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BODMAS Rule

BODMAS rule is an acronym that is 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 (whichever comes first from the left side of the expression), and then finally, subtraction or addition, whichever comes on the left side.

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. What is BODMAS?
2. BODMAS Rule Meaning and Definition
3. BODMAS vs PEMDAS
4. FAQs on BODMAS Rule

What is BODMAS?

BODMAS, which is 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 that have 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 a little trickier! Let us 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 we 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? In such cases, we use BODMAS to find the correct answer. Let us look at the example given below to get an idea of how BODMAS works:

BODMAS Rule Example

BODMAS Rule Meaning and Definition

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

BODMAS Definition

According to the BODMAS 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, in the end, work on the addition and subtraction. Following the order of operations in the BODMAS rule, always results in the correct answer. 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. 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.

Observe 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 that are 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 in different countries. While using the BODMAS rule or the PEMDAS rule we should remember that when we come to the step of division and multiplication, we solve the operation which comes first from the left side of the expression. The same rule applies to addition and subtraction, that is, we solve that operation that comes first on the left side.

BODMAS and PEMDAS Rule

When to Use BODMAS?

BODMAS is 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 the BODMAS Rule

The simple rules to remember the BODMAS rule are given below:

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

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. So, if there are multiple brackets in an expression, all the same types of brackets can be solved simultaneously.
  • An error occurs in certain cases because of lack of proper understanding of the addition and subtraction of integers. For example, 1-3+4 = -2+4 = 2. But sometimes the following errors are made that lead to the wrong answer, such as 1-3+4 =1-7 = -6.
  • An error by assuming that division has higher precedence over multiplication and addition has higher precedence over subtraction. Following the rule of left to right while choosing these operations helps to get the correct answer.
  • 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.

Topics Related to BODMAS

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

 

BODMAS Rule 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 finally, addition, 2 + 7 = 9

    ∴ The expression is simplified and the answer is 9.

  2. Example 2: Evaluate using the order of operations using the BODMAS rule: (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: Solve the given expression applying the BODMAS rule: [50-{3×(9+7)}]
    Solution:
    To solve this expression, [50-{3×(9+7)}], we will use the following steps:
    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.
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FAQs on BODMAS Rule

What is the Bodmas Rule in Mathematics?

The BODMAS rule refers to the rule that is followed to solve mathematical expressions. BODMAS is the order of operations for mathematical expressions that involve more than one operation. The acronym 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 in the order of the BODMAS rule. We solve the part 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 means simplifying exponents or roots in the expression, if any, before arithmetic operations. In certain countries, 'O' is used to represent 'of' which again means multiplication.

How to Apply the Bodmas Rule?

BODMAS rule can be applied in case of expressions that have more than one operator. In that case, we simplify the 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 the 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 is the Bodmas Rule not Applicable?

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

Who Invented the 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?

The full form of BODMAS is, 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.

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