BODMAS Explained - Order Of Mathematical Operations
When you're given a sum has two numbers and one operator, calculating an answer seems straight forward (25 × 3 = 75). But, what happens if someone throws in a couple more numbers and operators: (5 + 25 × 3 − 2 = .....)? Which part do you do first? Thankfully, there is a set of simple rules for solving mathematical sums. This is where BODMAS comes in.
What is the BODMAS rule?
BODMAS is an acronym that represents the order of mathematical operations. When a sum contains multiple numbers and operations, you need to know which part to solve first in order to solve it in the correct order. If you don't, you'll get an incorrect answer.
BODMAS stands for
- Brackets (any part contained in brackets comes first)
- Order (operations containing powers or square roots)
How well known is BODMAS?
In 2012, Dr Peter Price, co-founder of the Classroom Professor website, posted a mathematical question on his Facebook page. This is what he asked:
Can you answer this?
7 - 1 x 0 + 3 ÷ 3 = ?
The post quickly spread around Facebook, with over 70,000 people seeing the post and 6,000 people leaving answers and comments. After 2 weeks, Peter pulled together the results - results that surprised him. Only 26% of respondents gave the correct answer (the correct answer is 8).
When you consider that, psychologically, people are mostly likely to comment on something public like this if they are fairly confident of their answer, so as not to seem foolish, it appears to say a lot about mathematical understanding in the population as a whole. Indeed, it appears to demonstrate that the large majority of people (probably much more than 74%) don't understand the concept of BODMAS and the order of operations.
Sequencing sums: BODMAS
How often have you seen this kind of question doing the rounds on Facebook? The correct answer for this is 12.
In arithmetic, there are two types of components: the numbers themselves and the operators (also called operations) that tell you what to do with those numbers.So, in the sum 7 x 3 + 5 there are three numbers; 7, 3 and 5 and two operators, a multiplication (x) and an addition (+).
You can also see that this sum can produce two different answers depending on which order you use the operators.
- If you multiply seven by three and add five, the answer is 26.
- But if you multiply seven by the sum of three and five (eight), the answer comes out at 56.
So, how do you know in what order to proceed? Trained mathematicians know that there is a definite hierarchy of operations and a default order for performing basic arithmetical operations: adding, subtracting, multiplying and dividing).
BODMAS or PEMDAS?
The definitive order of operations is summed up in the acronym BODMAS, which stands for Brackets, Order, Divide, Multiply, Add, Subtract. It would be easier if BODMAS was recognised worldwide, but unfortunately it isn't.
In the USA it's normally called PEMDAS (Parenthesis, Exponent, Multiply, Divide, Add, Subtract) or PIDMAS (Parenthesis, Index, Divide, Multiply, Add, Subtract). Other places in the world might use BIDMAS (Brackets, Index, Divide, Multiply, Add, Subtract), while Canadians sit in the middle with BEMDAS (Brackets, Exponent, Multiply, Divide, Add, Subtract).
Applying the order of operations
The sequence of the order of operations (whether it be BODMAS, PEMDAS, PIDMAS, BIDMAS or BEMDAS) remains the same:
Step 1: Brackets
The highest level order is defined by anything contained in brackets. These sums are always calculated first. But what if there is more than one set of brackets? The rule then is to start at the innermost set and work outwards. Performing each bracketed calculation should leave you with a single number, allowing that set of brackets to be removed.
Step 2: Order or Index
The terms Order or Index all relate to operations containing powers or indices such as squaring or square rooting. These calculations are all performed second.
Steps 3 and 4: Divide and Multiply
The third and fourth steps, division and multiplication, have equal weight and so form a third level order of operations that are carried out at the same time. Importantly, when two or more operations of the same order appear one-after-another, the operations should be carried out from left to right.
So, if faced with a sum like:18 ÷ 6 × 4 ÷ 8
you just work from left to right. Eighteen over six is three, times four is twelve, divided by eight is 1.5.
Steps 5 and 6: Add and Subtract
Again, these carry equal weight. Therefore the addition and subtractions form the fourth and final level order of operations The third and fourth steps, division and multiplication, have equal weight and so form a third level order of operations that are carried out at the same time, again working from left to right.
In summary, once you have performed all the "B" and "O/E/I" calculations, in that order, just work from left to right doing any "Ds" or "Ms" as you find them, then go back to the beginning and work from left to right on all the "A" or "S" sums.
On the next page we look at some examples of BODMAS in action. We also examine exponents, brackets AND we end with a quick quiz of your BODMAS skills.
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Last update: 09 November 2018