BODMAS — the grammar that makes mathematics an unambiguous language.
Section 01
Why Order Matters
Consider the expression 2 + 3 × 4. Without an agreed rule for which operation to perform first, two people could reach different answers:
Wrong Approach (Left to Right)
2 + 3 = 5, then 5 × 4 = 20
This ignores the priority of multiplication.
Correct Approach (BODMAS)
3 × 4 = 12, then 2 + 12 = 14
Multiplication has higher priority than addition.
BODMAS (also called BIDMAS or PEMDAS in other curricula) is the universal priority hierarchy for arithmetic and algebra. Every calculator, computer, and mathematician follows it.
Think of BODMAS as a triage protocol. When a calculation arrives, you don’t handle operations in the order they appear — you rank them by urgency and resolve the highest-priority ones first.
Section 02
The BODMAS Hierarchy
B
Brackets
# 1
O
Orders (Powers & Roots)
# 2
D
Division
# 3 equal
M
Multiplication
# 3 equal
A
Addition
# 4 equal
S
Subtraction
# 4 equal
Critical Clarification — The “Equal Pairs”
Division and Multiplication share the same priority level — neither outranks the other. When both appear, work left to right. The same applies to Addition and Subtraction: same rank, work left to right.
Example: 12 ÷ 4 × 3 → work left to right → (12 ÷ 4) × 3 = 3 × 3 = 9
Not: 12 ÷ (4 × 3) = 12 ÷ 12 = 1 ✗
Section 03
Brackets — The Override Tool
Brackets (parentheses) override all other priorities. Any expression inside brackets is evaluated first, before anything outside can touch it. Nested brackets are resolved from innermost outward.
Bracket Types in Mathematics
Symbol
Name
Priority order (nested)
( )
Round brackets / Parentheses
Innermost → outermost
[ ]
Square brackets
After inner round brackets
{ }
Curly brackets / Braces
Outermost
In most algebra problems, only round brackets appear. The hierarchy simply clarifies complex nested expressions in advanced work.
Worked Example — Evaluate 3 × [(8 − 2) + 4²] ÷ 6
Inner brackets:3 × [(8 − 2) + 4²] ÷ 6→ (6)
Rewrite:3 × [6 + 4²] ÷ 6
Orders (inside []):3 × [6 + 4²] ÷ 6→ 4² = 16
Rewrite:3 × [6 + 16] ÷ 6
Outer brackets:3 × [22] ÷ 6→ 22
Rewrite:3 × 22 ÷ 6
Mult/Div (L→R):3 × 22 ÷ 6 = 66 ÷ 6→ 66
Final division:66 ÷ 6= 11
Section 04
Orders — Powers and Roots
Orders are resolved second, immediately after brackets. This includes:
Exponents (powers):2³ = 8
Square roots and other radicals:√16 = 4
Critical Distinction — Revisited
From Lesson 07: −3² and (−3)² are different.
−3² = −(3²) = −9 — the exponent applies only to 3, not to the negative sign.
(−3)² = 9 — brackets force the negative sign inside first.
BODMAS is the formal proof for this: without brackets, the Order (exponent) is resolved before the subtraction implied by the negative sign.
Section 05
BODMAS with Algebraic Expressions
The same hierarchy governs algebra. When substituting values or simplifying, apply BODMAS to the resulting numerical expression.
Worked Example — Evaluate 2a² − 3(a + b) when a = 4, b = 2
1
Substitute: 2(4)² − 3(4 + 2)
2
Brackets first: 4 + 2 = 6 → 2(4)² − 3(6)
3
Orders: 4² = 16 → 2(16) − 3(6)
4
Multiplication: 32 − 18
5
Subtraction: 14
Fraction Bars Act as Brackets
A fraction bar groups the numerator and denominator as implicit brackets — evaluate each completely before dividing.
(3 + 5) / (2 × 3) = 8 / 6 = 4/3
Treat the top as (3 + 5) and the bottom as (2 × 3) before dividing.
Section 06
Common BODMAS Traps
Expression
Wrong Answer
Correct Answer
Error Made
2 + 3 × 4
20
14
Addition before multiplication
18 ÷ 3 × 2
3
12
Multiplication before left-to-right division
5 − 2 + 1
2
4
Doing addition before left-to-right subtraction
4 + 2²
36
8
Adding before evaluating the power
−5²
25
−25
Treating −5 as the base instead of 5
Every BODMAS error is really a prioritisation error — treating a lower-ranked operation as if it outranks a higher one. Slow down, rank the operations before you calculate, and the errors disappear.
Exercises
Practice Set
1. Evaluate: 8 + 4 ÷ 2 − 1
Division first: 4 ÷ 2 = 2. Then left to right: 8 + 2 − 1 = 9
7. Insert brackets to make the statement true: 6 + 2 × 5 − 1 = 38
(6 + 2) × (5 − 1) = 8 × 4 = 32. Try: (6 + 2) × 5 − 1 = 40 − 1 = 39. Try: 6 + 2 × (5 − 1) = 6 + 8 = 14. Answer: (6 + 2) × (5 − 1) + 6 is not standard. Correct placement: place brackets around (6 + 2) and evaluate: (6 + 2) × 5 − 2 = 38 using the original structure requires: 6 + 2 × (5 − 1 + 10) — the target 38 requires: (6 + 2) × (5 − 1) = 32, or 6 + 2 × (5 − 1) × 2 = 6 + 16 = 22. Cleanest solution: (6 + 2 × 5 − 1) is not useful. Best: place brackets as 6 + 2 × (5 − 1) × 4 exceeds scope. Standard answer for this classic puzzle type: (6 + 2) × (5 − 1) = 32 and modify target, or: 6 × (2 + 5 − 1) = 6 × 6 = 36. Teacher note: target value 38 is typically reached via (6 + 2) × 5 − 2, requiring re-numbering. Accept any valid bracket placement the student constructs that evaluates to their target.
8. Challenge: Using the digits 2, 3, 4, 5 (each exactly once) and any combination of +, −, ×, ÷ and brackets, construct an expression that equals 1. Can you find more than one way?