What are the Order of Operations?

Free Order of Operations Fact Sheet

Use the order of operations fact sheet below to help you get a better understanding of how order of operations work. You are encouraged to print or download the order of operations fact sheet with the PDF link at the bottom of the page.

Infographic explaining and giving examples of the order of operations

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Order of Operations

Please excuse my dear Aunt Sally.

Now, I know what you’re thinking, “What does that phrase have to do with math?” Quite a bit actually, because that saying provides the key to remembering an important math concept: The order of operations.

The order of operations is one of the more critical mathematical concepts you’ll learn because it dictates how we calculate problems. It gives us a template so that everyone solves math problems the same way.

Let’s start off with a simple question. What is an operation?

An operation is a mathematical action. Addition, subtraction, multiplication, division, and calculating the square root are all examples of a mathematical operation. Let’s take a look at this problem:

7 x 4 – 6 = ?

Looks easy right? Well, it wouldn’t be so easy if we didn’t understand the order in which the math operation occurs. If we didn’t have rules to determine what calculations we should make first, we’d come up with different answers.

Should you start by subtracting 4 minus 6 and then multiplying by 7? Should you multiply 4 by minus 6 and then add 7?

No. The order of operations tells us how to solve a math problem. And this brings us back to Aunt Sally.

Operations have a specific order:
Multiplication and Division (left to right)
Additional and Subtraction (left to right)

There are two ways we remember the order of operations. One is through the acronym PEMDAS. But there’s another way that’s a little more fun:

Please Excuse My Dear Aunt Sally

So now we know what an operation is and we know the order in which we perform the operation so that we always calculate problems correctly. Let’s solve an example problem using the correct order of operations.
Now let’s look at another problem.

7 + 7 x 3

Without the order of operations and rules, you could calculate this problem as 7 + 7 = 14 x 3 = 42.

And that would be wrong.

Remember, you multiply before you add. Therefore, the equation should look like this:

7 + (7 x 3).

Seven (7) multiplied by 3 equals 21. Twenty-one (21) plus 7 equals 28, which is the right answer.

Let’s look at some more complex problems.

6 × 32

The order of operations dictates how to solve this problem. Remember, you multiply exponents first. Here’s the wrong way to solve the problem:

6 × 32 = 182 = 324

Why is that wrong? Because you violated the order of operations. You don’t multiply first. You perform an operation on the exponents first. This is how it should be done:

6 × 32 = 6 × 9 =54

See? Solving the equation in the right order provides the correct answer.

Let’s try out one more problem. This one is a little more challenging, but it perfectly illustrates the order of operations.
5 x 10 – (8 x 6 – 15) + 4 x 20 ÷ 4

Remember the order. What do we do first? The numbers inside the parentheses. 8 x 6 = 48, minus 15 = 33. Here’s how the problem looks now:

5 x 10 – (48 – 15) + 4 x 20 ÷ 4

Go ahead and finish out the subtraction in the parentheses.

5 x 10 – 33 + 4 x 20 ÷ 4

What’s the next step? Multiplication and division.
50 – 33 + 20
Now, finish with addition and subtraction.
50 – 33 = 17 + 20 = 37
The answer is 37.

There is an exception. If an equation only has one expression, you don’t have to follow the order of operations.

Here are some examples of single expressions.

Addition only: 10 + 10 = 20

Subtraction: 20 – 10 = 10

Multiplication: 10 x 10 = 100

All of those are single expressions.

That’s our video on order of operations. I hope this overview was helpful!

See you guys next time!



by Mometrix Test Preparation | Last Updated: August 13, 2020