Order of Operations & BODMAS
In this lecture, we are going to cover a basic math concept: “Order of Operations". This is one of the easier mathematical concepts, as it is simply the proper way in which people all over the world have accepted to use the mathematical operators. There are plenty of mathematical operators in use. The basic ones include addition, subtraction, multiplication, and division. Apart from these, there are many others like exponents and brackets which change the way we understand mathematical representations and bring in a level of complexity. The problems start arising when we have more than one operator in an equation. How does one decide whether to apply division first or whether to go with addition?
Order of Operations means that if in an equation, you are provided with more than one operator(s), you can apply the operators in a sequence that everyone agrees upon. Just imagine what can happen, if two friends solve the 3 × 2 + 4 expression in their own manner: One would solve the multiplication expression first, by multiplying 3 and 2 and then add 4. This would give the answer as 10. The other friend would add 2 and 4 first, and then multiply the answer (6) with 3 which would make the answer 18. Which of these is right? No one is wrong here, if you keep an open mind. The only issue is that we want the two friends to arrive at a single answer. Let's see, why it is so necessary to have a single answer to a mathematical expression.
What is the need for an order of operations? We want this, as much as we want anyone to think of a 'chameleon' when we say 'chameleon'. A mathematical expression is just like a word. It is a representation of something. Since we do not want people to think of two different things when we clearly mean one, therefore it makes sense to follow a certain order of operation. A set order of operations ensures that people everywhere in the world will understand a mathematical expression like this one: “3 × 2 + 4" in exactly the same way and that everyone will get the same result when they solve it.
Along with this lecture is a basic video detailing BODMAS concepts and more related ideas. If you believe in viewing rather than reading, take a look at it.
Theory Lecture
BODMAS rule
B = Brackets
O = Orders (exponents)
D = Division
M = Multiplication
A = Addition
S = Subtraction
Brackets > Orders > Division & Multiplication > Addition & Subtraction
This is the order in which the world agrees to use the mathematical operators in an equation. So in an expression that has all the operators, we would first solve the “Brackets", then the “exponents" (or the order), then “division" and “multiplication" operators would be resolved, after which we will solve for “addition" and “subtraction".
An important point here is that Division and Multiplication are at the same level of precedence. Therefore, in an equation, if you have division and multiplication both, they can be solved simultaneously or one after the other, it doesn't matter. Similar is the case with addition and subtraction.
For e.g. 10 × 2 + 4 ÷ 2 – 10 = ?
This equation would be solved using the BODMAS rule in the following manner:
= 10 × 2 + 2 – 10
= 20 + 2 – 10
= 22 – 10
= 12
Operands
An entity or a quantity upon which a mathematical operation is performed, is called as an operand. For e.g. in the expression 2 + 3, the numbers 2 and 3 are operands, while the sign '+' is the operator. Let's also take a look at what are mathematical operators:
Mathematical Operators
Let's talk briefly about some of these important mathematical operators. They are called operators because they 'operate on' the quantities or the numbers.
Addition
Addition is the process of combining two or more quantities. When you have two peaches, and you bring one more, you have basically added the two with the one to get three peaches.
The sign of the addition operator is '+' (pronounced plus).
Thus, we will say we have 2 + 1 = 3 peaches.
Subtraction
Subtraction is the process of removing one quantity from another. When you take three out of a group of five peaches, you are left with two. This is subtraction. Sad, but true! Just two peaches!
The sign of the subtraction operator is '−' (pronounced minus).
Therefore, we will say, we now have 5 − 3 = 2 peaches.
Multiplication
Multiplication is repeated addition. When you start adding more and more peaches to this small group of two peaches, it grows. Now, suppose you are adding three peaches every time you come and study. Suppose you came in ten times. Now, instead of saying that we have "2 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3" peaches, would it not be better to say we have "2 + 10 times 3" peaches? Oh yes, it would definitely be simpler and easier to read. This is why, we use multiplication.
The sign of the multiplication operator is '×'.
So, we will now say that we have 2 + 10 × 3 = 2 + 30 = 32 peaches.
Division
Division is separating something into equal parts. Suppose you had to separate 32 peaches into 4 equal groups. To achieve that, we would have to know how to divide a group of 32 entities in 4 equal parts.
The division sign is '÷'.
So, we will represent the division in this way:
32 ÷ 4 = 8
i.e. there will be 8 peaches in each group.
Orders/Exponents
Orders or exponents are the number of times the entity must be multiplied with itself. These are normally of the type x ^{2}, y^{3}, etc.
Brackets
Brackets are the operators used to break the general order of operations. Suppose in an equation, it is desired that multiplication be performed on the numbers before division, then we use brackets to group the operands. The commonly used brackets are (), {} and [].
For e.g.
10 × 2 + 4 ÷ 2 – 10 = ?
In the given equation, if we want to set a different order of solving the equation, we could do something like this
10 × (2 + 4) ÷ 2 – 10 = ?
Here, instead of doing the division/multiplication operation first, we would have to complete the addition operation first, and then proceed with the remaining operators in the usual sequence. If there are more than one bracket(s), we will have to work them all out first. There are different types of brackets too: Parentheses (), Curly brackets {} and Square brackets []. These are used for creating nested operations.
The order of prominence or the order of precedence states that we can nest multiple parentheses in curly brackets and multiple curly brackets in square brackets, viz.
[{(x + y) – (2x + 3)} × {(3y - 4) × 3}]
Here, we will solve the inner brackets (parentheses) first, then the curly brackets and then finally the square brackets.
Note: We can nest similar brackets too. For example, it is acceptable to nest parentheses in another pair of parentheses. It is a matter of convention and choice. But remember, the inner brackets always get solved first.
Suggestion: Brackets are utilized for not just their utility but sometimes also their aesthetic value in an equation. In a large equation, which has multiple operators, sometimes it becomes hard to keep track of the correct order. If you want to make it easier for others to understand your expression, you can use brackets. For e.g. in the expression, 10y + 4 × 3 + 12x + 3 × 12, through the BODMAS rule, we can ascertain the correct answer, but it can be tedious to solve. Instead, the same expression when written in this manner: 10y + (4 × 3) + 12x + (3 × 12) can increase the readability of the expression. Still, it should be kept in mind that bracket usage in this way is error-prone. If you are not careful, a misplaced bracket could change the correct answer. Choose with caution.
BODMAS or BEDMAS or PEMDAS or BIDMAS
In different countries, according to language differences, people refer to the Order of Operations differently. In the US, they call 'Orders' as 'Exponents'. Also, they use the word 'Parentheses' instead of the other choice - 'Brackets'. This is why, the acronym changes to PEMDAS. Also note that since division and multiplication are of the same rank/level, therefore it does not matter if they call it PEMDAS and not PEDMAS. Always remember to go from left to right and you are secure.
Note: In the case of exponents, it is a normal style to work from the top to the bottom. Don't confuse the left and right approach (all operators) with the top-down approach (exponents). For e.g. in the given expression,
we will first solve the y ^{z} part. Suppose the answer for this comes out to be 'm'. Once we have calculated this, we will then move to the lower part, i.e. x^{m}. Thus, you can see that this is a different type of movement. It can be considered to be a right-to-left movement (z, y and then x). This will be reverse of the normal order of calculation that we have seen till now. Or you could call it the top-down approach.
These are mnemonics created to remember the order. Depending upon different language preferences, people have named the order of operations differently:
BODMAS (Brackets > Order > Division & Multiplication > Addition & Subtraction) -- BODMAS is the common acronym in India and other countries.
BEDMAS (Brackets > Exponents > Division & Multiplication > Addition & Subtraction) – BEDMAS is commonly utilized by schools and colleges in countries like Canada, etc.
BIDMAS (Brackets > Indices > Division & Multiplication > Addition & Subtraction)
PEMDAS (Parentheses > Exponents > Multiplication & Division > Addition & Subtraction) PEMDAS is commonly utilized in the US.
There are even others, depending upon how you want to order the multiplication and division operators (BODMAS or BOMDAS), or whether you want to remove the confusion and keep it simple (PEMA). People can question that if division and multiplication are at the same level, and if addition and subtraction are at the same level of precedence, then why do we have D before M, and A before S in the acronym? Well, there can be multiple reasons. Some say, there is a historical justification to this fact. All we know is that it is easier to say BODMAS than BOMDSA or PEMDAS than PEDMSA. Probably, this was a reason too for the popularity of these acronyms over their counterparts.
Sometimes, because it is hard to remember an acronym that hardly makes any sense, people come up with familiar or funny expansions. For e.g. PEMDAS is often expanded as “ Please Excuse My Dear Aunt Sally". You can create your own expansion if you are feeling adventurous.
BODMAS implementation
Step 1: First of all, look for any brackets or parentheses in the equation. If you find some, solve the operations in them before anything else. For solving the brackets, work from the innermost one first. Once you have worked out the innermost one, whichever bracket is next, solve that. This way, you can solve all the brackets. Remember, brackets must be solved from inside out.
Step 2: Next, look out for any powers or exponents and work them out.
Step 3: After you have solved the above steps, focus on any division or multiplication operators in the equation. Working left to right, solve them one by one.
Step 4: Lastly, solve the addition and subtraction operators, working left to right.
What if there are similar operators?
Sometimes, we have similar operators in an equation. For e.g. you might have more than one multiplication sign in an equation. In these cases, how does one identify which sign to use first? The order of precedence helps here: it states that in case of similar operators, the operators on the left side are solved first. This is repeated till all the similar operators are exhausted.
For e.g. in (12 ÷ 6 – 2 ÷ 4), we would perform the (12 ÷ 6) operation first, and then the (-2 ÷ 4) operation.
This is just like when we are dealing with operators like division and multiplication (or addition and subtraction) in the same equation, it is a smart move to solve from Left to Right.
The strange case of the negative numbers
A negative number in an equation, or simply a negative sign can cause big problems with the BODMAS rule (if you are not careful). Since the rule looks like subtraction comes after addition, it can be confusing to follow the rule and still get the wrong answer. Take a look at this expression here:
18 ÷ 3 – 12 + 6 + 10
The next step would be
6 – 12 + 6 + 10
Now, if you forgot that subtraction and addition are on the same level and need to be solved simultaneously, then this is what will happen:
6 – 18 + 10
= 6 – 28
= – 22
This is the wrong answer. The right answer is 10, which can be reached at in two ways:
(a) First, if we follow what we have read till now and assume that we need to solve both addition and subtraction operations simultaneously, then this is what we will get:
18 ÷ 3 – 12 + 6 + 10
= 6 – 12 + 6 + 10
= –6 + 6 + 10
= 0 + 10
= 10
Here, we solved each operation singly and showed you the result by moving left to right. You can do all this together in one step.
(b) Second, if you want to solve subtraction after addition, then remember that in the given equation, you are not adding 12 and 6, instead you are adding (–12) and 6. In this way, we can be sure that we will get the same answer.
Type of questions
BODMAS questions are fairly basic in nature. They are asked in aptitude based examinations for a few reasons. First of all, this is a basic concept and so, the candidate should be completely sure about this. Secondly, these types of questions can be fairly long and cumbersome to solve. If a question throws in more than seven operators, it can start to look like a time-consuming question. But most often, these questions are put in to test the speed with which a candidate can solve them. So the next time you see a long BODMAS type question on a test, don't just skip it to look for a simpler question. Give it a try. It might turn out to be not all that long!
Sample Questions
Why don't we work out a few questions so that we can understand how BODMAS works in practice.
12 + 10 × 3 ÷ 3 – 12
Step 1: Solve the multiplication or the division operation first. Since we have both multiplication and division, we will begin solving from the one that comes first from left.
= 12 + 30 ÷ 3 – 12
Step 2: Next, we solve the division operation
= 12 + 10 – 12
Step 3: Now, we have an addition and a subtraction operation on our hands. We know that both of these are at the same level/rank and hence we need to work from left to right. From the left, the first operator we encounter is the addition one, so
= 22 – 12
Step 4: Finally,
= 10
Similarly, you can solve the following questions:
16 ÷ 2 × 2 ÷ 2 × 2 + 2
22 + 22 − 18 - 18
(62 + 38) ÷ [{(65 ÷ 5) - 3} × 2]
2 + 22 − 23 − (3 + 3) ^{2} + 32
Why don't calculators follow BODMAS?
Actually, they do! I mean, most of them do.
In a non-scientific calculator, most often there is no adherence to order of operations and whichever operator comes first in a left to right order, is solved first. So, if you input the expression, 2 + 4 × 6 in a basic calculator, you will get the answer as 36. Surprising, right! But that's how it is. The calculator is simply working on this expression from left to right without worrying about BODMAS or PEMDAS. This can create problems and hence care should be taken when working with these types of calculators.
On the other hand, most scientific calculators and even a lot of other calculators, follow BODMAS. The difference lies in the way these calculators are structured. The simpler calculators are programmed to accept two numbers or operands with an operator and as soon as you enter these three things, the calculator processes the answer. Whereas, in the case of more sophisticated calculators, we can enter the whole expression and the calculator will wait for us to complete the input. Only when we press the equal sign, will the calculator evaluate the mathematical expression. Working this way, the latter type of calculators can honour the BODMAS system.
The Basic Aptitude Course
A great way to build your aptitude is to join our comprehensive basic aptitude course. This course will not only teach you about BODMAS, but also all the other important mathematical concepts that are so vital to any job or course that you might be studying. Oh! And there's more in there apart from 'mathematical' aptitude.
And did I mention, this is a completely free course. Thank you very much for the applause.
And here's a presentation detailing the important concepts about BODMAS.
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This ebook also has quite a few practice questions along with concise theory.
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