# Order of Operations (BODMAS)

> Brackets, then powers, then multiply and divide, then add and subtract, so everyone gets the same answer.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/order-of-operations/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: Two calculators, two answers
Sam and Priya both type $2 + 3 \times 4$ into calculators. Sam's cheap calculator shows 20. Priya's scientific calculator shows 14.

Who is right? Sam's calculator did $2 + 3 = 5$, then $5 \times 4 = 20$. Priya's did $3 \times 4 = 12$ first, then $2 + 12 = 14$.

Mathematicians all over the world agree on one order, so everyone gets the same answer. In that order, multiplying comes before adding. Priya's answer of 14 is correct.

**In short:** The order of operations tells you which part of a calculation to do first: Brackets, Orders (powers and roots), Division and Multiplication, then Addition and Subtraction (BODMAS). Multiplying and dividing go left to right, and so do adding and subtracting.

## Before you start

- Your times tables up to $12 \times 12$, and division facts.
- A **square** is a number times itself: $3^2 = 9$. A **square root** goes backwards: $\sqrt{64} = 8$ because $8 \times 8 = 64$.

## BODMAS

### Key idea: The order
1. **B**rackets: work out anything inside brackets first.
2. **O**rders: powers and square roots, like $3^2$ or $\sqrt{64}$.
3. **D**ivision and **M**ultiplication: from **left to right**.
4. **A**ddition and **S**ubtraction: from **left to right**.

**Mind map**

```text
BODMAS
  Brackets
    ( ) first
    Inner brackets before outer
  Orders
    Powers $3^2$
    Roots $\sqrt{64}$
  Divide and Multiply
    Same level
    Left to right
  Add and Subtract
    Same level
    Left to right
```

### Watch out: D does not beat M
Division and multiplication are on the **same level**. Do whichever comes first from the left. The same goes for addition and subtraction.

$24 - 12 + 3$: left to right gives $12 + 3 = 15$. Doing the plus first would give $24 - 15 = 9$, which is wrong.

### Worked example: Multiply and divide before add and subtract
$24 + 3 \div 3 + 2$

Divide first: $3 \div 3 = 1$

Then left to right: $24 + 1 + 2 = 27$

### Worked example: Brackets first
$(24 - 12) \div 3 - 2$

Brackets: $12$. Then $12 \div 3 = 4$. Then $4 - 2 = 2$.

### Worked example: A power inside brackets
$1200 \div (2 \times 8 + 3^2)$

Inside the brackets, power first: $3^2 = 9$. Then $2 \times 8 = 16$. Then $16 + 9 = 25$.

Finally $1200 \div 25 = 48$.

## Brackets inside brackets

Square brackets $[\;]$ work just like round ones. When brackets are inside other brackets, start with the **innermost** pair.

### Worked example: Inside out
$11 \times [48 \div (4 + 2)]$

Inner brackets: $4 + 2 = 6$

Square brackets: $48 \div 6 = 8$

Last: $11 \times 8 = 88$

### Worked example: A longer one
$[(3 \times 16) - 3 \times 6 - 2] \div 7$

Inside the square brackets: $48 - 18 - 2 = 28$

Then $28 \div 7 = 4$

### Exam tip: Underline each step
Rewrite the whole line after every step, with the part you just worked out replaced by its answer. It is slower, but you will not lose track of a number.

## Turning words into a calculation

Selective test questions often hide the operations in words.

- "the **sum** of" means add
- "the **difference** between" means subtract (bigger minus smaller)
- "the **product** of" means multiply
- "the **quotient** of" means divide
- "**increase** by" means add, "**decrease** by" means subtract
- "a **dozen**" is 12

### Worked example: Words first
"Increase the product of 14.5 and 2 by $\sqrt{64}$."

Product: $14.5 \times 2 = 29$. $\sqrt{64} = 8$. Increase: $29 + 8 = 37$.

### Challenge: Selective level: clever shortcuts
$8 \times 37 \times 125 - 1000$

Multiplying can be done in any order, so pair $8 \times 125 = 1000$ first. Then $1000 \times 37 = 37\,000$, and $37\,000 - 1000 = 36\,000$.

"The total of 5 numbers is half of 10 dozen. What is their average?" Ten dozen is 120, half is 60, and $60 \div 5 = 12$.

## Quick summary

- BODMAS: Brackets, Orders (powers and roots), Division and Multiplication, Addition and Subtraction.
- $\times$ and $\div$ are equal partners: go left to right. So are $+$ and $-$.
- Work from the innermost brackets outwards.
- Learn the words: sum, difference, product, quotient, increase, decrease.
- Look for friendly pairs like $8 \times 125 = 1000$ to make sums easier.

## Quiz

Try these questions from the chapter quiz.

1. $5 + 4 - 2 \times 1 = $
   - A. 14
   - B. 7
   - C. 9
   - D. 3
   - Answer: B. 7. Multiply first: $2 \times 1 = 2$. Then $5 + 4 - 2 = 7$.

2. $12 \div 4 + 9 \div 3 = $
   - A. 4
   - B. 6
   - C. 1
   - D. 7
   - Answer: B. 6. Divide first: $3 + 3 = 6$.

3. $(5 + 4) \div (2 + 1) = $
   - A. 3
   - B. 6
   - C. 9
   - D. 7
   - Answer: A. 3. Brackets: $9 \div 3 = 3$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/order-of-operations/#quiz