Level 5Grade 5skill: order_of_ops· 7 min read

Order of Operations: the Basics

In short: Do brackets first, then multiplication and division (left to right), then addition and subtraction (left to right). The big idea: ×/÷ share one rank and +/− share another — within a rank you just go left to right.

The order every calculation follows

Here is a puzzle that splits a whole classroom in two. Ask ten people to work out 4 + 6 × 2 and some will say 20, some will say 16. They can't both be right — and maths would be useless if a single expression had two answers. So mathematicians agreed on a fixed running order, and once you know it, every one of these problems has exactly one correct answer.

The order is: brackets first, then multiplication and division, then addition and subtraction. In 4 + 6 × 2 the multiplication jumps the queue: 6 × 2 = 12, then 4 + 12 = 16. The person who got 20 just read the line like a sentence, left to right — and that is the single most common mistake in the whole topic.

Do anything inside brackets ( ) first.
Then do all multiplication (×) and division (÷), working left to right.
Then do all addition (+) and subtraction (−), working left to right.
When two operations share a rank, the left one goes first — never all of one before the other.

Why it's really two tiers, not four steps Why it works

Brackets

Whatever is inside brackets is one packaged number — finish it before anything else touches it.

2 × (3 + 4) = 2 × 7 = 14, not 2 × 3 + 4 = 10.

× and ÷ together

Multiply and divide are the SAME level. Don't do every × before every ÷ — go left to right.

12 ÷ 2 × 3 = 6 × 3 = 18. Doing 2 × 3 first gives the wrong 2.

+ and − together

Add and subtract are also one level. Left to right again — subtraction does not wait for addition.

10 − 4 + 3 = 6 + 3 = 9, not 10 − 7 = 3.

The insight most people miss: the famous rule is not four separate steps but really two tiers. Multiplication and division are one tier; addition and subtraction are the tier below. Inside a tier there is no priority at all — you simply read left to right, exactly like normal. Whichever operator sits furthest left on the page goes first; the tier is the only thing that ever overrides that.

Worked examples

8 + 3 × 4 − 5 = 15

Find the × or ÷ first: 3 × 4 = 12.
Rewrite: 8 + 12 − 5.
Now left to right: 8 + 12 = 20, then 20 − 5 = 15.

20 − 12 ÷ 4 = 17

Division outranks subtraction, so do 12 ÷ 4 = 3 first.
Rewrite: 20 − 3.
20 − 3 = 17.

24 ÷ 4 ÷ 2 = 3

Both are on the ÷/× tier, so go strictly left to right.
Leftmost first: 24 ÷ 4 = 6.
Then 6 ÷ 2 = 3. (Doing 4 ÷ 2 first would give the wrong 12.)

2 × (7 − 3) + 5 × 2 = 18

Brackets first: 7 − 3 = 4.
Do the multiplications: 2 × 4 = 8 and 5 × 2 = 10.
Add: 8 + 10 = 18.

Four mistakes to watch for

Reading straight across, left to right

✗ 22

For 3 + 8 × 2, this treats the line like a sentence: 3 + 8 = 11, then 11 × 2 = 22. It ignores that × outranks +.

Fix: Scan the whole expression first and circle every × and ÷. Do those before you touch any + or −. Here 8 × 2 = 16 comes first, giving 3 + 16 = 19.

order_of_ops_lr

Stopping one step early

✗ 6

For 5 + 2 × 3, this does the multiplication 2 × 3 = 6 correctly — and then writes 6 as the answer, forgetting the + 5 that was still waiting.

Fix: Multiplication was only step one. After 2 × 3 = 6, rewrite the whole line as 5 + 6 and finish it: the answer is 11. Never stop while an operation is left unused.

partial_computation

Practice: order of operations

Work through mixed expressions using brackets, ×, ÷, +, and −. Take each one in order and finish every step before you commit to an answer.

Frequently asked questions

Is it BODMAS, BIDMAS, or PEMDAS — and do they give different answers?

They are the same rule with different names. B/P means brackets or parentheses, O/I/E means orders (powers), DM means division and multiplication, AS means addition and subtraction. They always give the same answer. The one thing every version can hide is that division and multiplication share a rank, and so do addition and subtraction.

Do I always do multiplication before division?

No — that is the most common trap. Multiplication and division sit at the same level, so you work left to right. In 12 ÷ 2 × 3 the division is on the left, so 12 ÷ 2 = 6 first, then 6 × 3 = 18. Doing the multiply first would give the wrong answer.

What if there are no brackets and no × or ÷?

Then you just work left to right. For 10 − 4 + 3, do 10 − 4 = 6, then 6 + 3 = 9. Subtraction does not wait for addition; whichever comes first on the line goes first.

Why can't I just calculate left to right like reading?

Because everyone would get different answers to the same problem. The agreed order guarantees one correct result. Multiplication is a shortcut for repeated addition, so 4 + 6 × 2 really means 4 plus two sixes — the × has to happen first for the expression to mean what it says.