Order of Operations
Master BODMAS to correctly evaluate expressions with multiple operations, including integers, fractions, and mixed calculations.
What is BODMAS?
When a calculation has more than one operation, we need to agree on the order to follow. BODMAS gives us the rules everyone uses so we all get the same answer.
B
Brackets
( ) first
O
Orders
Powers & roots
D
Division
left to right
M
Multiplication
left to right
A
Addition
left to right
S
Subtraction
left to right
Note: Division and Multiplication have equal priority — work left to right. Same for Addition and Subtraction.
Why Order Matters
Without an agreed order, the same expression can give different answers. Compare these two approaches for 3 + 4 × 2:
Wrong (left to right):
3 + 4 = 7, then 7 × 2 = 14 ✗
Correct (BODMAS):
4 × 2 = 8 first, then 3 + 8 = 11 ✓
Using Brackets
Brackets tell us to compute what is inside them first, before anything else. They let us change the natural order of operations.
Compare: 4 + 6 × 3 vs (4 + 6) × 3
Without brackets:
6 × 3 = 18
4 + 18 = 22
With brackets:
(4 + 6) = 10
10 × 3 = 30
Nested Brackets: Work from the inside out
Example: 2 × [3 + (8 − 2)]
Step 1: Inner brackets: (8 − 2) = 6
Step 2: Outer brackets: [3 + 6] = 9
Step 3: Multiply: 2 × 9 = 18
BODMAS with Fractions and Negatives
The same BODMAS rules apply when fractions and negative numbers are involved. Treat fraction bars like brackets — calculate the numerator and denominator separately first.
Example: Evaluate −3 + 2 × 4 − (6 ÷ 2)
Step 1 (Brackets): (6 ÷ 2) = 3
Step 2 (Multiply): 2 × 4 = 8
Step 3 (Add/Subtract left to right): −3 + 8 − 3 = 2
Key Vocabulary
BODMAS
An acronym for the correct order of operations: Brackets, Orders, Division, Multiplication, Addition, Subtraction.
Orders
The 'O' in BODMAS stands for powers and roots, e.g., 3² = 9 or √16 = 4.
Expression
A mathematical phrase with numbers and operations but no equals sign, e.g., 3 + 4 × 2.
Evaluate
To calculate the value of an expression by following the order of operations.
Worked Examples
Evaluate: 5 + 3 × 4 − 2
Step 1 (M): 3 × 4 = 12
Step 2 (A/S): 5 + 12 − 2 = 15
Answer: 15
Evaluate: (8 − 3) × 2 + 3²
Step 1 (B): (8 − 3) = 5
Step 2 (O): 3² = 9
Step 3 (M): 5 × 2 = 10
Step 4 (A): 10 + 9 = 19
Evaluate: 20 ÷ 4 + 2 × (−3)
Step 1 (D): 20 ÷ 4 = 5
Step 2 (M): 2 × (−3) = −6
Step 3 (A): 5 + (−6) = 5 − 6 = −1
Knowledge Check
Select the correct answer for each question.
Question 1
Evaluate: 3 + 5 × 2
Question 2
Evaluate: (2 + 3) × 4
Question 3
Evaluate: 2² + 12 ÷ 4 − 1
Question 4
Evaluate: 10 − 2 × 3 + 1
Question 5
Which step should you do first in: 4 + (9 − 3) ÷ 2 × 5?
Key Concepts Summary
- ●BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, Subtraction.
- ●Always calculate brackets first, then powers/roots, then multiply/divide (left to right), then add/subtract (left to right).
- ●Division and Multiplication have equal priority; work from left to right when both appear.
- ●For nested brackets, work from the innermost set outwards.