Order of operations
If you have an expression with one operation, the answer is obvious: 3 + 5 is 8. But what about expressions with two or more operations?
2 + 3 × 4 = ?
You could try two ways:
- Left to right: 2 + 3 = 5, then 5 × 4 = 20.
- Multiply first: 3 × 4 = 12, then 2 + 12 = 14.
Both look reasonable — but only one is correct. Mathematicians long ago agreed on a single rule so everyone gets the same answer. That rule is called the order of operations.
The rule
1. Brackets first (anything inside `(...)`).
2. Multiplication and division, going from left to right.
3. Addition and subtraction, going from left to right.
Two famous mnemonics for the same idea:
- PEMDAS (US): Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
- BIDMAS (UK): Brackets, Indices, Division, Multiplication, Addition, Subtraction.
In Year 5 we don't usually meet exponents or indices, so the rule is just brackets → × ÷ → + −.
Applying the rule
2 + 3 × 4
No brackets. Multiplication before addition: 3 × 4 = 12, then 2 + 12 = 14.
(2 + 3) × 4
Brackets first: 2 + 3 = 5, then 5 × 4 = 20.
The same numbers, the same operations — but the brackets change everything.
Left to right when same priority
When two operations are at the same level (× and ÷ together, or + and − together), you do them left to right.
20 ÷ 4 × 5 = ?
Both ÷ and × are level 2. Left to right: 20 ÷ 4 = 5, then 5 × 5 = 25. (Not 4 × 5 = 20, then 20 ÷ 20 = 1 — that would be wrong.)
100 − 30 + 10 = ?
Both − and + are level 3. Left to right: 100 − 30 = 70, then 70 + 10 = 80. (Not 30 + 10 = 40, then 100 − 40 = 60 — that would be wrong.)
Worked examples
8 − 2 × 3
× first: 2 × 3 = 6. Then 8 − 6 = 2.
(8 − 2) × 3
Brackets first: 8 − 2 = 6. Then 6 × 3 = 18.
24 ÷ (4 + 2)
Brackets first: 4 + 2 = 6. Then 24 ÷ 6 = 4.
5 × (3 + 2) − 4
Brackets first: 3 + 2 = 5. Then × : 5 × 5 = 25. Then − : 25 − 4 = 21.
100 − 6 × 4 + 8 ÷ 2
Level 2 first (× and ÷ left to right):
- 6 × 4 = 24
- 8 ÷ 2 = 4
The expression now reads: 100 − 24 + 4.
Level 3 left to right:
- 100 − 24 = 76
- 76 + 4 = 80.
Common mistakes
- Going purely left to right. "2 + 3 × 4 = 5 × 4 = 20". No — × goes first.
- Ignoring brackets. Brackets aren't decoration; they change the order.
- Doing + before − because + is "stronger". They're at the same level — go left to right.
- Multiplying out of order. "20 ÷ 4 × 5 = 20 ÷ 20 = 1". No — left to right within level 2.
A puzzle
Where do brackets go so that 4 + 6 × 2 − 1 = 21?
Try `(4 + 6) × 2 − 1 = 10 × 2 − 1 = 20 − 1 = 19`. Not quite.
Try `4 + 6 × (2 − 1)` … wait, that's 4 + 6 × 1 = 10. Not right either.
Try `(4 + 6) × (2 − 1)` … that's 10 × 1 = 10. No.
Try `4 + 6 × 2 − 1 = 4 + 12 − 1 = 15` (no brackets at all). Not 21.
What gives 21? `(4 + 6 × 2 − 1)` doesn't change anything. Try arranging differently: there might be no solution with one bracket pair. (And that's part of the lesson — brackets only help when there's an actual choice to make.)