Order of Operations: The Rules + Worked Examples

A mixed calculation contains more than one operation — addition, subtraction, multiplication and division — in a single expression. The order of working cannot be arbitrary, because different orders give different answers. It is taught from Year 4 and is the prerequisite for all secondary algebra.

Why the Order Matters

Try 2 + 3 × 4.

• Working left to right: (2 + 3) × 4 = 20
• Following the rule: 2 + (3 × 4) = 14

Only one is correct: 14. Without a fixed rule, one expression would have many answers and mathematics would stop working. That is why the rule exists — not because teachers want to complicate things.

The Order of Operations

🔴 The most misunderstood part: multiplication does not come before division, and addition does not come before subtraction. They sit at the same level, so what decides is position from the left.

Example: 24 ÷ 6 × 2

• Wrong: 24 ÷ (6 × 2) = 24 ÷ 12 = 2
• Right: (24 ÷ 6) × 2 = 4 × 2 = 8

The division is written first, so it is done first. This is the number one error in the topic, and it usually comes from a memory acronym that puts multiplication before division.

Worked Examples

Example 1: 100 − 4 × 15

• Multiply first: 4 × 15 = 60
• Then subtract: 100 − 60 = 40

Example 2: 36 ÷ 4 + 5 × 3

• Divide and multiply first, left to right: 36 ÷ 4 = 9 and 5 × 3 = 15
• Then add: 9 + 15 = 24

Example 3: 12 + (8 − 3) × 4

• Brackets first: 8 − 3 = 5
• Multiply: 5 × 4 = 20
• Add: 12 + 20 = 32

Example 4: 50 − 12 + 8

• Equal rank, so left to right: (50 − 12) + 8 = 38 + 8 = 46
• Not 50 − (12 + 8) = 30

The fourth example is the classic trap, and it catches many children precisely because it looks easiest.

A Worked Word Problem

A shopper buys 3 boxes of eggs with 12 eggs in each. Then 5 break. How many are left?

3 × 12 − 5 = 36 − 5 = 31 eggs

Notice the expression already follows the rule naturally: the multiplication has to come first because the total number of eggs must be known before any break.

How to Practise at Home

Write one expression such as 6 + 6 ÷ 3 on paper and ask the child to explain the order first, before calculating. If they can say "divide first, then add", their understanding is sound even if the arithmetic later slips. That separates what needs fixing from what does not.

Operations with Negative Numbers

When a problem contains negative numbers, the order of working stays the same. Only the signs need care. Subtracting a negative means adding: 5 − (−3) = 8, because cancelling a debt of 3 thousand is the same as gaining 3 thousand. For multiplication and division one rule applies: matching signs give a positive, differing signs give a negative. So (−4) × (−3) = 12, while 4 × (−3) = −12.

Example: 10 − 2 × (−3)

• Multiply first: 2 × (−3) = −6
• Then subtract: 10 − (−6) = 10 + 6 = 16

The Mistakes That Actually Cost Marks

1. Working everything left to right. Answering 20 for 2 + 3 × 4. The number one cause in Years 4–5.

2. Thinking multiplication always precedes division. It comes from memory acronyms. They rank equally.

3. Thinking addition always precedes subtraction. The same error at level four, and it shows up most in expressions like 50 − 12 + 8.

4. Ignoring nested brackets. Work the innermost pair first.

5. Sign errors with negative integers. If a child understands the order and still gets the wrong answer, this is usually why — not the order itself.