Order of Operations: Making Sense of PEMDAS and BODMAS

What Does PEMDAS Mean?

PEMDAS is the acronym most US schools teach, and its whole job is to pin the correct sequence in your head. Each letter marks a stage in that sequence:

Letter	Stands For	What You Do
P	Parentheses	Clear out anything inside brackets first
E	Exponents	Handle powers and roots next
M	Multiplication	Same rank - move left to right, whichever shows up first
D	Division
A	Addition	Same rank - move left to right, whichever shows up first
S	Subtraction

You've likely run into "Please Excuse My Dear Aunt Sally" as a memory trick for this list. It's a little corny, but it's stuck around because it's easier to recall than six bare letters.

BODMAS vs. PEMDAS - What's Actually Different

Nothing, really. BODMAS - used across the UK, India, Australia, and much of the rest of the world - spells out Brackets, Orders, Division, Multiplication, Addition, Subtraction. Relabel the terms and you get the identical sequence: "Orders" is just another word for exponents, and "Brackets" is the British term for parentheses.

So if you were taught BODMAS and someone else learned PEMDAS, neither of you is wrong - you just picked up the same rule under different labels.

Revisiting the Opening Example

Look again at 8 + 2 × 3.

Work strictly left to right and you'll compute 8 + 2 = 10 first, then 10 × 3 = 30 - which is the wrong answer.

Work it correctly by handling the multiplication before anything else: 2 × 3 = 6, then 8 + 6 = 14.

It looks like a minor gotcha, yet it snags people over and over - particularly anyone entering it into a bare-bones four-function calculator, since those devices simply process numbers in the order you key them in, with no concept of priority at all.

Working Through Each Stage

Stage 1: Resolve the Parentheses

Whatever sits inside brackets gets fully worked out before anything outside them is touched. When brackets are nested one inside another, start with the deepest pair and move outward from there.

Stage 2: Deal With Exponents and Roots

With the brackets cleared, powers and square roots come next. This step is where a surprising number of people go wrong without noticing.

Stage 3: Multiplication and Division, Left to Right

Here's the part that trips up nearly everyone eventually: multiplication doesn't outrank division. They're tied. Whichever one you reach first while scanning left to right gets done first.

This is arguably the most common misreading of PEMDAS out there. The letter M coming before D in the acronym is just alphabetical bookkeeping - it doesn't mean multiplication wins ties. What actually decides the order is where each operation sits in the expression.

Stage 4: Addition and Subtraction, Left to Right

The same rule applies one level down. Addition and subtraction are equally ranked, so you just work through them in the order they appear.

Full Worked Examples

A few complete problems, broken down one move at a time:

Problem 1:   5 + 3² × (4 - 1)

Problem 2:   20 ÷ 4 + 6 × 2 - 8

Problem 3:   (2 + 3)² - 4 × 2

Errors to Watch Out For

Mistake	What Goes Wrong	The Fix
Adding before multiplying	2 + 3 × 4 = 20 ✗	3 × 4 = 12, then + 2 = 14 ✓
Assuming M outranks D	12 ÷ 2 × 3: doing 2 × 3 first = 2 ✗	12 ÷ 2 = 6, then × 3 = 18 ✓
Skipping the left-to-right rule	9 - 4 + 1 = 4 ✗	9 - 4 = 5, then + 1 = 6 ✓
Misreading an implied bracket	-3² treated as (-3)² = 9 ✗	-(3²) = -9 ✓

Do Calculators Handle This Automatically?

A proper scientific calculator applies order of operations on its own - key in the expression exactly as written and trust the output. Basic four-function calculators usually can't manage that. Curious about yours? Type in 2 + 3 × 4. Getting 20 back means it's working strictly left to right with no priority logic; getting 14 means it's applying the rules the way it should.

Want a second opinion on a tricky calculation? SolverCalc's calculator parses the whole expression at once instead of reading it left to right like a basic model would.

Why Have a Fixed Order in the First Place

This isn't an arbitrary hurdle schools invented. Without an agreed sequence, the same string of digits and symbols could be read multiple conflicting ways depending on who's looking at it - and in fields like engineering, software, or finance, that kind of ambiguity is precisely what causes expensive mistakes. Order of operations exists so a calculation resolves to one answer, regardless of who's solving it.

It's also built into how programming languages work. JavaScript, Python, and nearly every other language apply operator precedence rules that line up with PEMDAS, so getting this down now carries over well past math class.

Quick Reference Table

Priority	Operation	Direction
1st	Parentheses / Brackets	Inside → out
2nd	Exponents / Orders / Roots	Right → left
3rd	Multiplication and Division (equal rank)	Left → right
4th	Addition and Subtraction (equal rank)	Left → right

Where That Leaves You

None of this is conceptually difficult - it just needs enough repetition to become automatic. When people slip up, it's rarely a misunderstanding of the idea itself; it's forgetting that multiplication and division outrank addition and subtraction, or forgetting that M and D are tied with each other (and A and S are tied too). Push through enough practice problems and it eventually stops requiring conscious thought.

If you take away just three things: brackets always go first, exponents come next and beat everything except brackets, and multiplication and division are equal partners with neither one automatically ahead of the other.