The Order of Operations Explained with Infographic
You can use a variety of operations to check mathematical expressions. This includes addition, subtraction, multiplication, and division.
Consider the following expression 3+ 6 × 24 – 4
This expression has many operations. So where do we when we calculate it? Begin at either end will get you a completely different answer.
To avoid confusion, we follow a certain system when we calculate the answer. We call this system the Order of Operations.
What Is the Order of Operations in Math?
When an expression has many operations we must solve them in the correct sequence. That is the Order of Operations. In some expressions we only add, only subtract, only multiply, or only divide, and we should do so from left to right.
For example 2+6, 9 ÷ 3, 6 × 9, or 6-5.
But, for expressions with many operations, like 3+ 6 × 24 – 4, we must follow the order of operations.
PEMDAS is a mnemonic device that reminds us of the order of operations. It stands for Parenthesis, Exponents, Multiplication, Division, Addition, and Subtraction. You can use the phrase “Please Excuse My Dear Aunt Sally” to help you remember the order.
Order of Operations PEMDAS Steps:
Parentheses
The first thing to do is to solve the operation in parentheses or brackets. Grouping things together is what parentheses are for. Solve everything from the inside out.
2+6 × (4 + 5) ÷ 3 – 5 = 2 + 6 × 9 ÷ 3 – 5
Exponents
Solve the exponential expressions after the parentheses.
Multiplication and Division
Next, moving from left to right, multiply and/or divide, whichever comes first.
Multiplication: 2 + 6 × 9 ÷ 3 – 5 = 2 + 54 ÷ 3 – 5
Division: 2 + 54 ÷ 3 – 5 = 2 + 18 – 5
Addition and Subtraction
Finally, moving from left to right, add and/or subtract, whichever comes first.
Addition: 2 + 18 – 5 = 20 – 5
Subtraction: 20 – 5 = 15
Why Follow the Order of Operations?
We follow the order of operations to ensure that everyone arrives at the same answer. Here is an example of how we could arrive at different answers if do not follow the Order of Operations:
Example 1: Solve: 4 + 8 × (2 + 5) ÷ 4 – 5 using PEMDAS.
Solved Examples
Solution:
(1.) Parenthesis: 4 + 8 × (2 + 5) ÷ 4 – 5 = 4 + 8 × 7 ÷ 4 – 5
(2.) Multiplication: 4 + 8 × 7 ÷ 4 – 5 = 4 + 56 ÷ 4 – 5
(3.) Division: 4 + 56 ÷ 4 – 5 = 4 + 14 – 5
(4.) Addition: 4 + 14 – 5 = 18 – 5
(5.) Subtraction: 18 – 5 = 13
Example 2: Solve 4 – 5 ÷ (8 – 3) × 2 + 5 using PEMDAS.
Solution:
(1.) Parenthesis: 4 – 5 ÷ (8 – 3) × 2 + 5 = 4 – 5 ÷ 5 × 2 + 5
(2) Division: 4 – 5 ÷5 + 5 = 4 – 1 × 2+ 5
(3.) Multiplication: 4 – 1 × 2 + 5 = 4 – 2 + 5
(4.) Subtraction: 4 – 2 + 5 = 2 + 5
(5.) Addition: 2 + 5 = 7
Example 3: Solve 200 ÷ (6 + 7 × 2) – 5 using PEMDAS.
Solution:
(1.) Multiplicatin inside Parentheses: 200 ÷ (6 + 7 × 2) – 5 = 200 ÷ (6 + 14) – 5
(2) Addition inside Parentheses: 200 ÷ (6 + 14) – 5 = 200 ÷ 20 – 5
(3.) Division: 200 ÷ 20 – 5 = 10 – 5
(4.) Subtraction: 10 – 5 = 5
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