What is PEMDAS? Your Guide to Order of Operations

blogstutorology
Nov 2
5 min read

Have you ever looked at a math problem like 15 - (6 + 1) × 2 + 8 ÷ 4 and felt a little lost? Knowing where to start can be tricky. Do you add first? Or multiply? The order you choose completely changes the answer. Thankfully, there’s a simple rule to follow that ensures you get the right result every time: PEMDAS.

This guide will break down the order of operations, explaining what PEMDAS stands for and why it’s a fundamental concept in mathematics. Whether you're a student trying to ace your next exam, a parent helping with homework, or just someone wanting to refresh your math skills, this post will make PEMDAS easy to understand and apply.

What Does PEMDAS Stand For?

PEMDAS is an acronym that helps you remember the correct order to solve mathematical expressions. Each letter represents a step in the sequence:

Parentheses
Exponents
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

A common way to remember this is the phrase: "Please Excuse My Dear Aunt Sally." Let’s look at what each step involves.

1. Parentheses

The first rule is to solve any calculations inside parentheses ( ). This also includes other grouping symbols like brackets [ ] and braces { }. If you have multiple sets of parentheses nested within each other, start with the innermost set and work your way out.

Example: 3 × [4 + (2 × 1)]

First, solve the inner parentheses: (2 × 1) = 2
The expression becomes: 3 × [4 + 2]
Next, solve the brackets: [4 + 2] = 6
Finally, perform the last operation: 3 × 6 = 18

2. Exponents

After you’ve handled all the parentheses, the next step is to address any exponents. Exponents (also known as powers or indices) are the small numbers written to the upper right of a base number, like the 2 in 5².

Example: (2 + 3)² - 5

First, solve the parentheses: (2 + 3) = 5
The expression is now: 5² - 5
Next, calculate the exponent: 5² = 25
Finally, perform the subtraction: 25 - 5 = 20

3. Multiplication and Division

Once the parentheses and exponents are out of the way, it’s time for multiplication and division. These two operations are a pair—they have the same level of importance. You don't always do multiplication before division. Instead, you perform them as they appear from left to right in the expression.

Example: 10 ÷ 2 + 7

Working from left to right, the division comes first: 10 ÷ 2 = 5
The expression becomes: 5 + 7
Now, perform the addition: 5 + 7 = 12

4. Addition and Subtraction

The final step is to perform any addition and subtraction. Just like multiplication and division, these two operations are a team. You solve them as they appear from left to right.

Example: 15 - 14 + 2

Starting from the left, the subtraction is first: 15 - 14 = 1
The expression is now: 1 + 2
Finally, do the addition: 1 + 2 = 3

Why Is PEMDAS Important?

PEMDAS isn't just a rule for math class; it's a universal standard that ensures everyone who solves a problem arrives at the same, correct answer. Without a standard order of operations, an expression like 2 + 3 × 4 could have two different results.

If you add first: 2 + 3 = 5, then 5 × 4 = 20.
If you multiply first (the correct way): 3 × 4 = 12, then 2 + 12 = 14.

This consistency is crucial in fields like science, engineering, computer programming, and finance, where precise calculations are essential. Even in daily life, like calculating a discount or splitting a bill, the order of operations helps you get the right number.

Examples of PEMDAS in Action

Let's walk through a few more examples to see how PEMDAS works.

Expression with Multiple Parentheses: 2 × {3 + [5 - (1 + 1)]}
Innermost parentheses: (1 + 1) = 2
- Brackets: [5 - 2] = 3
- Braces: {3 + 3} = 6
- Multiplication: 2 × 6 = 12
Expression with Fractions: (1/2 + 1/4) × 8
Parentheses: 1/2 + 1/4 = 2/4 + 1/4 = 3/4
- Multiplication: (3/4) × 8 = 24/4 = 6
Expression with Negative Numbers: 10 - (5 × -2)
Parentheses: 5 × -2 = -10
- Subtraction: 10 - (-10) = 10 + 10 = 20
Complex Expression: 15 - (6 + 1) × 2 + 8 ÷ 4
Parentheses: (6 + 1) = 7
- Expression is now: 15 - 7 × 2 + 8 ÷ 4
- Multiplication (left to right): 7 × 2 = 14
- Expression is now: 15 - 14 + 8 ÷ 4
- Division (left to right): 8 ÷ 4 = 2
- Expression is now: 15 - 14 + 2
- Subtraction (left to right): 15 - 14 = 1
- Expression is now: 1 + 2
- Addition: 1 + 2 = 3

Common Mistakes to Avoid

Applying PEMDAS can be simple, but a few common slip-ups can lead to the wrong answer.

Thinking Multiplication Always Comes Before Division: This is a big one. Remember, multiplication and division are partners. You solve them from left to right as they appear. The same goes for addition and subtraction.
Ignoring Parentheses: Always start with what's inside the parentheses. Their job is to tell you what to do first.
Skipping Steps: It can be tempting to do calculations in your head, but writing each step down helps prevent errors, especially with complex problems.
Not Double-Checking Your Work: A quick review can help you catch small mistakes, like forgetting a negative sign or misapplying the order.

PEMDAS for Parents and Tutors

If you're helping a student learn PEMDAS, focus on making it relatable. Use the "Please Excuse My Dear Aunt Sally" mnemonic, but also explain why the order matters. Work through examples together, starting with simple ones and gradually increasing the difficulty. Encourage them to write out each step and talk through their process. This builds a strong foundation and boosts their confidence.

Master PEMDAS for Math Success

Understanding the order of operations is a building block for almost all higher-level mathematics. By mastering PEMDAS, you unlock the ability to tackle complex equations with confidence. It transforms confusing problems into a clear, step-by-step process.

If you or your child needs extra help making sense of PEMDAS or other math concepts, Tutor-ology's expert math tutors are here to guide you. Our personalized lessons are designed to build skills and confidence. Enroll today, and let's conquer math together!