Order of Operations (BODMAS/PEMDAS) — The Rules That Make Maths Unambiguous

Mathematics needs agreed rules for the order in which operations are done, otherwise the same expression could mean two different things. The expression 10 + 5 × 2 is the classic example: if you work left to right, you get 30, but the agreed answer is 20, because multiplication is done before addition. The rules that settle this are called the order of operations, known as BODMAS in the UK and PEMDAS in the US. The letters stand for slightly different words, but the rules are the same, and learning them once removes a huge source of confusion from algebra, science and everyday calculation.

What BODMAS and PEMDAS mean

BODMAS stands for Brackets, Orders, Division and Multiplication, Addition and Subtraction. PEMDAS stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Brackets and parentheses are the same thing, and orders and exponents are the same thing: powers and roots. The key point that beginners often miss is that division and multiplication are equal partners, done left to right, and so are addition and subtraction. BODMAS does not mean division always comes before multiplication, and it does not mean addition always comes before subtraction. Within each pair, go left to right, and that resolves expressions like 100 ÷ 10 × 2, which is 20, not 5, because the division comes first when reading left to right.

The rules, step by step

First, do anything inside brackets, starting with the innermost brackets if they are nested. Second, do powers and roots, such as squares, cubes and square roots. Third, do multiplication and division, working left to right. Fourth, do addition and subtraction, working left to right. A useful way to remember the order is to practise with short expressions and to write each step on a new line, so you can see the expression shrinking. For example, 3 + 4 × (6 − 2)²: brackets first give 6 − 2 = 4, then the power gives 4² = 16, then multiplication gives 4 × 16 = 64, then addition gives 3 + 64 = 67. Skipping a step or doing multiplication before brackets would give a different, wrong answer.

Worked examples

Example one: 10 + 5 × 2 = 10 + 10 = 20, because multiplication comes before addition. Example two: (10 + 5) × 2 = 15 × 2 = 30, because brackets come first and change the meaning entirely. Example three: 2 × 3² = 2 × 9 = 18, because the power applies to the 3 only, not to the 2 × 3. If you meant the whole product squared, you would write (2 × 3)² = 36, and the brackets make that intention clear. Example four: 20 − 12 ÷ 4 + 3 = 20 − 3 + 3 = 20, because division comes before addition and subtraction, and then addition and subtraction are done left to right. Example five, with nested brackets: 2 × (3 + (4 × 5)) = 2 × (3 + 20) = 2 × 23 = 46, working from the innermost brackets outwards.

Why the rules exist

Without agreed rules, every formula in science, engineering, finance and computing would need brackets everywhere, and even then people would disagree about what was meant. The order of operations is a convention, like driving on one side of the road: it works because everyone follows the same convention, not because nature forces it. Calculators and computer programs follow the same rules, which is why a scientific calculator gives 20 for 10 + 5 × 2, and why programming languages give the same result. When you write a formula for someone else, though, do not rely on them remembering every subtlety: extra brackets cost nothing and remove doubt, so (10 + 5) × 2 is better written than 10 + 5 × 2 ever could be if you meant 30.

Common mistakes

The most common mistake is working strictly left to right and ignoring the order. The second is treating BODMAS as a strict top-to-bottom list, doing all division before any multiplication, or all addition before any subtraction. Remember the pairs: multiplication and division together, left to right; addition and subtraction together, left to right. The third is applying a power to too much: in 3 × 2², only the 2 is squared. The fourth is forgetting that a fraction bar acts like brackets: in (10 + 5) ÷ 3, the top must be added before dividing, even though no brackets are drawn. The fifth is sign errors with subtraction: 10 − 3 + 2 is 9, not 5, because you add 2 after subtracting 3, working left to right.

How to practise

Start with two-operation expressions, then add brackets, then powers, then nested brackets. Always write one step per line, and check your answer with a scientific calculator, making sure you enter brackets exactly as written. Try inventing expressions and swapping the brackets to see how the answer changes: that single exercise teaches more than memorising the acronym. In algebra, the same rules apply to letters: 2x + 3x² means 2 times x plus 3 times x squared, not (2x + 3x) squared, and that distinction matters for every equation you will solve later. Mastering the order of operations is therefore not a small school topic; it is the grammar of all the maths that follows, and it repays careful practice many times over.

Try it yourself

Use the related calculator on this site to check your own examples, then try one by hand to make sure the method makes sense.