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Grades 5 and 6 (Years 6 and 7 in England)

Order of operations: PEMDAS and BODMAS

The order of operations tells you which part of a calculation to do first, so that everyone gets the same answer. Here is the rule step by step (PEMDAS in the US, BODMAS or BIDMAS in the UK), with worked examples and the traps that catch people out. Try the free practice set below, then print an order of operations worksheet with an answer key.

The method, step by step

  1. Parentheses (brackets) first: work out anything inside them, starting with the innermost set.
  2. Exponents (powers) next, such as 3² or 2³.
  3. Multiplication and division next, working from left to right.
  4. Addition and subtraction last, working from left to right.
  5. Rewrite the whole calculation after each step, so you only ever do one operation at a time.

CCSS 5.OA.A.1CCSS 6.EE.A.2.CCCSS 6.EE.A.1England: Y6 Addition, subtraction, multiplication and divisionEngland: KS3 NumberEngland: Y6 Number – multiplication and division

Worked example

50 − (3 + 1)² ÷ 2 × 3 = ?

  1. Parentheses: 3 + 1 = 4, so the calculation becomes 50 − 4² ÷ 2 × 3.
  2. Exponents: 4² = 16, giving 50 − 16 ÷ 2 × 3.
  3. Division and multiplication from left to right: 16 ÷ 2 = 8, then 8 × 3 = 24.
  4. That leaves 50 − 24.
  5. Subtraction: 50 − 24 = 26.
  6. 50 − (3 + 1)² ÷ 2 × 3 = 26

Practice now

Question 1 of 100 correct

8 − 3 × 2 = ?

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Why there is an order at all

Without an agreed order, 2 + 3 × 4 could mean 5 × 4 = 20 or 2 + 12 = 14. Mathematicians settled on one convention so that a written calculation means the same thing to everyone, and the agreed answer is 14: multiplication comes before addition unless parentheses say otherwise.

There is a sensible idea behind it. Multiplication is repeated addition, so 3 × 4 is one bundle: three groups of four. Think of 2 + 3 × 4 as 2 loose apples plus 3 bags of 4 apples. You need to know how many apples are in the bags before you can add the loose ones, and that gives 2 + 12 = 14.

PEMDAS, BODMAS and BIDMAS mean the same thing

In the US, children usually learn PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Many remember it as 'Please Excuse My Dear Aunt Sally'. In the UK the same rule is called BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) or BIDMAS, where the I stands for Indices. In Canada you may also see BEDMAS.

Every version describes one rule. Exponents, orders and indices are all names for powers like 5². PEMDAS puts M before D and BODMAS puts D before M, but neither actually comes first: multiplication and division share a level and are done from left to right, and so are addition and subtraction.

Order of operations examples

  • Multiplication before addition: 20 + 4 × 5 = 20 + 20 = 40.
  • Parentheses first: (9 + 3) × 2 − 5 = 12 × 2 − 5 = 24 − 5 = 19.
  • Division and multiplication from left to right: 24 ÷ 4 × 2 = 6 × 2 = 12, not 24 ÷ 8 = 3.
  • Subtraction and addition from left to right: 10 − 4 + 3 = 6 + 3 = 9, not 10 − 7 = 3.
  • Exponents before multiplication: 20 + 3 × 3² = 20 + 3 × 9 = 20 + 27 = 47.
  • Nested brackets, from the inside out: 6 × [13 − (2 + 4)] = 6 × [13 − 6] = 6 × 7 = 42.
  • A bracket raised to a power: (9 + 9)² ÷ 6 − 17 = 18² ÷ 6 − 17 = 324 ÷ 6 − 17 = 54 − 17 = 37.

The left-to-right rule

A very common slip comes from reading PEMDAS as six separate steps in that exact order. Take 12 − 5 + 2. Doing the addition first gives 12 − 7 = 5, but the correct answer is 7 + 2 = 9, because addition and subtraction are worked from left to right. In the same way, 18 ÷ 3 × 2 is 6 × 2 = 12, not 18 ÷ 6 = 3.

A tidy way to remember it is to think of four levels rather than six letters: parentheses, then exponents, then multiplication and division together, then addition and subtraction together. Within a level, go from left to right.

Brackets change the answer

Parentheses let you override the usual order. 2 + 1 × 3 = 5, but (2 + 1) × 3 = 9. A good puzzle is to give a calculation and a target and ask where the brackets must go. For 40 − 5 × 3 + 2 = 107, the brackets go around 40 − 5: (40 − 5) × 3 + 2 = 35 × 3 + 2 = 105 + 2 = 107. Without any brackets, the same calculation is 40 − 15 + 2 = 27.

Square brackets [ ] and braces { } work exactly like parentheses. They are used when one set sits inside another, and you always work from the innermost set outwards.

How it is taught in school

In the US, the order of operations first appears in grade 3, in a note to standard 3.OA.D.8 saying students should know the conventional order when there are no parentheses. In grade 5, students use parentheses, brackets and braces in expressions and evaluate them (5.OA.A.1). Grade 6 adds whole-number exponents (6.EE.A.1) and applies the full order of operations (6.EE.A.2.c).

In England, the national curriculum asks Year 6 pupils to use their knowledge of the order of operations to carry out calculations involving the four operations. Its guidance suggests exploring brackets with examples like 2 + 1 × 3 = 5 and (2 + 1) × 3 = 9.

How parents can help

  • Ask your child to teach you the rule, including the left-to-right part. If they say multiplication always comes before division, that is the gap to fix.
  • Encourage one step per line: copy the whole calculation each time and change only the part being worked out.
  • Play target games. Give four numbers and a target, and let your child build calculations with brackets to hit it.
  • Try a calculator together. Many basic calculators work from left to right as you type, while scientific ones follow the order of operations, so 2 + 3 × 4 can show 20 on one and 14 on the other.
  • Print a fresh worksheet, mark it with the answer key, and read the worked steps for any question that went wrong.

Common mistakes

  • Doing all multiplication before any division, or all addition before any subtraction, because of the letter order in PEMDAS. Work from left to right: 24 ÷ 4 × 2 = 12.
  • Working straight through from left to right and ignoring the rule: 20 + 4 × 5 is 40, not 120.
  • Squaring more than one number. In 7 + 3², only the 3 is squared, so the answer is 7 + 9 = 16, not 10² = 100.
  • Reading 3² as 3 × 2. It means 3 × 3, which is 9.
  • Dropping part of the calculation when rewriting it. Copy everything you have not worked on yet onto each new line.

Questions people ask

What is the order of operations?
Parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. In the US this is remembered as PEMDAS.
What is the difference between PEMDAS and BODMAS?
Only the names. PEMDAS (US) says Parentheses and Exponents; BODMAS (UK) says Brackets and Orders. Both put multiplication and division on one level and addition and subtraction on the next, each worked from left to right.
Does multiplication always come before division?
No. They have equal priority, so you do whichever comes first reading from left to right. For example, 24 ÷ 4 × 2 = 6 × 2 = 12.
What does BIDMAS stand for?
Brackets, Indices, Division and Multiplication, Addition and Subtraction. Indices are powers such as 2³, so BIDMAS is the same rule as BODMAS and PEMDAS.
What is 8 ÷ 2(2 + 2)?
Following the order of operations strictly, the brackets give 8 ÷ 2 × 4, and working left to right gives 4 × 4 = 16. Some people treat 2(2 + 2) as a single term and get 1, which is why the question causes so many arguments. Writing it clearly, as (8 ÷ 2) × (2 + 2) or 8 ÷ [2 × (2 + 2)], removes the doubt.
What grade learns the order of operations?
In the US Common Core, it starts in grade 3 for calculations without parentheses, grade 5 adds parentheses, brackets and braces (5.OA.A.1), and grade 6 adds exponents (6.EE.A.2.c). In England it is a Year 6 objective.
Is there a free order of operations worksheet with answers?
Yes. Use the Print a worksheet button on this page. Each worksheet is freshly made, and the answer key can show answers only or answers with worked steps.

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Order of Operations (PEMDAS and BODMAS) | Eurekami