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BODMAS Rule

BODMAS rule is an acronym used to remember the order of operations to be followed while solving expressions in mathematics. It stands for B - Brackets, O - Order of powers or roots, D - Division, M - Multiplication A - Addition, and S - Subtraction. It means that expressions having multiple operators need to be simplified from left to right in this order only. First, we solve brackets, then powers or roots, then division or multiplication (whatever comes first from the left side of the expression), and then at last subtraction or addition.

In this lesson, we will be learning about the BODMAS rule which helps to solve arithmetic expressions, containing many operations, like, addition (+), subtraction (-), multiplication (×), division (÷), and brackets ( ).

1. BODMAS Introduction
2. What is BODMAS?
3. BODMAS vs PEMDAS
4. When to Use BODMAS?
5. Easy Ways to Remember BODMAS Rule
6. Common Errors While Using the BODMAS Rule
7. FAQs on BODMAS

BODMAS Introduction

BODMAS, referred to as the order of operations, is a sequence to perform operations in an arithmetic expression. Math is all about logic and some standard rules that make our calculations easier. So, BODMAS is one of those standard rules for simplifying expressions having multiple operators.

In arithmetic, an expression or an equation involves two components:

  • Numbers
  • Operators

Numbers

Numbers are mathematical values used for counting and representing quantities, and for making calculations. In math, numbers can be classified as natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, complex numbers, imaginary numbers.

Operators or Operations

An operator is a character that combines two numbers and produces an expression or equation. In math, the most common operators are Addition (+), Subtraction (-), Multiplication (×), Division (÷). For mathematical expressions or equations, in which only a single operator is involved, finding the answer is fairly simple. In the case of multiple operators, finding a solution becomes difficult or a little trickier! Let’s understand this with an example. Jenny and Ron solved a mathematical expression 6 × 3 + 2  separately. The following are the two different methods by which Jenny and Ron solved the expression:

Jenny's Method: 6 × 3 + 2 = 6 × 5 = 30, Ron's Method: 6 × 3 + 2 = 18 + 2 = 20.

As you can observe, Jenny and Ron got different answers. In mathematics, we know that there can only be one correct answer to this expression. How to decide who is correct? Don’t worry! BODMAS is here to help you find the correct answer. Let's look at the example given below to get an idea of how BODMAS works:

BODMAS Example

What is BODMAS Rule?

BODMAS used to evaluate mathematical expressions and to deal with complex calculations in a much easier and standard way.

BODMAS Definition:  According to this rule, to solve any arithmetic expression, we first solve the terms written in brackets, and then we simplify exponential terms and move ahead to division and multiplication operations, and then, at last, moving ahead to addition and subtraction. Here, addition and subtraction can be considered as level 2 operations, and multiplication and division can be considered as level 1 operations as they have to be solved first. Simplification of terms inside the brackets can be done directly. This means we can perform the operations inside the bracket in the order of division, multiplication, addition, and subtraction. Sticking to this order of operations in the BODMAS rule, always give the correct answer. If there are multiple brackets in an expression, all the same types of brackets can be solved simultaneously. For example, (14+19)÷(13-2) = 33 ÷ 11 = 3.

Look at the table given below to understand the terms and operations denoted by the BODMAS acronym in the proper order.

 

B [{( )}] Brackets
O x² Order of Powers or Roots
D ÷ Division
M × Multiplication
A + Addition
S - Subtraction

BODMAS vs PEMDAS

BODMAS and PEMDAS are two acronyms used to remember the order of operations. The BODMAS rule is almost similar to the PEMDAS rule. There is a difference in the abbreviation because certain terms are known by different names at different locations. 

BODMAS and PEMDAS

When to Use BODMAS?

BODMAS has to be used when there is more than one operation in a mathematical expression. There is a sequence of certain rules that need to be followed when using the BODMAS method. This gives a proper structure to produce a unique answer for every mathematical expression.

Conditions to follow:

  • If there is any bracket, open the bracket, then add or subtract the terms. a + (b + c) = a + b + c,  a + (b - c) = a + b - c
  • If there is a negative sign just open the bracket, multiply the negative sign with each term inside the bracket. a – (b + c) ⇒ a – b – c   
  • If there is any term just outside the bracket, multiply that outside term with each term inside the bracket. a(b + c) ⇒ ab + ac    

Easy Ways to Remember BODMAS Rule

One easy way to remember the BODMAS rule is to perform these steps:

  • Simplify brackets first
  • Solve all exponential terms
  • Perform division or multiplication (go from left to right)
  • Perform addition or subtraction (go from left to right)

Topics Related to BODMAS

Click on any of the topics from the list below to learn more about BODMAS or other related articles.

Common Errors While Using the BODMAS Rule

One can make some common errors while applying the BODMAS rule to simplify expressions and those errors are given below:

  • The presence of multiple brackets may cause confusion and thus we may end up getting a wrong answer.
  • Error occurs in certain cases because of lack of proper understanding of addition and subtraction of integers. For example, 1-3+4 = -2+4 = 2. But if you simplify it like this 1-3+4=1-7= -6, you will get a wrong answer.
  • Error by assuming that division has higher precedence over multiplication and addition has higher precedence over subtraction.

Multiplication and division are same level operations and have to be performed in left to right sequence (whichever comes first in the expression) and same with addition and subtraction which are same levels operations to be performed after multiplication and division. If one solves division first before multiplication (which is on the left side of division operation) as D comes before M in BODMAS, they might end up getting the wrong answer.

 

Solved Examples

 

  1. Example 1: Simplify the expression by using the BODMAS rule: [18 –2(5 + 1)] ÷ 3 + 7
    Solution:
    The given expression is [18 –2(5 + 1)] ÷ 3 + 7

    We begin with solving the innermost bracket first. Starting with 5 + 1 = 6. Thus, [18 - 2(6)] ÷ 3 + 7

    Next, we work with the order. thereby multiplying 2 (6) or 2 × 6 = 12. Thus, [18 - 12] ÷ 3 + 7

    There is one bracket left, [18 – 12] = 6. So, 6 ÷ 3 + 7

    After B and O comes D, hence, 6 ÷ 3 = 2. So, 2 + 7

    And lastly, addition, 2 + 7 = 9

    ∴ The expression is simplified and the answer is 9.

  2. Example 2: Evaluate using the order of operations: (1 + 20 − 16 ÷ 4²) ÷ ((5 − 3)² + 12 ÷ 2)
    Solution:
    Step 1: First, we need to simplify the innermost bracket, (1 + 20 − 16 ÷ 4²) ÷ (2² + 12 ÷ 2)
    Step 2: Now we have to evaluate exponents, (1 + 20 − 16 ÷ 16) ÷ (4 + 12 ÷ 2)
    Step 3: Now, we need to divide 16 by 16 and 12 by 2 inside the brackets, and we get, (1 + 20 − 1) ÷ (4 + 6)
    Step 4: Add 1 to 20 and 4 to 6, (21 − 1) ÷ 10
    Step 5: Subtract 1 from 21 to solve the bracket, we get, 20 ÷ 10
    Step 6: Divide 20 by 10 to get the final answer, we get, 2.
    ∴ ​(1 + 20 − 16 ÷ 4²) ÷ ((5 − 3)² + 12 ÷ 2) = 2

  3. Example 3: Simplify the expression by using the BODMAS rule: ​(9 × 3 ÷ 9 + 1) × 3

    Solution:

    Step 1: Using Bodmas Rule (left to right whichever operations come first we will follow that) here, first we need to multiply 9 by 3 in the given expression, ​​​​​(9×3÷9+1)×3, and we get, ​​(27÷9+1)×3
    Step 2: Now, we need to divide 27 by 9 inside the bracket, and we get, (3+1)×3
    Step 3: Remove the parentheses after adding 3 and 1, we get, 4 × 3
    Step 4: Multiply 4 by 3 to get the final answer, which is 12.

    ∴ ​(9 × 3 ÷ 9 + 1) × 3= 12

  4. Example 4: Mellisa has to buy three school uniform sets. If one pair of pants costs $9 and one school t-shirt costs $7, and she pays a total amount of $50 to the shopkeeper, write an expression for finding the money that she will get back after buying 3 sets of uniforms. Also, find the amount she will get back after simplifying the expression.
    Solution:
    Given that cost of one pair of pants = $9 and the cost of one t-shirt = $7.
    So, cost of one set of uniform = $9+$7 = $16
    Expression to find the amount that Mellisa will get back from the shop owner is $[50-{3×(9+7)}]
    To solve this expression, we have to apply the BODMAS rule.
    Step 1: Solve the innermost bracket by adding 9 to 7, that is, 16. So, the simplified expression is $[50-{3×16}]
    Step 2: Multiply 3 by 16, to get $[50-48]
    Step 3: Subtract 48 from 50 to get the final answer, i.e, $2.
    Therefore, she will get $2 back from the shopkeeper after buying 3 sets of school uniforms.
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FAQs on BODMAS

What is the Bodmas Rule in Mathematics?

BODMAS means the order of operations for mathematical expressions involving more than one operation. It stands for B - Brackets, O - Order of powers, D - Division, M - Multiplication, A - Addition, and S - Subtraction.

How Does BODMAS Rule Work?

In any arithmetic expression, if there are multiple operations used, then we need to solve the terms written in brackets first. After solving the brackets, we carry out the multiplication and division operations, whichever comes first in the expression from left to right. Then, we get a simplified expression with only addition and subtraction operations. We solve addition and subtraction from left to right and get the final answer. This is how BODMAS works.

Does BODMAS Apply When there Are No Brackets?

Yes, even if there are no brackets, the BODMAS rule is still used. We need to solve the other operations in the same order. The next step after Brackets (B) is the order of powers or roots, followed by division, multiplication, addition, and then subtraction.

What Does the O in Bodmas Stand for?

O in Bodmas stands for Order which basically means simplifying exponents or roots in the expression, if any, before arithmetic operations.

How to Apply Bodmas Rule?

BODMAS rule can be applied in case of expressions having more than one operator. In that case, we simplify brackets first from the innermost bracket to the outermost [{()}], then we evaluate the values of exponents or roots followed by simplifying multiplication and division, and then, at last, perform addition and subtraction operations while moving from left to right.

Why is Order of Operations Important In Real Life?

The order of operations is a shorthand rule that enables you to follow the right order to solve different parts of a mathematical expression. It is a universal rule to solve all mathematical operations to get the correct answer.

When Bodmas Rule is Not Applicable?

BODMAS rule is not applicable to equations. It is applicable to mathematical expressions having more than one operator.

Who Invented Bodmas Rule? When Was It Introduced?

BODMAS rule was introduced by a mathematician, Achilles Reselfelt in the 1800s.

What is the Full Form of Bodmas Rule?

BODMAS refers to "brackets, order, division, multiplication, addition, subtraction".

Do Calculators Use BODMAS?

Calculators also use the BODMAS rule. The scientific calculators automatically apply the operations in the correct order. Calculations using calculators totally depend on the placement of the digits and operatoers by the user.

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