Algebra/Order of Operations

Algebra

Order of Operations: A way of evaluating expressions with more than one operation. These rules govern precedence in mathematical operations.

For Example: When faced with $4+2 \times 3$, how do you proceed?

There are two apparent options:

$4 + 2 \times 3 = (4 + 2) \times 3$

$4 + 2 \times 3 = 6 \times 3$

$4 + 2 \times 3= 18$

OR

$4 + 2 \times 3 = 4 + (2 \times 3)$

$4 + 2 \times 3 = 4 + 6$

$4 + 2 \times 3 = 10$

Which is correct?

In order to communicate using mathematical expressions we must agree on an order of operations so that each expression has a unique value.

For the above example all mathematicians agree the correct answer is 10.

The key question is: what is the order upon which mathematicians have agreed?

The Standard Order of Operations

Evaluate expressions in this order.

• Parentheses (evaluate what's inside them)
• Exponents
• Multiplication and/or division from left to right
• Addition and/or subtraction from left to right

An Easy Way of Remembering

Use this memory tool to help remember the order! Please Excuse My Dear Aunt Sally

An alternative form of this is brackets, indices, division, multiplication, addition, subtraction (BIDMAS).

It is also commonly called by its acronym, PEMDAS.

Examples

Order of Operations - Examples

Expression	Evaluation	Operation
4 × 2 + 1	= 4 × 2 + 1	Multiplication
	= 8 + 1	Addition
	= 9
12 - 9 ÷ 3	= 12 - 9 ÷ 3	Division
	= 12 - 3	Subtraction
	= 9
2 × 9 ÷ 3	= 2 × 9 ÷ 3	Left to Right
	= 18 ÷ 3	division
	= 6
9 ÷ 3 × 3	= 9 ÷ 3 × 3	Left to Right
	= 3 ×3	multiplication
	= 9
3 + 12 ÷ (5 - 2)	= 3 + 12 ÷ (5 - 2)	Parentheses
	= 3 + 12 ÷ 3	Division
	= 3 + 4	Addition
	= 7
7 × 10 - (2 × 4)2	= 7 × 10 - (2 × 4)2	Parentheses
	= 7 × 10 - 82	Exponents
	= 7 × 10 - 64	Multiplication
	= 70 - 64	Subtraction
	= 6

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Practice Problems

1. Evaluate the numerical expression 2 + 4 * 3
2. Evaluate the numerical expression 2 * 4 + 3
3. Evaluate the numerical expression (2 + 4) * 3
4. Evaluate the numerical expression 9² + 1 − 7 * (8 + 4) / 2

Practice Answers

Playing with Mathematics

To get yourself in the mindset of math, try this simple mathematical game:

Take the numbers 1 through 10 on the left side of an equation, and pick a number for the right side.

E.G. 1 2 3 4 5 6 7 8 9 10 = 1

Now put operators between those numbers. Only use parentheses when necessary.

E.G. 1 + 2 - 3 + 4 - 5 + 6 + 7 + 8 - 9 - 10 = 1

Change the number on the right-hand side. Can you generate an expression for this number? If not, can you prove why not?

Does this change if you change the order of the numbers?