# The Order of Operations in Math — A Complete & Kid-Friendly Guide

Jul 18, 2024 | Ankeny

Whether you're getting ready for an exam or simply want to refresh your knowledge of the order of operations in math, we've created a comprehensive and easy-to-follow guide for you.

Read on to find simple definitions of the order of operations and related terms, practical examples, and a fun quiz to test your knowledge.

## What Is the Order of Operations?

Have you ever tried to solve a math problem, where you didn't know where to start or what to do next?

For example, when you see 3 + 8 × 2 - 6, what should you do first?

• Multiply 8 and 2
• Subtract 6 from 2

The order of operations is a set of rules that tells us what math operation to do first in an expression with multiple operations like addition, subtraction, multiplication, and division.

Following the order of operations, when solving 3 + 8 × 2 - 6, we would first do the:

• Multiplication: 8 x 2 = 16, so we get 3 + 16 – 6.
• Addition: 3 + 16 = 19, so we get 19 – 6.
• Subtraction: 19 – 6 = 13.

3 + 8 × 2 – 6 = 13

One way to remember the order of operations is by the acronym PEMDAS.

## What Does “PEMDAS” Mean?

PEMDAS is an acronym that reminds us of the sequence of steps to follow when solving mathematical expressions with multiple operations.

• stands for Parentheses (sometimes called Brackets).
• stands for Exponents (sometimes called Indices).
• M stands for Multiplication.
• stands for Division.
• stands for Subtraction.

Much like grammatical rules that govern how we build sentences, PEMDAS is a set of rules mathematicians came up with and agree to follow so that we can have consistency and clarity when performing calculations.

Different countries have their own ways of remembering the order of math operations.

• In Canada, it's called BEDMAS: Brackets, Exponents, Division, Multiplication, Addition, and Subtraction.
• Some prefer BODMAS: Brackets, Order, Division, Multiplication, Addition, and Subtraction.
• Then there's GEMDAS: Grouping, Exponents, Multiplication, Division, Addition, and Subtraction.

When solving math expressions with multiple operations, we follow these steps from left to right.

There is a fun way of remembering the PEMDAS rule with the left to right rule using a playful mnemonic:

 “Please Excuse My Dear Aunt Sally and Let her Rest”

 Parentheses, Exponents, (Multiplication, Division), (Addition, Subtraction) – from Left to Right

Now, let’s work through each step of PEMDAS.

### 1. Parentheses

The first step to the order of operations is parentheses. This means that when we’re solving an expression with multiple operations, we first solve the operations within parentheses (). Here’s a simple example to illustrate.

3 x (4 + 2)

• First, we solve the expression within the parentheses: 4 + 2 =6, so we get 3 x 6.
• Then, we multiply 3 by the result of the expression inside the parentheses (which, as we've seen, is 6): 3 x 6.
• Finally, we get the answer 18.

Sometimes, math expressions contain a set of parentheses inside another set. To solve them, start by solving the operations inside the innermost parentheses, then work your way outward.

For example: {4 x [5 +(6 – 3)]} – (8 – 2)

• First, we subtract numbers inside the innermost parentheses: 6 – 2 = 3, so we get {4 x [5 + 3]} – (8 – 2).
• Then, add 5 to the answer inside the square brackets: 5 + 3 = 8, so we get {4 x 8} – (8 – 2).
• Next, multiply the result by 4 inside the curly brackets: 4 x 8 = 32, so we get 32 — (8 – 2).
• Now, solve the expression inside the parantheses: 8 – 2 = 6, so we get 32 – 6.
• Finally, subtract the result of step 4 from the result of step 3: 32 – 6 = 26

### 2. Exponents

The second step in the order of operations is exponents. Exponents, also known as indices, show how many times a number is multiplied by itself.

To illustrate, 3⁴ is an exponent form that means multiply 3 by itself 4 times, or 3 x 3 x 3 x 3.

Let’s look at this example:

22 + 4 x 2

• First, we look at the exponent 23 which means we multiply 2 by itself 3 times, or 2 × 2 × 2 = 8 , so we get: 8 + 4 × 2.
• Then we multiply: 4 × 2 = 8 , so we get 8 + 8.
• Finally, we add: 8 + 8 = 16.

Refresh Your Memory: What Does Squared Mean in Math?

### 3. Multiplication & Division

In the order of operations, multiplication and division are prioritized equally and are performed from left to right as they appear in the expression. This is why we consider them together as the third step in PEMDAS.

Let’s see that in practice with: 8 ÷ 2 × 4 + 3

• First, we perform the division because it appears first, before the multiplication: 8 ÷ 2 = 4, so we get 4 x 4 + 3.
• Then, we perform the multiplication: 4 x 4 = 16, so we get 16 + 3.
• Next, we perform the addition: 16 + 3 = 19.
• Finally, we get the result 19.

As the last steps of PEDMAS, we work on the addition and subtraction.

Just like multiplication and division, addition and subtraction are prioritized equally and are performed from left to right as they appear in the expression.

For example:

10 – 4 + 3

• First, we subtract: 10 – 4 = 6, so we get 6 + 3
• Then, we add: 6 + 3 = 9
• Thus, the result is 9.

Now that we have covered each step of PEMDAS, let’s look at a more challenging example:

Find more tips in our Order of Operations Series Part 2.

## Common Mistakes with the Order of Operations in Math

PEMDAS makes solving math expressions much easier than simply memorizing the order of operations, but students sometimes forget the steps to take.

Here are some common mistakes to keep an eye on when solving math expressions with multiple operations:

### 1. Going Only Left to Right

Some students forget PEMDAS and just focus on the left-to-right approach, which might feel natural because of how we read.

It's important to remember that while going left to right can work, it's only suitable when you're solving expressions that only contain addition and subtraction, or multiplication and division.

For example, using the left to right approach works if you are trying to solve:

8 – 6 + 9 – 3

Or, if you are trying to solve:

8 ÷ 2 × 9 ÷ 3

But let's say you are trying to solve:

10 – 3 x 2 + 5 ÷ 5

As you can see, this expression contains subtraction, multiplication, addition, and division.

If we go from left to right – first subtracting 3 from 10 to get 7, then multiplying 7 by 2 to get 14, then adding 5 to it to get 19, and finally diving 19 by 5, we get 3.8 which is an incorrect result.

But if we follow PEMDAS, which tells us to start with multiplication and division (left to right) before we get to addition and subtraction, we get:

10 – 3 x 2 + 5 ÷ 5

= 10 – 6 + 1

= 5

### 2. Doing Addition Before Subtraction

Many students tend to do addition before subtraction in math problems because they learned addition before subtraction or because they're close together in the order of operations. Some also think that because A comes before S in PEMDAS you should always do addition before subtraction.

For instance, in the expression 8 2 + 3  you might tackle the addition before the subtraction and get: 8 – 5 = 3.

In the order of operations, addition and subtraction have the same priority which is why we have to use left-to-right approach and perform subtraction first:

8 – 2 + 3

= 6 + 3

= 9

### 3. Doing Multiplication Before Division

Similarly to addition vs. subtractions, students often think multiplication comes before division because that's what they've learned before or because they're used to seeing multiplication listed first in math problems. Another mistake is believing M comes before D in PEMDAS because you always do multiplication before division.

For example, in the expression 6 ÷ 2 × 3 = 6 ÷ 6 = 1 , a student might do the multiplication first and get a wrong result:

6 ÷ 2 × 3 = 6 ÷ 6 = 1

But we know by now that multiplication and division are equally important. So, we need to work from left to right and do the division first:

6 ÷ 2 × 3 = 3 × 3 = 9

## Solved Examples for the Order of Operations

### Example 1

Using PEMDAS, let’s solve this one:

3 x (8 – 4)2 + 6 ÷ 2

• First, we do the subtraction inside the parentheses: 8 – 4 = 4, so we get 3 x 4+ 6 ÷ 2.
• Next, we square the result: 4= 16, so we get 3 x 16 + 6 ÷ 2.
• Then, we multiply: 3 x 16 = 48, so we get 48 + 6 ÷ 2.
• Next, we divide: 6 ÷ 2 = 3, so we get 48 + 3.
• Finally, we add: 48 + 3 = 51.

### Example 2

Let’s do the same for this expression:

3 x (7 – 2) + 6 ÷ 2

• First, we do the subtraction inside the parentheses: 7 – 2 = 5, so we get 3 x 5 +  6 ÷ 2.
• Next, we multiply: 3 x 5 = 15, so we get 15 +  6 ÷ 2.
• Then, we divide: so we get 6 ÷ 2 = 3, so we get 15 + 3.
• Finally, we add:15 + 3 = 18.

### Example 3

Finally, we can work on this one:

80  ÷ (6 + 7 x 2) – 5

• First, we perform the multiplication inside the parentheses: 7 x 2 = 14, so we get 80 ÷ (6 + 14) – 5.
• Next, we perform the addition inside the parentheses: 6 + 14 = 20, so we get 80 ÷ 20 – 5.
• Then, we divide: 80 ÷ 20 = 4, so we get 4 – 5.
• Finally, we subtract and get: 4 – 5 = –1.

## Flash Quiz: How Well Do You Know the Order of Operations?

Create your own user feedback survey

Discover clear and easy-to-understand explanations about the rules and steps for solving math problems using the order of operations.

### 1. Can PEMDAS be applied to all math operations?

Yes, PEMDAS applies to all mathematical operations involving multiple steps, including addition, subtraction, multiplication, division, and exponentiation

### 2. What are BODMAS and BEDMAS?

BODMAS and BEDMAS, just like PEDMAS are mnemonic devices used to remember the order of operations in mathematics:

BODMAS:

• B: Brackets
• O: Orders
• D: Division
• M: Multiplication
• S: Subtraction

BEDMAS:

• B: Brackets
• E: Exponents
• D: Division
• M: Multiplication
• S: Subtraction

### 3. What should I do when math expressions have parentheses within parentheses?

When there are parentheses within one another (nested parentheses), you should start by doing the innermost expressions first and then work your way outwards.

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