BODMAS is a memory aid for the order of operations, the agreed set of rules for simplifying expressions that contain more than one operation. It stands for Brackets, Orders, Division, Multiplication, Addition, and Subtraction. These rules matter because the same numbers can give different answers if operations are done in a random order.
BODMAS makes mathematical writing unambiguous, so everyone reads an expression the same way.
Understanding Math: Order of operations (UK form) (BODMAS)
The order is best understood as a set of layers. Brackets create a self contained calculation inside a larger one. When brackets are nested, start with the innermost group and work outward.
Different bracket shapes can be used to make nested work easier to read, but they have the same job. A fraction bar has a similar effect because everything above it forms one group and everything below it forms another group.
This is important when simplifying algebra, where a long numerator or denominator can contain several operations. Careful grouping prevents a small early mistake from changing every later step.
Some parts of BODMAS are often misunderstood. Division does not come before multiplication simply because its letter appears first. They belong to one level, so calculations move from left to right across that level.
The same rule applies to addition and subtraction. For example, twenty divided by five multiplied by two becomes four multiplied by two, giving eight. Doing the multiplication first would change the meaning of the written calculation.
Left to right is not a shortcut. It is the rule that keeps a chain of equal priority operations consistent.
Negative numbers need extra care because a minus sign can have two roles. It may mean subtract, or it may be attached to a number to show that the number is negative. Brackets make the intended meaning clear.
The square of negative three means negative three multiplied by negative three, which gives positive nine. In contrast, negative three squared means the negative of three squared, which gives negative nine.
This difference appears in algebra, graph work, and calculator questions. Students should write brackets around negative values before applying a power unless the meaning is completely clear.
Order of operations appears outside textbook exercises whenever a calculator, spreadsheet, computer program, or formula processes a calculation. A phone calculator usually follows these rules, but entering a calculation in the wrong order can still give an unintended result. In spreadsheets, a missing pair of brackets can alter a budget, a score total, or a scientific result across many rows.
Good working helps catch errors. Rewrite one operation at each step, keep groups together, and estimate the likely size and sign of the answer. If a calculation involving only positive quantities gives a negative result, or a small change produces a huge answer, check the grouping and the order used.
Key Facts
- BODMAS means Brackets, Orders, Division, Multiplication, Addition, Subtraction.
- Brackets are solved first: 3 x (8 - 5) = 3 x 3 = 9.
- Orders include powers, roots, and indices: 2^3 = 8 and sqrt(25) = 5.
- Multiplication and division have equal priority, so work from left to right: 12 ÷ 3 x 2 = 4 x 2 = 8.
- Addition and subtraction have equal priority, so work from left to right: 10 - 4 + 1 = 6 + 1 = 7.
- Example: 3 + 2^2 x (8 - 5) = 3 + 4 x 3 = 3 + 12 = 15.
Vocabulary
- Brackets
- Brackets group part of an expression so that the grouped calculation is completed before operations outside it.
- Orders
- Orders are powers, roots, and indices that tell you to repeat multiplication or find a related root.
- Expression
- An expression is a mathematical phrase made from numbers, variables, and operations, without an equals sign.
- Operation
- An operation is a mathematical action such as adding, subtracting, multiplying, dividing, or raising to a power.
- Left to right
- Left to right means performing operations of equal priority in the order they appear from the start of the expression to the end.
Common Mistakes to Avoid
- Doing addition before multiplication because it appears first is wrong because multiplication has higher priority than addition. For example, 3 + 4 x 2 is 3 + 8 = 11, not 7 x 2 = 14.
- Treating division as always before multiplication is wrong because division and multiplication have equal priority. Work from left to right, so 20 ÷ 5 x 2 = 4 x 2 = 8.
- Treating addition as always before subtraction is wrong because addition and subtraction have equal priority. Work from left to right, so 10 - 3 + 2 = 7 + 2 = 9.
- Thinking BODMAS and PEMDAS give different answers is wrong because they describe the same rules using different words. Orders in BODMAS means the same idea as exponents in PEMDAS.
Practice Questions
- 1 Evaluate 7 + 3 x (10 - 6)^2.
- 2 Evaluate 48 ÷ 6 x 2 + 5^2 - 9.
- 3 Explain why 18 - 6 ÷ 3 x 2 must be worked using left to right after the division and multiplication step, rather than doing all division before all multiplication.