Order of Operations

Why Order of Operations Matters

Without agreed rules, the same expression could give many different answers depending on the order you work in. BODMAS gives everyone the same rulebook so that any expression has exactly one correct value.

Example: 3 + 4 × 2

• If you add first: (3 + 4) × 2 = 7 × 2 = 14
• If you multiply first: 3 + (4 × 2) = 3 + 8 = 11

The correct answer is 11 — multiplication comes before addition in BODMAS.

Key Terms

• BODMAS — the UK acronym for order of operations (Brackets, Orders, Division, Multiplication, Addition, Subtraction)
• PEMDAS — the US equivalent (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction)
• Expression — a mathematical phrase combining numbers and operations (e.g. 3 + 4 × 2)
• Evaluate — to work out the value of an expression

BODMAS in Full

Letter	Operation	Examples
B	Brackets (parentheses)	(3 + 4), (2 × 5 - 1)
O	Orders (powers and roots)	4², √25, 2³
D	Division	12 ÷ 4, 15/3
M	Multiplication	3 × 7, 5(4)
A	Addition	6 + 9
S	Subtraction	10 - 3

Critical rule: D and M have equal priority — work left to right when both appear. Critical rule: A and S have equal priority — work left to right when both appear.

Worked Examples

Example 1: 3 + 4 × 2

• No brackets or orders
• Multiply: 3 + 8
• Add: 11 ✓

Example 2: (3 + 4) × 2

• Brackets first: 7 × 2
• Multiply: 14 ✓

Example 3: 20 - 4² + 3

• Orders first: 20 - 16 + 3
• Left to right (subtraction then addition): 4 + 3 = 7 ✓

Example 4: 18 ÷ (2 + 1) × 3

• Brackets first: 18 ÷ 3 × 3
• Division and multiplication left to right: 6 × 3 = 18 ✓

Example 5: 5 + 3² × 2 - 1

• Orders first: 5 + 9 × 2 - 1
• Multiply: 5 + 18 - 1
• Left to right (add then subtract): 23 - 1 = 22 ✓

Nested Brackets

When expressions have brackets inside brackets, work from the innermost set outward:

Example: 2 × ((3 + 1) × 5 - 2)

• Innermost: 2 × (4 × 5 - 2)
• Next bracket: 2 × (20 - 2) = 2 × 18
• Multiply: 36 ✓

In some textbooks, square brackets [ ] and curly braces are used to make nested brackets clearer, but mathematically they all work the same way.

The "O" in BODMAS: Orders

"Orders" means powers (exponents) and roots:

• 4² = 16 (4 squared)
• 2³ = 8 (2 cubed)
• √25 = 5 (square root)

Example: 2 × (3 + 4)² - 5

• Brackets first: 2 × 7² - 5
• Orders: 2 × 49 - 5
• Multiply: 98 - 5
• Subtract: 93 ✓

Equal Priority: Left to Right

When two operations have the same priority, always work left to right:

Division and Multiplication (same level):

• 12 ÷ 3 × 2 → (12 ÷ 3) × 2 = 4 × 2 = 8 (not 12 ÷ 6 = 2)

Addition and Subtraction (same level):

• 10 - 3 + 2 → (10 - 3) + 2 = 7 + 2 = 9 (not 10 - 5 = 5)

BODMAS vs PEMDAS

Both acronyms describe the same rules:

• Brackets = Parentheses
• Orders = Exponents
• Division and Multiplication (same level, left to right)
• Addition and Subtraction (same level, left to right)

There is no difference in the actual order — just different letter names.

Common Mistakes

• Mistake: doing addition before multiplication: 3 + 4 × 2 = 14. Fix: multiplication always comes before addition unless brackets say otherwise.
• Mistake: in 20 - 16 + 3, doing the addition first to get 20 - 19 = 1. Fix: work left to right: 20 - 16 = 4, then 4 + 3 = 7.
• Mistake: with 18 ÷ 3 × 3, doing multiplication first to get 18 ÷ 9 = 2. Fix: division and multiplication are equal priority — go left to right: 18 ÷ 3 = 6, then 6 × 3 = 18.
• Mistake: forgetting that the "O" in BODMAS applies to square roots as well as powers.

Tips and Tricks

• When in doubt, add brackets to make the intended order completely explicit — extra brackets never change the answer when placed correctly, but they make your working much clearer.
• Work through expressions one operation at a time and rewrite the expression at each step.
• Write out all steps: skipping steps is the most common cause of order-of-operations errors.
• A handy check: if you get two different answers when you try a calculation two different ways, one of them violates BODMAS — find out which.