Order of Operations (BIDMAS)

Brackets, indices, multiplication, division, addition, subtraction · Grade 1–3.

The rule that tells you what to do first.

This booklet helps you learn the order to do a calculation when there is more than one operation in it. The rule is called BIDMASThe agreed order to do operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction. BIDMAS or BODMAS for short..

What does BIDMAS mean?

BIDMASThe agreed order to do operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction. BIDMAS or BODMAS for short. is a way of remembering the order. Each letter stands for an operation:

• B — BracketsRound brackets ( ). Whatever is inside ( ) is worked out FIRST.
• I — IndicesA small raised number meaning repeated multiplication. 2³ = 2 × 2 × 2 = 8. Also called a power. (powersA small raised number meaning repeated multiplication. 2³ = 2 × 2 × 2 = 8. Also called a power. like 2², 3³)
• D — DivisionSplitting into equal groups, written as ÷ or /. 12 ÷ 3 = 4.
• M — MultiplicationCombining groups, written as × or *. 4 × 3 = 12.
• A — AdditionPutting things together: 3 + 4 = 7.
• S — SubtractionTaking away: 7 − 4 = 3.

Top of the list goes first. Bottom of the list goes last.

Example: 4 + 2 × 3

If you just read left to right, you would do 4 + 2 = 6 first, then 6 × 3 = 18. But that’s wrong.

BIDMASThe agreed order to do operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction. BIDMAS or BODMAS for short. says do Multiplication before Addition. So: 2 × 3 = 6, then 4 + 6 = 10. That’s the right answer.

A note about D, M, A, and S

D and M tie — they have the same priority. A and S tie too. When two operations tie, work from left to right.

That’s why we’ll sometimes call BIDMASThe agreed order to do operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction. BIDMAS or BODMAS for short. four priority levels, not six steps.

What you’ll learn

We start with simple two-operation calculations. Then we add bracketsRound brackets ( ). Whatever is inside ( ) is worked out FIRST., then indicesA small raised number meaning repeated multiplication. 2³ = 2 × 2 × 2 = 8. Also called a power. (powersA small raised number meaning repeated multiplication. 2³ = 2 × 2 × 2 = 8. Also called a power.), then harder questions and word problems.

How to use the booklet

Work through the questions in order. If you get stuck, that’s OK — stop there. Where you stop tells your tutor exactly where you need help.

How the priority levels work

Here's the ladder. Top of the ladder goes first. Bottom of the ladder goes last. If two things sit on the same rung, you read them left to right in the expressionA mathematical phrase with numbers and operations, but no equals sign. 3 + 4 × 2 is an expression..

That's the whole thing. Four levels. Two ties. Left to right when tied.

Lesson one — Brackets first

BracketsRound brackets ( ). Whatever is inside ( ) is worked out FIRST. are the override switch. Whatever sits inside the bracketsRound brackets ( ). Whatever is inside ( ) is worked out FIRST. gets calculated first, no matter what else is going on.

Compare that with the same expressionA mathematical phrase with numbers and operations, but no equals sign. 3 + 4 × 2 is an expression. without the bracketsRound brackets ( ). Whatever is inside ( ) is worked out FIRST.:

Same numbers, same operations, different answer. That's what bracketsRound brackets ( ). Whatever is inside ( ) is worked out FIRST. do.

Implicit multiplication outside brackets

Sometimes a number sits right next to a bracket with no × sign between them. Like this:

The 3 and the bracket are multipliedCombining groups, written as × or *. 4 × 3 = 12.. The expressionA mathematical phrase with numbers and operations, but no equals sign. 3 + 4 × 2 is an expression. is the same as 3 × (4 + 2). BracketsRound brackets ( ). Whatever is inside ( ) is worked out FIRST. first — that gives 3 × 6 = 18.

This trips a lot of pupils up because the multiplicationCombining groups, written as × or *. 4 × 3 = 12. sign is invisible. If you see a number stuck to the outside of a bracket, slip a × in mentally before you do anything else.

Fraction bars hide brackets

A fraction bar acts like a bracket around the top and another bracket around the bottom:

You work out the top (10 + 2 = 12) and the bottom (3 − 1 = 2) before you divideSplitting into equal groups, written as ÷ or /. 12 ÷ 3 = 4.. Answer: 12 ÷ 2 = 6.

If you wrote this as a fraction with a horizontal bar, you'd see why — the bar visually groups the top and the bottom. When it's typed sideways on a phone, the bracketsRound brackets ( ). Whatever is inside ( ) is worked out FIRST. become essential.

Sense-check your answer

After working out something with bracketsRound brackets ( ). Whatever is inside ( ) is worked out FIRST., ask: did the bracketsRound brackets ( ). Whatever is inside ( ) is worked out FIRST. actually change anything? If (5 + 3) × 2 gave you 11, you've probably ignored the bracketsRound brackets ( ). Whatever is inside ( ) is worked out FIRST. — that's the answer for 5 + 3 × 2. BracketsRound brackets ( ). Whatever is inside ( ) is worked out FIRST. exist for a reason. They should pull the answer somewhere it wouldn't have gone otherwise.

Lesson two — Indices next

IndicesA small raised number meaning repeated multiplication. 2³ = 2 × 2 × 2 = 8. Also called a power. (also called powersA small raised number meaning repeated multiplication. 2³ = 2 × 2 × 2 = 8. Also called a power. or exponentsA small raised number meaning repeated multiplication. 2³ = 2 × 2 × 2 = 8. Also called a power.) are the little raised numbers, like 2³ or 5². After bracketsRound brackets ( ). Whatever is inside ( ) is worked out FIRST., they get top priority.

Before we go further, let's settle what an indexA small raised number meaning repeated multiplication. 2³ = 2 × 2 × 2 = 8. Also called a power. actually means — because this is where another common BIDMASThe agreed order to do operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction. BIDMAS or BODMAS for short. trap lives.

What an index actually means

2³ does not mean 2 × 3. It means 2 × 2 × 2. The little 3 tells you how many times the 2 appears in a multiplicationCombining groups, written as × or *. 4 × 3 = 12., not what to multiplyCombining groups, written as × or *. 4 × 3 = 12. the 2 by.

• 2² = 2 × 2 = 4
• 2³ = 2 × 2 × 2 = 8
• 2&sup4 = 2 × 2 × 2 × 2 = 16
• 3² = 3 × 3 = 9
• 5² = 5 × 5 = 25

If you ever catch yourself thinking 3² is 6, slow down. 3² is 9. 3 × 2 is 6. They look similar on paper but they mean completely different things.

Powers in expressions

Once you know the indexA small raised number meaning repeated multiplication. 2³ = 2 × 2 × 2 = 8. Also called a power. value, you handle it before any divisionSplitting into equal groups, written as ÷ or /. 12 ÷ 3 = 4., multiplicationCombining groups, written as × or *. 4 × 3 = 12., additionPutting things together: 3 + 4 = 7. or subtractionTaking away: 7 − 4 = 3..

Not 3 × 2 + 4 = 10. Not (3 + 4)² = 49. The answer is 13.

Sense-check your answer

IndicesA small raised number meaning repeated multiplication. 2³ = 2 × 2 × 2 = 8. Also called a power. grow numbers fast. 2³ already beats 2 × 3. 5² is 25 while 5 × 2 is only 10. If you've calculated something with an indexA small raised number meaning repeated multiplication. 2³ = 2 × 2 × 2 = 8. Also called a power. in it and the answer feels small, you've probably read the indexA small raised number meaning repeated multiplication. 2³ = 2 × 2 × 2 = 8. Also called a power. as a multiplicationCombining groups, written as × or *. 4 × 3 = 12..

Lesson three — Times-and-divide (they tie)

This is the lesson that matters most. DivisionSplitting into equal groups, written as ÷ or /. 12 ÷ 3 = 4. and multiplicationCombining groups, written as × or *. 4 × 3 = 12. have equal priority. Neither one goes first. When you have both in an expressionA mathematical phrase with numbers and operations, but no equals sign. 3 + 4 × 2 is an expression., you read left to right.

The pupil who thinks "M before D because M comes before D in BIDMASThe agreed order to do operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction. BIDMAS or BODMAS for short." does 3 × 2 = 6 first, then 12 ÷ 6 = 2. Wrong answer. Wrong reasoning.

D and M tie because they're really the same operation in two outfits. Dividing by 2 is the same as multiplying by ½. They do the same kind of work, just written differently. The order they're written in is the order you use them.

Sense-check your answer

When an expressionA mathematical phrase with numbers and operations, but no equals sign. 3 + 4 × 2 is an expression. has both × and ÷ in it, those tied operations should have been done left to right. If you swapped their order (M before D or D before M), you've slipped — recompute.

When DM mixes with AS

Most expressionsA mathematical phrase with numbers and operations, but no equals sign. 3 + 4 × 2 is an expression. you'll meet have both kinds of operation: something from the DM rung and something from the AS rung. DM goes first (it's higher up the ladder), then AS.

The pupil who adds first gets 6 + 4 = 10, then 10 × 2 = 20, then 20 − 3 = 17. Wrong by a long way. × and ÷ always do their work before any + or −.

Lesson four — Plus-and-minus (they tie too)

Last level. AdditionPutting things together: 3 + 4 = 7. and subtractionTaking away: 7 − 4 = 3. have equal priority. Read left to right when both are present.

The "A before S" trap says 3 + 2 = 5 first, then 8 − 5 = 3. Same wrong reasoning as the M-before-D version. A and S are inverse operations — subtracting 3 is the same as adding −3. They're tied. Read left to right.

Subtraction is where most mistakes happen

SubtractionTaking away: 7 − 4 = 3. is where most BIDMASThe agreed order to do operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction. BIDMAS or BODMAS for short. errors happen at this level. The trick is to never let yourself slow down on a subtractionTaking away: 7 − 4 = 3.. Treat it exactly like an additionPutting things together: 3 + 4 = 7. with a negative sign and keep reading left to right.

That means 10 − 4 + 3 is (10) + (−4) + (3) = 9. The signs travel with the numbers they're attached to. The expressionA mathematical phrase with numbers and operations, but no equals sign. 3 + 4 × 2 is an expression. isn't 10 − (4 + 3) = 3. That would only be true if there were bracketsRound brackets ( ). Whatever is inside ( ) is worked out FIRST..

Sense-check your answer

If you did your subtractionTaking away: 7 − 4 = 3. before your additionPutting things together: 3 + 4 = 7. (when they sit on the same level), your answer will usually come out wrong by a noticeable amount. Re-read left to right and check.

Where this actually matters

BIDMASThe agreed order to do operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction. BIDMAS or BODMAS for short. isn't a maths-class novelty. It's the rule every calculator, every spreadsheet, every formula and every algebra problem follows. Once you know it, a lot of things stop being mysterious.

Calculators

When you type 3 + 4 × 2 into a scientific calculator, you get 11, not 14. The calculator is applying BIDMASThe agreed order to do operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction. BIDMAS or BODMAS for short.. Older or simpler calculators that read left to right without precedence would give 14 — that's part of why scientific calculators exist.

Formulas in geometry and science

The area of a trapezium is A = ½(a + b)h. The bracketsRound brackets ( ). Whatever is inside ( ) is worked out FIRST. aren't there for decoration — without them you'd compute ½ × a + b × h, which is something else entirely. BIDMASThe agreed order to do operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction. BIDMAS or BODMAS for short. is what tells you to add a + b first, then multiplyCombining groups, written as × or *. 4 × 3 = 12. by ½, then multiplyCombining groups, written as × or *. 4 × 3 = 12. by h.

The perimeter of a rectangle is P = 2(L + W). The 2 sits next to a bracket — implicit multiplicationCombining groups, written as × or *. 4 × 3 = 12.. Add the length and the width first, then double it.

Algebra and substitution

If you're given a = 4 and b = 3 and asked for a² + 2b, you do 4² = 16 first, then 2 × 3 = 6, then 16 + 6 = 22. BIDMASThe agreed order to do operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction. BIDMAS or BODMAS for short. again — same rules, just with letters instead of digits.

Real life

A shop selling 3 items at £20 each with 25% off works it out as 3 × 20 × 0.75 — because a 25% discount means you pay 75% of the price. 3 × 20 = 60, then 60 × 0.75 = 45. Total £45. BIDMASThe agreed order to do operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction. BIDMAS or BODMAS for short. tells the calculator to read those multiplications left to right.

Some shops show the same discount as 3 × 20 × (1 − 0.25). Same answer — the bracket works out the 0.75 for you. Once you've met BIDMASThe agreed order to do operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction. BIDMAS or BODMAS for short., both versions read identically.

Different countries teach different mnemonics — BIDMASThe agreed order to do operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction. BIDMAS or BODMAS for short. in the UK, PEMDAS in the US (ParenthesesRound brackets ( ). Whatever is inside ( ) is worked out FIRST., ExponentsA small raised number meaning repeated multiplication. 2³ = 2 × 2 × 2 = 8. Also called a power., MultiplicationCombining groups, written as × or *. 4 × 3 = 12., DivisionSplitting into equal groups, written as ÷ or /. 12 ÷ 3 = 4., AdditionPutting things together: 3 + 4 = 7., SubtractionTaking away: 7 − 4 = 3.) — but the underlying rule is identical. The mnemonic is just a memory aid; the priority levels are the maths.