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Year 7 Mathematics · Number and Algebra · 9m read

Order of Operations (BIDMAS)

Ever wondered why 2 + 3 × 4 doesn't equal 20? BIDMAS is the set of rules that tells you the order to do operations: Brackets, Indices, Division and Multiplication, then Addition and Subtraction. Think of it like following a recipe—do things in the wrong order and you'll mess up the result. Mastering BIDMAS means you'll solve complex expressions correctly every time, which is essential for algebra, physics, and pretty much any maths you'll encounter in VCE.

Learning objective: Apply BIDMAS to evaluate multi-step expressions correctly.

By the end of this lesson you can:

  • I recall the BIDMAS order.
  • I evaluate expressions with brackets, powers, ×, ÷, +, −.
  • I identify and fix common BIDMAS errors.

BIDMAS — the order hierarchy

B — Brackets ()
I — Indices powers and roots e.g. 3², √9
D — Division ÷ ← same priority as M: LEFT to RIGHT
M — Multiplication ×
A — Addition + ← same priority as S: LEFT to RIGHT
S — Subtraction −

KEY FACTS:
• D and M have EQUAL priority. Work LEFT to RIGHT when both appear.
• A and S have EQUAL priority. Work LEFT to RIGHT when both appear.
• Brackets ALWAYS first — even nested brackets (innermost first).

EXAMPLES:
6 + 2 × 3 = 6 + 6 = 12 (× before +)
(6 + 2) × 3 = 8 × 3 = 24 (brackets first → different answer!)
12 ÷ 4 × 2 = 3 × 2 = 6 (left to right for ÷ and ×)
20 − 3 + 4 = 17 + 4 = 21 (left to right for − and +)

Traffic light analogy. BIDMAS is like traffic rules. Brackets are emergency vehicles — they always go first. Indices are traffic lights — second priority. Multiplication/Division are like give-way signs. Addition/Subtraction? Park and wait your turn.

Common BIDMAS mistakes

  • Left-to-right tie: 12 ÷ 4 × 2 is NOT 12 ÷ 8. Do left to right: (12 ÷ 4) × 2 = 3 × 2 = 6.
  • Subtraction tie: 10 − 3 + 2 = 9, NOT 5. Left to right: (10−3)+2 = 7+2 = 9.
  • Nested brackets: 2 × (3 + (4 − 1)): solve innermost (4−1)=3 first, then (3+3)=6, then ×2=12.

BIDMAS order

  • Order: B \,>\, I \,>\, D/M \,>\, A/S
  • Same precedence: \text{evaluate left to right}

Quick reference

| Order | Letter | Means |
|---|---|---|
| 1 | B | Brackets |
| 2 | I | Indices (powers, roots) |
| 3 | D M | Division and multiplication, left to right |
| 4 | A S | Addition and subtraction, left to right |
| Example | 3 + 4 × 2² | = 3 + 4 × 4 = 3 + 16 = 19 |

Worked example (Medium)

Evaluate: 20 − 4 × (3 + 2)²

  1. B — Brackets. 3 + 2 = 5
  2. I — Indices. 5² = 25
  3. M — Multiply. 4 × 25 = 100
  4. S — Subtract. 20 − 100 = −80

Answer: −80

Worked example (Hard)

Evaluate: $\frac{(8 - 2)^2}{3} + 4 \times (10 - 7)$

  1. B — Brackets. $(8 - 2) = 6$ and $(10 - 7) = 3$.
  2. I — Indices. $6^2 = 36$.
  3. D/M — Division and Multiplication. $36 \div 3 = 12$ and $4 \times 3 = 12$.
  4. A — Addition. $12 + 12 = 24$.

Answer: $24$

Common mistakes

  • Doing addition before multiplication Students read left to right and add before multiplying: $3 + 4 \times 2 = 14$ (wrong) instead of $3 + 8 = 11$ (correct). Fix: Always scan the ENTIRE expression for × and ÷ BEFORE doing + and −. Multiplication and division have higher priority.
  • Treating ÷ and × as sequential instead of equal priority Students do all division first, then all multiplication, instead of working left to right. Fix: ÷ and × have EQUAL priority. Work left to right: $12 ÷ 4 \times 2 = 3 \times 2 = 6$, NOT $12 ÷ 8 = 1.5$.
  • Forgetting to apply indices before multiplication In $2 \times 3^2$, students calculate $2 \times 3 = 6$, then $6^2 = 36$ instead of $3^2 = 9$, then $2 \times 9 = 18$. Fix: Indices come BEFORE multiplication in BIDMAS. Always resolve powers first, then multiply.

Why BIDMAS exists — and what happens without it

Without an agreed order of operations, the same expression would give different answers depending on who calculated it. Consider $2 + 3 \times 4$. Without rules: some people get $20$ (adding first), others get $14$ (multiplying first). Mathematics cannot function if the same expression has multiple valid answers.

BIDMAS (or PEMDAS/BODMAS in other countries) is not a law of nature — it is a convention agreed upon by mathematicians worldwide to ensure consistency. It works because it reflects how mathematical expressions are intended to be read.

Interestingly, computers and calculators follow BIDMAS internally. When you type $2 + 3 \times 4$ into a scientific calculator, it returns $14$ because it is programmed with the correct order. A basic four-function calculator, however, might return $20$ because it processes left to right — which is why scientific calculators exist!

Programmers must understand order of operations deeply because a single misplaced bracket in code can crash an entire program or produce catastrophically wrong results — from financial calculations to spacecraft trajectories.

Practice questions

  1. Evaluate: 3 + 4² × 2

    1. 98
    2. 35
    3. 35
    4. 22

    (1 marks)

  2. Evaluate: (8 − 2)² ÷ 3 + 4

    (2 marks)

Summary

  • Recall the BIDMAS order.
  • Evaluate expressions with brackets, powers, ×, ÷, +, −.
  • Identify and fix common BIDMAS errors.

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