PEMDAS and the order of operations
Memes involving ambiguous mathematical equations have become a common sight on Facebook news feeds everywhere. You've probably seen them - commonly featuring a mixture of additions, subtractions, multiplications and divisions and baiting you that "90%" of people get the answers wrong.
What these memes rely on is the human desire to be part of the superior minority that know the correct answer. As you'll discover, however, most people get the answers wrong (assuming there is actually a correct answer, of course).
In 2012, Dr Peter Price of the Classroom Professor website decided to try an experiment and posted the following mathematical question on his Facebook page:
The post quickly gained popularity and spread. Over 70,000 people ended up seeing it and 6,000 people responded with answers. The question was left up for 2 weeks before answers were collated. Having examined the results, Peter was surprised to find that only 26% of respondents gave the correct answer (see the article here). The correct answer is 8.
So, what does this say about the mathematical education of the population as a whole? Well, it could be suggested that a large majority of people around the world don't understand the concept of PEMDAS and the order of operations.
In order to help teach the order of operations, I have put this article together. With the help of Nick Valentine, we will take you gently through the world of PEMDAS.
PEMDAS and sequencing sums
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 PEMDAS.
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 6 x 2 + 8 there are three numbers; 6, 2 and 8 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 six by two and add eight, the answer is 20.
- But if you multiply six by the sum of two and eight (ten), the answer comes out at 60.
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).
PEMDAS or BODMAS
The definitive order of operations is summed up in the acronym PEMDAS, which stands for Parenthesis, Exponent, Multiply, Divide, Add, Subtract. It would be easier if PEMDAS was recognised worldwide, but unfortunately it isn’t.
Some think of PIDMAS (Parenthesis, Index, Divide, Multiply, Add, Subtract) while in the UK it’s normally called BODMAS (Brackets, Order, Divide, Multiply, Add, Subtract) or BIDMAS (Brackets, Index, Divide, Multiply, Add, Subtract). Canadians sit in the middle with BEMDAS (Brackets, Exponent, Multiply, Divide, Add, Subtract).
How does PEMDAS work?
Regardless of the exact terminology, the sequence remains the same:
Step 1: Parenthesis
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: Exponent (or Index)
The terms Exponent 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: Multiply and Divide
The third and fourth steps, multiplication and division, 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:20 ÷ 5 × 3 ÷ 6 18 ÷ 6 × 4 ÷ 8
you just work from left to right. Twenty over five is four, times three is twelve, divided by six is 2.
Steps 5 and 6: Add and Subtract
Once 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 "P" and "E/I" calculations, in that order, just work from left to right doing any "Ms" or "Ds" 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 demonstrate some examples of PEMDAS calculations, we examine exponents, parenthesis AND we finish with a quick quiz of your PEMDAS skills.
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Last update: 15 March 2016
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