What is Order of Operations in Math (PEMDAS vs BODMAS Explained)

In this blog, we understand what order of operation is, why it exists and how PEMDAS and BODMAS help students remember the rules.

Have you ever solved a math problem and got a different answer than your friend, even though you both used the same numbers? Well, the reason can be the order of operations. In math, the order in which you solve each part of an expression matters significantly, and without following the order of operations, the same expression can produce different answers. In the next secction we are going to discuss “what is order of Operations?”

What is the Order of Operations?

Order of operations in math is a set of rules that tells us the correct sequence to solve a math problem. As most expressions contain more than one operation, including addition, subtraction, multiplication, division, exponents and parentheses, mathematicians agreed on a standard sequence so that everyone got the same answer. With the rule of order of operations in math, math could be ambiguous. Take the order of operations examples.

6+4 x 3

If we solve left to right, we get 6+4=10, then 10×3= 30.

But if you follow the order of operations and multiply first,

4 x 3= 12, then 6 +12= 18

The correct answer is 18. Multiplication should always be solved before addition, irrespective of where it appears in an expression. The order of operations exists to eliminate confusion and get the correct answer every time.

What Is a Mathematical Expression?

A mathematical expression is a combination of:

Numbers: Such as 1, 4, 7, 10 or 100
Operators: The symbols that tell you what to do: +(addition), – (subtraction), x (multiplication), ÷ (division)
Grouping symbols: Parentheses (), brackets [] and braces {} signal which part to solve first.
Exponents: A small raised number that tells you how many times to multiply a number by itself, like 2²= 4

It is important to understand what a mathematical expression actually contains, before diving into the order of operations step by step.

Let’s look at two examples of expressions.

6+ 4 x 3

8 ÷ 2 (2 + 2)

These look simple, but without knowing the order of operations in math, one can get the wrong answer. That is exactly why the rule is designed to prevent it.

How to Solve Order of Operations Step by Step

The correct sequence of solving a mathematical expression is:

Parentheses/Brackets: Solve anything inside grouping symbols first
Exponents/orders: Evaluate power and square roots next
Multiplication & Division: Solve these left to right
Addition & Subtraction: Solve these left to right

Now, let’s walk through an example step-by-step.

3 + 6 x (5 + 4) 3 – 7

Step 1: Parentheses/Brackets: (5 + 4) = 9

Step 2: Exponents/orders: None in this expression

Step 3: Multiple/divide (left to right): 6 x 9 = 54, then 54 ÷ 3 = 18 → 3 + 18 – 7

Step 4: Add/subtract (left to right): 3 + 18 = 21, then 21- 7 = 14

The final answer = 14

PEMDAS Rule Explained

PEMDAS is an acronym used to help students remember the order of operations. Each of the letters stands for one step in the sequence.

Letter	Stands For What its Means	Table Header
P	Parentheses	Solve the expression inside () first
E	Exponents	Evaluate powers like 3^2 or 2^3
M	Multiplication	Solve x → left to right with division
D	Division	Solve ÷ → left to right with multiplication
A	Addition	Solve + → left to right with subtraction
S	Subtraction	Solve – → left to right with addition

Important note: M and D share equal priority; solve whichever comes first, reading left to right. The same goes for A and S.

Example Using PEDMAS

(2 + 3)² x 4 – 10 ÷ 2

P: (2 + 3)= 5 → 5² x 4 -10 ÷ 2

E: 5² = 25 → 25 x 4 -10 ÷ 2

M/D (left to right): 25 x 4 = 100, then 10 ÷ 2 = 5 → 100 – 5

A/S: 100 – 5 = 95

Final answer = 95

BODMAS, BEDMAS, and GEMDAS Explained

Different countries use different acronyms to teach the same rule. Here is what these acronyms mean.

Acronym	What Stand for	Used in
BODMAS	Brackets, Orders, Division, Multiplication, Addition, Subtraction	UK, India, Australia
BEDMAS	Brackets, Exponents, Division, Multiplication, Addition, Subtraction	Canada
GEMDAS	Grouping, Exponents, Multiplication, Division, Addition, Subtraction	Some international schools
PEMDAS	Parentheses, Exponents, Multiplication, Division, Addition, Subtraction	USA

Despite the different letters, all four acronyms represent the same mathematical rule. “Brackets” and “Parentheses” mean the same grouping symbol. “Orders” and “Exponents” both mean power. The sequence of operations is universal.

PEMDAS vs BODMAS (Are They Different?)

No, PEMDAS and BODMAS are not different rules. They are simply two different names for the same sequence used around different parts of the world. PEMDAS is commonly used in the USA, while BODMAS is widely used in the UK, India and other countries. Students are often confused and assume BODMAS puts Division before Multiplication. In BODMAS, Division and Multiplication are still solved in the same priority level, viz., left to right, similar to PEDMAS.

Grouping Symbols in the Order of Operations

Grouping symbols tell you to solve what’s inside before anything else. The most common types are:

Parentheses (): The most common grouping symbol
Brackets []: Used outside of parentheses in nested expressions
Braces {}: Used as the outermost layer in complex expressions

When expressions are nested, one grouping symbol inside another, always solve the innermost group first and work outwards.

Like, for instance, the following expression:

6 + [4(3 + 1)]

Step 1: Innermost parentheses: (3 + 1) = 4 → 6 + [4 + 4]

Step 2: Brackets: 4 x 4 = 16 → 6 + 16

Step 3: Addition: 6 + 16 = 22

Final answer = 22

Why Multiplication and Division Are Solved Left to Right

One of the most common mistakes students make is assuming Multiplication before Division because M comes before D in PEMDAS, which is incorrect. Multiplication and Division are equal in priority, and whichever appears first, reading left to right, gets solved first.

Important Rule: Multiplication does NOT always come before Division. Solve them left to right.

Let’s understand this with an example.

20 ÷ 5 × 2

Left to right, ÷ comes first: 20 ÷ 5 = 4 → 4 x 2 = 8

Final answer = 8

If you have multiplied first instead, 5 × 2 = 10, then 20 ÷ 10 = 2

The answer would be wrong.

That is why always read left to right when Multiplication and Division appear together.

Easy Tricks to Remember the Order of Operations

Memory tricks called mnemonics help students easily recall the correct order. Here are some popular orders of operations for kids to memorize.

Acronym	Mnemonic Phrase
PEMDAS	“Please Excuse My Dear Aunt Sally”
BODMAS	“Brackets Open Doors, Multiple and Subtract”
BEDMAS	“Big Elephants Destroy Massive Angry Snakes”

Each word in these phrases starts with the same letter as the corresponding operation. Students can easily memorize these phrases to recall the correct order during exams. Pick whichever phrase matches the acronym your teacher uses, and practice it before solving problems.

Order of Operations Examples

Example1: Basic Expression

6 + 4 x 3

Multiple first: 4 x 3 = 12 → 6 + 12 = 18

Answer = 18

Example 2: Expression with Parentheses

(8 + 2) x 5

Parentheses first: (8 + 2) = 10 → 10 x 5 = 50

Answer = 50

Example 3: Expression with Exponents

7 + 3²

Exponent first: 3² = 9 → 7 + 9 = 16

Answer = 16

Common Mistakes with Order of Operations

Even students who are well-versed with PEMDAS or BODMAS and solving order of operations practice problems tend to make predictable mistakes. The following are some of the most common problems and how to avoid them.

Mistake	Example of Error	Correct Approach
Solving addition before multiplication	6 + 3 x 3 = 30	Multiply first: 6 + 12 = 18
Ignoring Parentheses	(2 + 3) x 4 = 2 + 12 = 14	Parentheses first: 5 x 4 = 20
Misunderstanding multiplication vs division	20 ÷ 5 x 2 (multiplying 5 x 2) first = 2	Left to right, 20 ÷ 5 = 4, then 4 x 2 = 8
Mistakes with Exponents	-32 = 9	-32 = -9 (only 3 is squared, not the negative sign)

Order of Operations Word Problems

Real-time situations can directly translate to expressions that require order of operations. Here are some examples.

Word Problem 1

A student buys 3 notebooks costing $5 each and a pen for $2. How much does the student spend in total?Expression:

3 × 5 + 2

Multiply first: 3 x 5 = 15 → 15 + 2 = 17

Answer = $17

Word Problem 2

A baker makes 4 trays of cookies with 12 cookies each. He gives away 6 cookies and packs the rest into 3 boxes. How many cookies does she fill in the boxes?

Expression:

(4 x 12 – 6) ÷ 3

Parentheses (left to right inside): 4 x 12 = 48, then 48 – 6 = 42 → 42 ÷ 3 = 14

Answer = 14 cookies in each of 3 boxes

Word Problem 3

A shop has 5 shelves with 8 items each. 3 items fall off the two shelves. How many items remain?

Expression:

5 x 8 – 2 x 3

Multiply first (both left to right): 5 x 8 = 40, then 2 x 3 = 6 → 40 – 6 = 34

Answer = 34

Do Calculators Follow the Order of Operations?

No. Not all calculators follow the order of operations the same way, which might be surprising for many students.

Scientific calculators and graphing calculators like the TI-84 follow PEMDAS/BODMAS. One needs to type the full expression, and the calculator evaluates it correctly.
In contrast, the basic four-function calculators (simple ones without many buttons) often solve expressions left to right, regardless of rules. This implies they can give the wrong answer for complex expressions.

Pro Tip: When using a calculator for multi-step problems, always check whether it follows the order of operations in math or simply break expressions into smaller steps manually.

History of the Order of Operations

The order of operations as we know it today was not always standardized. Mathematical notations have evolved over a period of time. Different mathematicians use different conventions for writing and solving expressions. By the late 19th and 20th centuries, as Algebra became a standard part of education, there felt a need for a universal agreement. This gave birth to PEMDAS and BODMAS and similar acronyms as modern teaching tools. These acronyms were invented to help students remember the agreed-upon rule and not to define the rule itself.

The underlying mathematical convention predates these acronyms by many decades. Today, these rules are consistent worldwide, even though different acronyms are used in different countries.

Order of Operations for Kids

There is a simple way to go about it. Whenever you see a math problem with more than one operation, don’t go to solve it from left to right. Instead, follow this order:

Solve anything inside the brackets or parentheses first
Handle exponents (numbers with raised powers)
Do multiplication and division, left to right
Do addition and subtraction, left to right

Let’s take an easy example:

2 + 3 × 4

Should you add 2 + 3 first? No! Multiply first: 3 × 4 = 12. Then add 2 + 12 = 14

Remember: Multiplication and division are stronger than addition and subtraction; they always get to go first.

Order of Operations by Grade Level

Students are introduced to the order of operations gradually. Here is what is typically covered at each grade:

Order of Operations Grade 3

At Grade 3, students are taught basic arithmetic operations, viz., addition, subtraction, multiplication and division. They are taught the order of operations Grade 3, with the introduction of the idea that multiplication is solved before addition. Simple two-operation expressions like 4 + 3 x 2 are used without parentheses or exponents.

Order of Operations Grade 4

At Grade 4, students are introduced to parentheses. They learn that expressions inside parentheses are always solved first, and practice with problems like (3 + 5) x 2. The order of operations Grade 4 teaches them the concept that grouping symbols changes the result, which is practised through comparison exercises.

Order of Operations Grade 5

By Grade 5, students apply the full order of operations across all four arithmetic operations with parentheses. Multi-step expressions like 3 x (4 + 2) – 5 become standard. In order of operations Grade 5, PEMDAS or BODMAS are introduced as memory tools.

Order of Operations Grade 6

At Grade 6, exponents are introduced into order of operations problems. Students work with expressions like 2³ + 4 x 5 and learn to place exponents in the correct sequence, after parentheses, but before multiplication and division. Students are taught the order of operations Grade 6, which requires multi-step expressions to be common.

Order of Operations Grade 7

In the order of operations Grade 7 students work with integers (positive and negative numbers), fractions and decimals within order of operations problems. Expressions become complex, and students work with nested parentheses and brackets. At this level, students also get to learn about distributive property in the context of grouped expressions.

Practice Problems

Now that we’ve learnt about order of operations, try solving the expressions using order of operations.

6 + 4 × 3
(8 + 2) × 5
12 ÷ 3 + 6
7 + 3²
8 ÷ 2(2 + 2)
(5 − 2)² + 4 × 3
18 ÷ (3 + 3) × 2
50 − 4 × 3 + 2²

Answers

Problem	Working	Answer
6 + 4 x 3	Multiply first: 4 x 3 = 12, then 6 + 12	18
(8 + 2) x 5	Parentheses: 8 + 2 = 10, then 10 x 5	50
12 ÷ 3 + 6	Divide first: 12 ÷ 3 = 4, then 6 + 4	10
7 + 3^2	Exponent: 3^2 = 9, then 7 + 9	16
8 ÷ 2(2 + 2)	P: (2 + 2)= 4, then left to right: 8 ÷ 2, then 4 x 4	16
(5 – 2)^2 + 4 x 3	P: (5 -2) =3, E: 3^2 = 9, M: 4 x 3 = 12, A: 9 + 12	21
18 ÷ (3 +3) x 2	P: (3 +3)= 6, left to right: 18 ÷6 = 3, then 3 x 2	6
50 – 4 x 3 + 2^2	E: 22= 4, M: 4 x 3= 12, left to right: 50 – 12 + 4	42

Frequently Asked Questions

What is the BODMAS rule?

BODMAS rule stands for Brackets, Orders, Division, Multiplication, Addition and Subtraction. This acronym is popularly used in the UK and India to help students understand the standard order of mathematical operations in the correct sequence. In simple words, the sequence in which parts of a mathematical expression must be solved first to get the correct answer.

Which is correct, PEMDAS or BODMAS?

Both are correct. PEMAS is used in the US, and BODMAS in the UK and India. Both these acronyms represent the same mathematical rule. Different letters are used in different countries, but the underlying order of operations is universal.

What is 8 ÷ 2(2+2)?

16. This is one of the most debated expressions online. Let’s solve it with the correct order of operations:

Step 1 Parentheses: (2 + 2) = 4

Step 2 left to right: 8 ÷ 2 = 4, then 4 x 4 = 16

The answer is 16. The confusion comes from treating 2(4) as a grouped item, but standard PEMDAS/BODMAS evaluates division and multiplication from left to right.

Is it PEMDAS or PEDMAS?

The standard acronym is PEMDAS- Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. PEDMAS is merely a spelling mistake.

Why does BODMAS use the word “orders”?

In the UK math, powers and roots are referred to as orders instead of exponents. So the ‘O ’ in BODMAS stands for Order, which covers the same operations as ‘E’ (Exponents) in PEMDAS.