The order of operations, worked out step by step

What is the rule, in one sentence?

The order of operations works through an expression in one fixed order: brackets first, then exponents, then multiplication and division together from left to right, then addition and subtraction together from left to right. That sentence is the entire rule.

The rule is a writing convention, agreed so that every reader of 4 + 3 × 5 arrives at the same number. The Common Core mathematics standards say as much, asking sixth graders (6.EE.2c) to perform arithmetic "in the conventional order when there are no parentheses to specify a particular order". Without that agreement one person adds first and gets 35, another multiplies first and gets 19. With it, the answer is 19: 3 × 5 first, then add 4.

Brackets are the override switch: when a line is confusing, adding brackets is always the right fix.

Are PEMDAS, BODMAS and BIDMAS different rules?

PEMDAS, BODMAS and BIDMAS are one convention under different mnemonics, and which you met depends on where you went to school.

Three details matter:

• Parentheses and brackets are the same thing, and exponents, orders, indices and powers all mean 3² or 2³.
• PEMDAS puts M before D and BODMAS puts D before M, which changes nothing, because the two operations are equal rank.
• Every version hides the most important part of the rule, the left-to-right tiebreaker.

The mnemonics are a classroom habit and do not appear in curriculum text. The national curriculum for mathematics in England never prints BIDMAS: it asks Year 6 pupils to "use their knowledge of the order of operations to carry out calculations involving the 4 operations".

What do most people get wrong?

The most common order-of-operations mistake is to read 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 left to right. Three short lines make the point, each with the wrong answer the acronym invites:

• 10 − 4 + 3 = 6 + 3 = 9. The trap: 10 − 7 = 3.
• 100 ÷ 5 ÷ 2 = 20 ÷ 2 = 10. The trap: 100 ÷ 2.5 = 40.
• 20 − 12 ÷ 4 × 2 = 20 − 3 × 2 = 20 − 6 = 14. The trap: 20 − 12 ÷ 8 = 18.5.

Each wrong answer comes from obeying the acronym letter by letter instead of reading left to right. When a line holds both × and ÷, or both + and −, name which stands further left before working anything out.

Five worked examples, step by step

These five worked examples run in order of difficulty, from a single multiplication up to nested brackets with an exponent, each written one step per line.

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 come first, whatever is inside them.

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

Answer: 16.

3. 20 − 12 ÷ 4 × 2

The expression 20 − 12 ÷ 4 × 2 has no brackets and no exponents, so the work starts at step 3 and reads left to right. The division stands further left.

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 and does not reach the 2.

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

Answer: 23. Squares appear constantly once exponents are in play; the square numbers game drills them.

5. 2 × (3 + (10 − 4) ÷ 2)² − 7

Nested brackets are worked from the inside out: the innermost pair, then 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 stops the left-to-right slips, and the order of operations game uses that format when it shows a missed answer.

Play free in your browserOrder of OperationsGrade 5+

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

The expression 6 ÷ 2(1 + 2) splits the internet because it is written ambiguously, while the arithmetic inside it is easy. Every couple of years such a line 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 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

In both lines the bracket carries a multiplication with the sign left out. The standard reading gives it equal rank with 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, and the reading has serious support: Oliver Knill of Harvard's mathematics department traces both conventions and concludes that such a line is genuinely ambiguous. Texas Instruments documents that its TI-80, TI-81, TI-82 and TI-85 gave implied multiplication the higher priority, while the TI-83, TI-84 Plus, TI-89 and TI-Nspire families rank it equally, which is why two calculators in the same room show different answers.

The fight is about a gap in the notation that no acronym covered. Anyone writing mathematics seriously would use one of two unambiguous forms instead:

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

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

The four common mistakes

Four errors account for almost every wrong answer, and three come from trusting the acronym over left to right.

1. Treating M and D as two steps. In 20 − 12 ÷ 4 × 2, doing 4 × 2 = 8 first gives 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 neighbor. In 2 × 3² the exponent belongs to the 3 only, so the line is 2 × 9 = 18. Squaring 2 × 3 first gives 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 3 + 1 underneath and you get 5 instead of 3. Type 12 ÷ (3 + 1).

How do you make the rule automatic?

The rule becomes automatic through short mixed expressions drilled with the four steps in front of you. Reading the rule takes two minutes; using it under pressure is a separate skill.

Four habits speed that up:

1. Practice three-operation lines. 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, the way the five worked examples do.
3. Get the underlying facts fast first. If 12 ÷ 4 costs three seconds, no attention is left for the rule, and the mental math tricks that work cover that layer.
4. Mix the operations in one session. A page of pure multiplication never tests ordering; a mixed page does, and the printable mixed worksheet for fifth graders comes with an answer key.

Free printable worksheetMixed Operations — 5th GradersWorksheet · PDF

Curricula introduce this around fifth and sixth grade, just before algebra makes it unavoidable: 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 expressions get longer only once the short ones are right.