Aptitude(Preparation & Quiz)-Order of Operations & BODMAS

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).

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.
Thus, we will say we have 2 + 1 = 3 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, y3, 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.
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.
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.

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.

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!

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