The order of operations, worked out step by step

What is the rule, in one sentence?

Work through an expression in this order: brackets first, then exponents, then multiplication and division together from left to right, then addition and subtraction together from left to right. That is the entire rule. Everything below is that sentence applied carefully.

It helps to know what the rule is not. Nothing here is proved, and no law of nature is involved. It is a writing convention, agreed so that every reader of 4 + 3 × 5 arrives at the same number. Without it, one person adds first and gets 35, another multiplies first and gets 19, and nobody can say who misread the line. With it, the answer is 19: 3 × 5 first, then add 4.

Brackets are the override switch. Anything the rule can express, brackets can express more plainly — so when a line is confusing, adding brackets is always the right fix.

Are PEMDAS, BODMAS and BIDMAS different rules?

No. They are one convention under different mnemonics, depending on where you went to school. The letters change; the order of the work does not.

Three things worth noticing:

• Parentheses and brackets are the same thing. So are exponents, orders, indices and powers: all mean 3² or 2³.
• PEMDAS puts M before D and BODMAS puts D before M. That changes nothing: the two operations are equal rank. The acronym had to print one letter first; the mathematics does not care.
• Every version hides the most important part of the rule: the left-to-right tiebreaker.

What do most people get wrong?

They count the acronym as six steps. There are four, because multiplication and division share one, and addition and subtraction share another.

1. Brackets, innermost pair first.
2. Exponents.
3. Multiplication and division, left to right.
4. Addition and subtraction, left to right.

Inside a step you read the line like a sentence, left to right, taking whichever operation you meet first. Three short lines make the point, each with the wrong answer the acronym invites:

• 10 − 4 + 3 = 6 + 3 = 9. Not 10 − 7 = 3.
• 100 ÷ 5 ÷ 2 = 20 ÷ 2 = 10. Not 100 ÷ 2.5 = 40.
• 20 − 12 ÷ 4 × 2 = 20 − 3 × 2 = 20 − 6 = 14. Not 20 − 12 ÷ 8 = 18.5.

In each case the wrong answer comes from obeying the acronym letter by letter instead of reading left to right. To check yourself: when a line holds both × and ÷, or both + and −, name which stands further left before working anything out.

Five worked examples, step by step

Five lines in order of difficulty, from one multiplication up to nested brackets with an exponent. Each is written one step per line — the format to copy.

1. 4 + 3 × 5

Multiplication outranks addition, so it goes first.

1. 3 × 5 = 15
2. 4 + 15 = 19

Answer: 19.

2. 8 + 2 × (7 − 3)

Brackets first, whatever is inside them.

1. 7 − 3 = 4
2. 2 × 4 = 8
3. 8 + 8 = 16

Answer: 16.

3. 20 − 12 ÷ 4 × 2

No brackets and no exponents, so start at step 3 and read left to right. The division stands further left, so it goes first.

1. 12 ÷ 4 = 3
2. 3 × 2 = 6
3. 20 − 6 = 14

Answer: 14.

4. 5 + 2 × 3²

The exponent is attached to the 3 alone, not to 2 × 3.

1. 3² = 9
2. 2 × 9 = 18
3. 5 + 18 = 23

Answer: 23. Squares appear constantly once exponents are in play, so they are worth knowing by sight; the square numbers game drills them.

5. 2 × (3 + (10 − 4) ÷ 2)² − 7

Nested brackets: resolve the innermost pair, finish the outer bracket with the same four steps, then the exponent, the multiplication, the subtraction.

1. 10 − 4 = 6, so the line becomes 2 × (3 + 6 ÷ 2)² − 7
2. Inside the bracket, division before addition: 6 ÷ 2 = 3
3. 3 + 3 = 6, so the line becomes 2 × 6² − 7
4. 6² = 36
5. 2 × 36 = 72
6. 72 − 7 = 65

Answer: 65.

Writing one line per step is not busywork: it is the habit that stops the left-to-right slips, and it is the format the order of operations game uses when it shows a missed answer.

Why does 6 ÷ 2(1 + 2) split the internet?

Because the expression is written badly, not because the arithmetic is hard. Every couple of years a line like 6 ÷ 2(1 + 2) or 8 ÷ 2(2 + 2) goes viral, half the comments saying 1 and half saying 9 or 16, and both sides can quote a rule at you.

Apply the convention exactly as it is taught and you get:

• 6 ÷ 2(1 + 2) becomes 6 ÷ 2 × 3, then 3 × 3 = 9
• 8 ÷ 2(2 + 2) becomes 8 ÷ 2 × 4, then 4 × 4 = 16

Here 2(1 + 2) is a multiplication with the sign left out. The standard reading treats it as ordinary multiplication, equal in rank to the division, so left to right decides the order.

The other camp reads 2(1 + 2) as one glued-together quantity, because a multiplication written without a sign looks tighter than a division sign. That gives 6 ÷ 6 = 1 and 8 ÷ 8 = 1. It is not invented out of nowhere: it is how people read algebra, where 2x is one thing, and calculator models genuinely disagree, which is why two phones in the same room show different answers.

So the fight is not about arithmetic but about a gap in the notation that no acronym covered. Nobody writing mathematics seriously would publish that line; they would write one of these, and there is nothing left to argue about:

• (6 ÷ 2) × (1 + 2) = 9
• 6 ÷ (2 × (1 + 2)) = 1

The practical lesson of the whole topic: if a reader could reasonably get two answers, add brackets.

The four common mistakes

Four errors account for almost every wrong answer. Three come from trusting the acronym over the left-to-right rule.

1. Treating M and D as two steps. In 20 − 12 ÷ 4 × 2, doing 4 × 2 = 8 first gives 20 − 1.5 = 18.5. The answer is 14.
2. Treating A and S as two steps. In 10 − 4 + 3, adding 4 + 3 first gives 3. The answer is 9.
3. Letting an exponent swallow its neighbour. In 2 × 3² the exponent belongs to the 3 only, so it is 2 × 9 = 18, not (2 × 3)² = 36.
4. Leaving out a bracket you meant to write. A fraction bar groups everything above and below it; a division sign on one line does not. Write 12 ÷ 3 + 1 when you meant the whole of 3 + 1 underneath and you get 5 instead of 3. Type 12 ÷ (3 + 1).

How do you make the rule automatic?

By drilling short mixed expressions, not long ones, with the four steps in front of you. Reading the rule takes two minutes; using it under time pressure is a separate skill.

Four things speed that up:

1. Practise three-operation lines, not monsters. 20 − 12 ÷ 4 × 2 teaches more than a half-page expression, because ordering is the only thing it tests.
2. Write the step you just did, one line each, as in the examples above.
3. Get the underlying facts fast first. If 12 ÷ 4 costs three seconds, no attention is left for the rule itself, and the mental math tricks that actually work cover that layer.
4. Mix the operations in one session. A page of pure multiplication never tests ordering; a mixed page does, which is what the printable mixed worksheet for fifth graders is for, answer key included.

Most curricula introduce this around fifth and sixth grade, just before algebra makes it unavoidable: once letters arrive, a line like 3(x + 4) − 2 can be read only one way, and the solve for x game assumes you can read it. The fifth-grade math games hub collects the browser games at that level.

Twenty-two of MathIt's roughly 90 game types run free in a browser with no sign-up, and the free iOS and Android app adds adaptive difficulty, so the expressions get longer and more deeply nested only once you are getting the short ones right.