Order of Operations: BODMAS

Order of Operations

In mathematics, an operation is an action such as addition, subtraction, multiplication and division.

In a given mathematical expression, the order in which we carry out a calculation is important. The wrong order of operations will often lead to the wrong answer.

For example, consider the expression \[4\div5+6\times2.\]

  • If we first divide $4$ by $5$ to get $0.8$, then multiply $6$ by $2$ to get $12$, and then add $0.8$ to $12$ we get $12.8$.
  • If instead we first add $5$ and $6$ to get $11$, then multiply $11$ by $2$ to get $22$, and then divide the $4$ by $22$ we get $0.182$ (to $3$ d.p.).

We can see that the result is very different when we perform the operations in a different order.

BODMAS

BODMAS is an acronym which tells us the correct order in which we should carry out mathematical operations:

  Brackets
  Order
  Division
  Multiplication
  Addition
  Subtraction

Division and multiplication, and addition and subtraction, have the same priority - the convention is to work from left to right when the order of operations would be unclear.

Note: An alternative form of this mnemonic is BIDMAS, where the I stands for indices. PEMDAS (“Parentheses, exponents, ...”) and BEDMAS are also used in the USA and Australia.

Returning to the above example, the correct answer would be the first answer as it follows the rules of BODMAS: division can be done before multiplication and must be done before addition, and multiplication comes before addition. So the answer is $12.8$.

We will now look at more examples to practice using BODMAS.

Example 1

Evaluate the following expression: $20\times(100+1)$.

Solution

Applying the BODMAS rule, we know that we must first consider everything inside the brackets. Since the only operation inside the brackets is a single sum, we first add $1$ to $100$ to get $101$. The expression then becomes: \[20\times101\].

All that is left is to multiply the two numbers together. This gives $2020$ and so we have: \[20\times(100+1)=2020\].

Example 2

Evaluate the following expression: $(-4)\times102$.

Solution

First notice that although this expression contains a pair of brackets, the brackets are only there to indicate that $-4$ is a negative number.

Applying the BODMAS rule, we must first evaluate the exponent: $10^2=100$.

Finally, we multiply $100$ by $(-4)$ to get $-400$. We thus have

\[(-4)\times10^2=-400\]

Example 3

Evaluate $2+4\times3-1$

Solution

Applying BODMAS, we do the multiplication first. $3\times4=12$ so we have: \[2+12-1\] Then addition and subtraction have the same priority, so we can do either next. Performing the addition first we have: \[14-1=13.\] Check for yourself that doing the subtraction before the addition gives the same answer.

Example 4

Evaluate $3+2^2$

Solution

Applying BODMAS, we evaluate the power first, then the addition. $2^2=4$ so we have: \[3+4=7.\]

Test Yourself

Try our Numbas test: Arithmetic operations

External Resources