Order of mathematical processes

When we are given a question such as 4×54×5, we would get the same answer even if we swapped the numbers around and did 5×45×4. The same is true if we were to try to calculate 2+19+82+19+8; it is the exact same as 8+19+28+19+2 or any other combination of the same numbers.

This is because addition and multiplication are commutative, meaning that no matter what order they are placed in, you will always get the same answer. However, if we tried the same for division or subtraction, we would find that this is not the case. For example, 5−35−3 does not give the same answer as 3−53−5, and neither do 5÷35÷3 and 3÷53÷5. This is because subtraction and division are what is called non-commutative; the order in which they occur is vital to getting the correct answer.

BIDMAS

When you are faced with a large sum in mathematics it is extremely important that you do certain operations in the correct order. For example, doing the sum 4÷3+54÷3+5 could be calculated as (4÷3)+5=193(4÷3)+5=193 OR 4÷(3+5)=484÷(3+5)=48

Clearly only one of these can be correct, yet we need to know a fool-proof way of working out which. This is where we will make use of the BIDMAS rule. BIDMAS stands for the order in which calculations should be done. It stands for Brackets, Indices, Division, Multiplication, Addition and Subtraction. Sometimes you will see this written as BODMAS, which is the exact same thing but with the O standing for order (which is just another name for indices).

This rule tells us the exact order in which calculations should be carried out to ensure we get the correct answer. Normally brackets would be inserted into questions so that it is obvious which calculation should be carried out first, but this is not always the case so the BIDMAS rule is of great help.

Example

Calculate7×3+6÷27×3+6÷2

To do this we must work through the BIDMAS rule. Firstly, we do the calculations in brackets first, of which there are none, then the indices and so on. The first calculation that we must then do is the DIVISION:

So we get 7×3+(6÷2)=7×3+37×3+(6÷2)=7×3+3

And next we do the MULTIPLICATION:

So we get (7×3)+3=21+3(7×3)+3=21+3

And finally the ADDITION:

So we have 21+3=2421+3=24

To work through using the BIDMAS rule, we must do small calculations in the correct order, skipping any that do not appear in our problem (such as indices in this example).

BIDMAS on a calculator

Your calculator will already be programmed to use the BIDMAS rule, so you can always directly type a question in and you will get the correct answer.