Order of Operations | PEMDAS, BODMAS

Ever wondered about those irritating ‘Solve it if you are a genius‘ or ‘90% will get it wrong‘ puzzles that flood your Facebook and LinkedIn pages?

Often on the same image, they also have a confused looking Einstein staring back at you wondering why he’s being dragged into this frivolous race for social media domination.

Most of these ‘genius’ level puzzles are basic maths problems that you feel compelled to solve and share, just because 5 of your friends did the same.

Apart from the little kick you get from cracking them, there’s more to it.

Some of the concepts are building blocks based on which more complex problems are designed in aptitude tests like the GMAT, GRE, CAT etc. The most basic rule among them is the order of operations.
 

Order of Operations in Maths | PEMDAS, PEDMAS, BODMAS

The basic operations in mathematics, as everyone knows, are addition, subtraction, multiplication and division.

In algebra, these operations are used with numbers or letters or a combination of both.

A number or variable or number multiplied by a variable is called a ‘term’.

A combination of such terms with operators results in an ‘expression’.

For example: In 2a+b, 2a and b are called terms and 2a+b is called an expression.

Some typical examples of algebraic expressions look like this

1)         2+3(8-4)-6/3

2)         10+7(3-1)*8/2²-1

3)         x+2(4x-5)+3(2(x+6))

Definition: The order of operations is defined as the sequence in which operations are performed on a given mathematical expression.

Ever wondered why we need to follow a sequence while calculating?
 

Why is the order of operations important?

Consider the expression: 2+3(8-4)-6/3

Which one of these is the correct way of solving it?

Method 1:

2+3(8-4)-6/3    →         5(8-4)-6/3        →         5(4)-6/3           →         20-6/3             →         14/3     →         4.6666

Method 2:

2+3(8-4)-6/3    →         2+3(4)-6/3       →         2+12-6/3         →         14-6/3             →         8/3       →         2.6666

Method 3:

2+3(8-4)-6/3    →         2+3(4)–6/3       →         2+12-2             →         12

Every expression can be calculated in more than one way and can result in more than one answer. Of course, not every answer is correct. This is where and why we need to apply the ‘order of operations’.
 

Order of operations rules

The Order of operations for any given expression is governed by the following rule:

Parentheses → Exponent → Division → Multiplication → Addition → Subtraction

OR                

Brackets → Orders → Division → Multiplication → Addition → Subtraction

According to the US learning system, it is ‘PEDMAS’ (Some remember it as ‘PEMDAS’ too). According to the UK learning system, it is ‘BODMAS’.

1. Parentheses or Brackets are always the first to be resolved in a given expression and are evaluated starting from the innermost ones.
2. Exponents or Orders are given the next priority.
3. Division and Multiplication are the next and are treated to be on the same level of precedence.
4. Addition and Subtraction are the last and are treated to be on the same level of precedence.

When operators on the same level of precedence are found, as a thumb rule, we work from left to right.

 

Order of Operations Examples

Nothing like a few ridiculously simple examples to get some clarity.
 

1. Parentheses or Brackets are always the first to be resolved in a given expression and are evaluated starting from the innermost ones.

Consider an expression like this:

4(2+(7(5-3)))

Here the innermost parentheses has (5-3). That is the first to be evaluated.

4(2+(7(2)))

Then comes the 7(2) which is in a parentheses

4(2+14)

Next in the parentheses is 2+14

4(16)

Result is 64
 

2. Exponents or Orders are given the next priority.

Consider an expression like this

5(2²+3)+(2³)²

The expression inside the parentheses is 2²+3 (starting from left to right). Following PEDMAS, we need to evaluate the exponent first before we do the addition, which results in (4+3).

It is important to note here that 2²+3 is different from (2+3)².

Now, our expression looks like this:

5(4+3)+(2³)²

Next, we see (4+3) inside a parentheses.

5(7)+(2³)²

There is an exponent to be evaluated before further operation is performed.

5(7)+8²

5(7)+64

Next comes the multiplication.

35+64

And finally the addition.

Result is 99
 

3. Division and Multiplication are the next and are treated to be on the same level of precedence.

Consider the expression

6*2+5*1+4/2-1

Applying our thumb rule of working from left to right,

6*2+5*1+4/2-1

results in

12+5+2-1

Now, all that’s left is additions/subtractions

18 is our answer.

Note here that performing the addition/subtraction before any multiplication/division is performed would result in wrong answer.
 

4. Addition and Subtraction are the last and are treated to be on the same level of precedence.

Consider the expression:

1+(2(4-3+1)+7)-2

Evaluating the parentheses first,

1+(2(2)+7)-2

Next there is a multiplication

1+(4+7)-2

Evaluating the expression inside the parentheses.

1+11-2

Finally the addition/subtraction.

Our answer is 10.

Sometimes, operations performed in any order would yield the same result.

For instance, in the above example, the calculation of 1+(4+7)-2 would yield the same result when performed in any order.

However, when performing the operation from right to left, it is important to note the ‘-‘ sign before 2, forgetting which, results in erroneous answer.
 

Order of operations examples

Example 1:         2+3(8-4)-6/3

Applying PEDMAS, going from left to right, first is the parentheses

2+3(4)-6/3

There are no exponents here. Performing the multiplication/division

2+12-2

And finally the addition

12 is the result.

 

Example 2:         10+7(3-1)*8/2²-1

Going from left to right, parentheses is the first

10+7(2)*8/2²-1

Next, there is an exponent to be evaluated

10+14*8/4-1

Next comes the multiplication/division

10+28-1

Last comes the addition/subtraction

37        is the result.

 

Example 3:         x+2(4x-5)+3(2(x+6))

Working on the innermost parentheses first and then the outer ones,

x+2(4x-5)+3(2x+12)

There are no exponents and no divisions.

Performing the multiplication

x+8x-10+6x+36

And finally, addition/subtraction

15x+26 is the result.
 

Quiz: Order of Operations Problems

Solve these to see if you’ve got the concepts and rules right.
 

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Problem 1: Click here

Answer 1: Click here

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Problem 2: Click here

Answer 2: Click here

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