BODMAS - order of operations explained
There have been Facebook memes aplenty in recent years involving mathematical questions containing a mixture of additions, subtractions, multiplications and divisions to challenge your brain. They prove popular and spread quickly, as people try to show off their mathematical prowess. In the majority of cases, however, the opposite occurs - people actually get their calculations wrong.
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 post on something public like this if they are fairly confident of their answer, so as not to make a fool of themselves, this 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 the order of operations.
And so we come to the purpose of this article. For those of you wondering which part of the calculation to do first, fear not. Nick Valentine is here to take you gently through the world of BODMAS.
Sequencing sums: BODMAS
One of the trickiest aspects of maths is to know in what order to perform operations within complex sums. In this article, we will show how to navigate the order of operations, also known as 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.
Some think of BIDMAS (Brackets, Index, Divide, Multiply, Add, Subtract) while in the USA it’s normally called PEMDAS (Parenthesis, Exponent, Multiply, Divide, Add, Subtract) or PIDMAS (Parenthesis, Index, Divide, Multiply, Add, Subtract). Canadians sit in the middle with BEMDAS (Brackets, Exponent, Multiply, Divide, Add, Subtract).
Applying the order of operations
Regardless of the exact terminology, the sequence 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: 13 February 2016
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