Order of Operations in Math: PEMDAS Rules

Have you ever looked at a math problem like 2 + 5 × 4 and wondered what to do first? One friend might add first and get 28, while another multiplies first and gets 22. Who’s right? The answer is: follow the order of operations in math! This guide breaks down PEMDAS step by step with fun examples.

What Is the Order of Operations in Math?

The order of operations in math is a set of agreed-upon rules that dictate the sequence for evaluating mathematical expressions containing multiple operations, such as addition, subtraction, multiplication, division, exponents, and grouping symbols like parentheses. These rules ensure that everyone solving the same problem arrives at the same result, preventing confusion and inconsistency in calculations.

Without these rules, even simple expressions could yield wildly different answers depending on personal preference. For instance, consider the expression 2 + 5 × 4. If you add first (2 + 5 = 7, then 7 × 4 = 28), you get one answer. But if you multiply first (5 × 4 = 20, then 2 + 20 = 22), you get another. Both approaches might seem logical, but only one is correct under the standard rules—highlighting why we need a universal system.

Think of the order of operations like following a recipe in cooking: you can’t randomly mix ingredients; you must add them in the right sequence to get the desired outcome. This “recipe” for math is captured in the acronym PEMDAS, which stands for:

• P – Parentheses (or brackets): Solve inside these grouping symbols first.
• E – Exponents: Handle powers and roots next.
• M/D – Multiplication and Division: Perform these from left to right (they share equal priority).
• A/S – Addition and Subtraction: Do these last, also from left to right (equal priority).

Fun Mnemonic: “Please Excuse My Dear Aunt Sally”

In some regions, like the UK, Australia, or India, it’s called BODMAS (Brackets, Orders/exponents, Division/Multiplication, Addition/Subtraction). Regardless of the name, both acronyms describe the identical process, adapting to local terminology while maintaining the core hierarchy.

Why Do We Need the Order of Operations in Math?

The primary goal is consistency and clarity in mathematics. As demonstrated in educational resources like the Math Antics video, imagine two students tackling 2 + 5 × 4:

• Student A prefers addition: 2 + 5 = 7, then 7 × 4 = 28.
• Student B prefers multiplication: 5 × 4 = 20, then 2 + 20 = 22.

Neither made calculation errors, but their differing orders led to conflicting results. The order of operations resolves this by prioritizing multiplication and division over addition and subtraction, making 22 the universally accepted answer.

This system isn’t arbitrary, it’s rooted in mathematical conventions that make complex expressions predictable. For K-12 learners, mastering it builds a strong foundation for algebra, geometry, and beyond, turning potential frustration into confident problem-solving.

Step-by-Step PEMDAS Rules

1. Parentheses (or Brackets) First

Always solve inside parentheses () or brackets [] before anything else. They group numbers like a “package.”

Example: 10 × (4 + 5)

• Inside parentheses: 4 + 5 = 9
• Now: 10 × 9 = 90

Multiple Parentheses Example: (5 − 3) + (6 × 2)

• First group: 5 − 3 = 2
• Second group: 6 × 2 = 12
• Then: 2 + 12 = 14

Tip: If exponents are inside parentheses, do them as part of the group!

2. Exponents Next

Exponents mean repeated multiplication, like 5² = 5 × 5 = 25.

Example: 3 × 5²

• Exponent first: 5² = 25
• Then: 3 × 25 = 75

With Parentheses: (3² × 4) + 6

• Inside: 3² = 9, then 9 × 4 = 36
• Then: 36 + 6 = 42

3. Multiplication and Division (Left to Right)

These are equal priority—do them from left to right, whichever comes first.

Example: 40 ÷ 4 × 5

• Left to right: 40 ÷ 4 = 10, then 10 × 5 = 50
  (Not 4 × 5 first, which would wrongly give 2!)

More Examples:

• 3 × 5 − 1 = 15 − 1 = 14 (Multiply first)
• 20 − 10 ÷ 5 = 20 − 2 = 18 (Divide first)
• 12 ÷ 6 + 5 = 2 + 5 = 7

4. Addition and Subtraction (Left to Right)

Last step, and again left to right.

Example: 9 – 24 ÷ 8 × 2 + 3

• Division: 24 ÷ 8 = 3
• Multiplication: 3 × 2 = 6
• Subtraction: 9 − 6 = 3
• Addition: 3 + 3 = 6

Practice Problems: Test Your Order of Operations in Math Skills

(1) 2 + 6 × (4 + 5) ÷ 3 – 5

• Parentheses: (4 + 5) = 9 → 2 + 6 × 9 ÷ 3 – 5
• Multiplication: 6 × 9 = 54 → 2 + 54 ÷ 3 – 5
• Division: 54 ÷ 3 = 18 → 2 + 18 – 5
• Addition/Subtraction: 2 + 18 = 20, 20 – 5 = 15

(2) 4 – 5 ÷ (8 – 3) × 2 + 5

• Parentheses: (8 – 3) = 5 → 4 – 5 ÷ 5 × 2 + 5
• Division: 5 ÷ 5 = 1 → 4 – 1 × 2 + 5
• Multiplication: 1 × 2 = 2 → 4 – 2 + 5
• Subtraction/Addition: 4 – 2 = 2, 2 + 5 = 7

(3) 100 ÷ (6 + 7 × 2) – 5

• Inside parentheses (multiplication first): 7 × 2 = 14 → (6 + 14) = 20
• Division: 100 ÷ 20 = 5 → 5 – 5 = 0

Quiz Question: Simplify 4 + (5 × 3² + 2)
(Answer: 51 – Check your work with PEMDAS!)

Common Mistakes to Avoid in Order of Operations in Math

Even when students remember the PEMDAS rule, small misunderstandings can lead to big errors. Here are some common mistakes to watch out for:

1. Reading strictly from top to bottom
   Many students make the mistake of solving operations exactly in the order they appear in PEMDAS. Remember that the acronym shows priority, not strict sequence. Multiplication and division share the same level of importance, as do addition and subtraction.
2. Forgetting to go left to right for M/D and A/S
   When multiplication and division (or addition and subtraction) appear together, always work from left to right. For example, in 20÷5×220 ÷ 5 × 220÷5×2, you divide first (20 ÷ 5 = 4) and then multiply (4 × 2 = 8). Doing it the other way would give you the wrong result.
3. Ignoring parentheses
   Parentheses always come first and can completely change the meaning of an expression. For instance, 8+(6÷3)8 + (6 ÷ 3)8+(6÷3) is not the same as (8+6)÷3(8 + 6) ÷ 3(8+6)÷3. Always handle what’s inside the parentheses before moving on.

By keeping these points in mind, students can avoid the most common calculation mistakes and apply the order of operations accurately every time.

Master Order of Operations with WuKong Math

A solid grasp of PEMDAS helps students build the foundation for higher-level math. WuKong Math offers structured online courses designed to strengthen this understanding through clear instruction and guided practice.

Why choose WuKong Math:

• Expert teachers with years of K–12 math teaching experience
• Step-by-step lessons that reinforce logic and accuracy
• Interactive exercises that build problem-solving confidence
• Progressive curriculum aligned with international standards

With WuKong Math, students move beyond memorizing rules, they learn to think critically and apply math concepts with confidence.

Conclusion

Mastering the order of operations is a key step toward developing strong problem-solving skills in math. By understanding how PEMDAS works and avoiding common mistakes, students can approach complex equations with clarity and confidence. Consistent practice and guidance from skilled teachers help turn these rules into second nature, setting a solid foundation for future success in higher-level math.

FAQ: Order of Operations in Math

James | WuKong Math Teacher

Graduated from Columbia University in the United States and has rich practical experience in mathematics competitions’ teaching, including Math Kangaroo, AMC… He teaches students the ways to flexible thinking and quick thinking in sloving math questions, and he is good at inspiring and guiding students to think about mathematical problems and find solutions.