheses, Exponent, Multiplication, Division, Addition, Subtraction.

Parentheses is another word for brackets.

A useful memory device for PEMDAS is "Please Excuse My Dear Aunt Sally."

Note

WHAT IS AN OPERATION

An operation takes two values (called operands) and produces another value from them.

The type of operation is denoted by an operator.

For example, the addition operation is given shown by:



LEFT TO RIGHT

Operations on the same level (× and ÷, + and -) need to be evaluated left to right.

That is the order is determined by its position, not whether it is a × or ÷.

As an example, consider the mutliplications and divisions shown below:



The left-most operation is the division between 12 and 6. Do this first:



Not the left-most operation is the multiplication between 2 and 3:



The only operation left is 6 ÷ 2 = 3.

BRACKETS

Brackets are the first operation that is performed according to BODMAS.

However, the brackets may itself contain brackets, orders, divisions, multiplications, additions and subtractions.

In the case, use BODMAS on the expression within the Bracket, simplifying the bracket expression, before applying ODMAS to the rest of the original expression.

See Also...