Order of Operations – BODMAS / PEMDAS / BIDMAS
When a calculation contains more than one operation, you cannot just work from left to right. There is a strict, universally agreed set of rules that tells you which operation to perform first — and it is one of the most important rules in the whole of mathematics, because without it the same string of numbers and symbols could honestly mean several different things. Consider 3 + 4 × 2: read strictly left to right it looks like 7 × 2 = 14, but the accepted rule says multiply first, giving 3 + 8 = 11. Only one of those is correct, and the entire purpose of a fixed order of operations is to make sure everybody — students, teachers, calculators, and computer programs alike — always lands on the same answer.
These rules go by several names depending on where you learned maths: BODMAS in the UK and much of the Commonwealth, PEMDAS (often remembered as “Please Excuse My Dear Aunt Sally”) in the United States, and BIDMAS in some other countries. All three describe exactly the same priority order — they are simply different mnemonics built from slightly different words for the same mathematical ideas.
What Does BODMAS Stand For?
| Letter | Stands for | Example |
|---|---|---|
| B | Brackets (parentheses) | (3 + 4) |
| O | Orders (powers and roots) | 2³, √9 |
| D | Division | 12 ÷ 4 |
| M | Multiplication | 3 × 5 |
| A | Addition | 7 + 2 |
| S | Subtraction | 10 − 3 |
Important: Division and Multiplication have equal priority and are worked left to right. The same applies to Addition and Subtraction. This is the single most misunderstood part of BODMAS — the order the letters are written in does not mean division always happens before multiplication, or addition always happens before subtraction.
BODMAS, PEMDAS, and BIDMAS Compared
These three mnemonics are taught in different countries but describe an identical hierarchy — each letter in one column lines up exactly with the letter in the same row of the others.
| BODMAS (UK) | PEMDAS (US) | BIDMAS (also UK/elsewhere) | What it means |
|---|---|---|---|
| B – Brackets | P – Parentheses | B – Brackets | ( ), [ ], { } — always evaluated first |
| O – Orders | E – Exponents | I – Indices | Powers and roots — three different words for the same idea |
| D – Division | M – Multiplication | D – Division | Equal precedence — left to right |
| M – Multiplication | D – Division | M – Multiplication | Equal precedence — left to right |
| A – Addition | A – Addition | A – Addition | Equal precedence — left to right |
| S – Subtraction | S – Subtraction | S – Subtraction | Equal precedence — left to right |
Notice that PEMDAS writes M before D, and BODMAS/BIDMAS write D before M — yet all three are taught identically in practice: multiplication and division are worked in whichever order they appear, left to right. The mnemonics were built to be pronounceable words, not to imply one operation always outranks the other.
Step-by-Step Method
- Work out everything inside brackets first.
- Evaluate any powers or roots.
- Work any division and multiplication from left to right.
- Finally, work any addition and subtraction from left to right.
Worked Examples
Calculate 3 + 4 × 2
Step 1: No brackets or powers.
Step 2: Multiply first: 4 × 2 = 8
Step 3: Add: 3 + 8 = 11
Calculate (5 + 3) × 2 − 4 ÷ 2
Step 1: Brackets: (5 + 3) = 8
Step 2: 8 × 2 − 4 ÷ 2
Step 3: Multiply and divide left to right: 8 × 2 = 16, then 4 ÷ 2 = 2
Step 4: Subtract: 16 − 2 = 14
Calculate 2³ + (12 − 4) ÷ 4 × 3
Step 1: Brackets: (12 − 4) = 8 → gives 2³ + 8 ÷ 4 × 3
Step 2: Powers: 2³ = 8 → gives 8 + 8 ÷ 4 × 3
Step 3: Division and multiplication left to right: 8 ÷ 4 = 2, then 2 × 3 = 6
Step 4: Addition: 8 + 6 = 14
Calculate √16 + 3 × 2²
Step 1: Orders rank above multiplication and addition, so evaluate both the root and the power first: √16 = 4 and 2² = 4.
Step 2: The expression is now 4 + 3 × 4.
Step 3: Multiply before adding: 3 × 4 = 12.
Step 4: Add: 4 + 12 = 16
Calculate 24 ÷ 4 × 3
Step 1: Division and multiplication are equal precedence, so BODMAS does not mean “always divide before you multiply” — it means work through them in the order they are written, left to right.
Step 2: The leftmost operation is division: 24 ÷ 4 = 6.
Step 3: Now multiply: 6 × 3 = 18
Common mistake: multiplying first instead (4 × 3 = 12, then 24 ÷ 12 = 2) gives the wrong answer of 2, because it ignores the left-to-right rule.
Calculate 2 × (3 + (4 × 2))
Step 1: Work from the innermost bracket outward: (4 × 2) = 8.
Step 2: The expression is now 2 × (3 + 8).
Step 3: Finish the outer bracket: 3 + 8 = 11.
Step 4: Multiply: 2 × 11 = 22
Common Mistakes
- Working strictly left to right without checking priorities. 3 + 4 × 2 is 11, not 14 — multiplication must happen before the addition, regardless of which one is written first.
- Treating “D before M” and “A before S” literally. They have equal priority within each pair and must be worked left to right, not in the order the mnemonic’s letters happen to appear.
- Evaluating powers or roots after multiplication instead of before it. 3 × 2² means 3 × 4 = 12, not 6² = 36.
- Forgetting that operations inside nested brackets must be resolved from the innermost bracket outwards. Always finish the deepest bracket pair before touching anything outside it.
- Assuming every calculator applies BODMAS automatically. Simple four-function calculators sometimes evaluate strictly left to right with no operator priority at all, which can produce a different — and mathematically incorrect — result from the same button presses.
Where Order of Operations Actually Gets Used
BODMAS is not just a classroom exercise. Every device or formula that performs a calculation on your behalf relies on this exact same hierarchy.
- Scientific and graphing calculators apply the full BODMAS hierarchy automatically, which is why typing 3 + 4 × 2 correctly returns 11.
- Spreadsheet formulas in Excel or Google Sheets (for example =3+4*2) follow BODMAS/PEMDAS exactly, and brackets are used constantly to force a particular calculation order.
- Programming languages such as Python, JavaScript, and C all define an “operator precedence” table that is, at its core, the same BODMAS hierarchy extended to cover additional operators.
- Algebra and formulae you will meet later — substituting numbers into an equation always relies on BODMAS to evaluate the result correctly.
Key Takeaways
- BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, Subtraction — PEMDAS and BIDMAS describe the identical hierarchy under different names.
- Always resolve brackets first (innermost first if nested), then powers/roots, then division/multiplication, then addition/subtraction.
- Division and multiplication share equal priority; so do addition and subtraction — work these left to right, not in mnemonic-letter order.
- Genuinely ambiguous expressions exist — when in doubt, add extra brackets to make your intended order explicit.
- Every calculator, spreadsheet, and programming language you use evaluates expressions using this same hierarchy.
Practice: Evaluate with BODMAS
A fresh expression is generated below every time — sometimes with brackets, sometimes without. Work it out using BODMAS. If you get it wrong (or leave it blank), we'll go through it step by step.