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 + 3^{2} − (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 + 3^{2} − (8 × 2) ÷ 2
= 2 + 3^{2} − 16 ÷ 2
2
Order.
Evaluate numbers with exponents second.
In our example, there is one exponent: 3^{2} = 9.
2 + 3^{2} − 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 + 3^{2} − (8 × 2) ÷ 2 = 3Top 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.