The Lesson

The order of operations tells us what order to perform operations in. A calculation may have several operations, such as: adding, subtracting, multiplying, dividing and squaring.

Why Do We Need the Order of Operations?

Imagine we wanted to find the answer to the calculation below:

This calculation contains two operations: adding and multiplying. There are two orders to doing this calculation and two answers. Do we add then multiply, or multiply then add?

Order 1

Add the first two numbers, then multiply the result with the third number.
1 + 2 × 3 = 3 × 3 = 9

Order 2

Multiply the last two numbers, then add the result to the first number.
1 + 2 × 3 = 1 + 6 = 7
Which answer is the correct one? It turns out the second order of operations is the correct one. Luckily, there is a simple way to use the correct order.

BODMAS

BODMAS is an acronym for the order of operations. It stands for:

The order of operations is read from top to bottom. The operations with a curly bracket ({) are on the same level, and can be performed in any order.
  • Brackets. Evaluate brackets first.
  • Order. Evaluate exponents (such as squares and square roots) second.
  • Division and Multiplication. Evaluate numbers that are divided and multiplied third.
  • Addition and Subtraction. Evaluate numbers that are added and subtracted fourth.

How to Use the Order of Operations

Using the order of operations is easy.

Question

Find 2 + 32 − (8 × 2) ÷ 2.

Step-by-Step:

1

Brackets. Evaluate expressions within brackets first. In our example, there is one pair of brackets: (8 × 2) = 16.
2 + 32 − (8 × 2) ÷ 2 = 2 + 32 − 16 ÷ 2

2

Order. Evaluate numbers with exponents second. In our example, there is one exponent: 32 = 9.
2 + 32 − 16 ÷ 2 = 2 + 9 − 16 ÷ 2

3

Division and Multiplication. Evaluate numbers that are divided and multiplied third. In our example, there is one division: 16 ÷ 2 = 8.
2 + 9 − 16 ÷ 2 = 2 + 9 − 8

4

Addition and Subtraction. Evaluate numbers that are added and subtracted fourth. In our example, there is one +'s and one −. Addition and subtraction take the same precedence, so it does not matter which order we do them in. We will do them left to right.
2 + 9 − 8 = 11 − 8 \(\:\:\:\:\:\:\:\:\:\:\:\:\) as 2 + 9 = 11
11 − 8 = 3

Answer:

2 + 32 − (8 × 2) ÷ 2 = 3

Lesson Slides

The slider below shows another real example of how to use the order of operations. Open the slider in a new tab In another example, there is a pair of brackets which contains a long expression that itself needs to use the order of operations. Open the slider in a new tab

Top Tip

BODMAS, BIDMAS, BEDMAS, PEMDAS

BODMAS is one acronym used to remember the order of operations, but different parts of the world will use different ones.
  • BODMAS is used in the United Kingdom and Australia. Sometimes BIDMAS is used, where I is for Indices rather than Ofor Order.
  • BEDMAS is used in Canada, where E is for Exponent. The words (Order, Indices, Exponents) are all different words for the same thing.
  • PEMDAS is used in the United States of America: 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 values and calculates a new value from them.
  • The numbers operated upon are called operands.
  • The symbol which shows what type of operation is taking place is called the operator.