Order Of Operations – Definition, Rules & Examples

Order of Operations is a collection of mathematical principles that determine the order in which computations are to be executed in an expression. The order of operations and rules are expressed here:

• Brackets ( ), { }, [ ]
• Exponents
• Division (÷) and Multiplication (×) (from left to right)
• Addition (+) and Subtraction (-)

These guidelines guarantee that everyone gets the same solution while solving a problem.

PEMDAS vs BODMAS

Order of Operations principles specify the order in which mathematical equations are solved, maintaining consistency and correctness throughout calculations.

• These criteria are critical for preventing misunderstanding and producing accurate outcomes.
• They include parentheses, exponents, multiplication and division, and addition and subtraction, which are often known by acronyms like as PEMDAS or BODMAS.

PEMDAS Rule

PEMDAS is an abbreviation for:

• P stands for Parentheses ( ), { }, [ ]
• E stands for Exponents (a^b)
• M stands for Multiplication(×)
• D stands for Division(÷)
• A stands for Addition(+)
• S stands for Subtraction(-)

This rule prioritizes calculations in brackets first, then exponents, multiplication and division, and finally addition and subtraction.

Examples of PEMDAS

Let's solve the expression (3 + 2) × 4 - 6 ÷ 2 using PEMDAS

Step 1: Inside Parentheses: (3 + 2) × 4 - 6 ÷ 2 = 5 × 4 - 6 ÷ 2
Step 2: Multiplication: 5 × 4 - 6 ÷ 2 = 20 - 6 ÷ 2
Step 3: Division: 20 - 6 ÷ 2 = 20 - 3
Step 4: Subtraction: 20 - 3 = 17

So, the result is 17

BODMAS Rule

BODMAS is an abbreviation for:

• B stands for Brackets ( ), { }, [ ]
• O stands for Order
• D stands for Division (÷)
• M stands for Multiplication (×)
• A stands for Addition (+)
• S stands for Subtraction (-)

It defines the correct order for solving mathematical expressions to ensure consistent results.

Examples of BODMAS:

Let's solve the expression 6 + 3 × 2 - 4 ÷ 2 using BODMAS

Step 1: Multiplication: 6 + 3 × 2 - 4 ÷ 2 = 6 + 6 - 4 ÷ 2
Step 2: Division: 6 + 6 - 4 ÷ 2 = 6 + 6 - 2
Step 3: Addition: 6 + 6 - 2 = 12 - 2
Step 4: Subtraction: 12 - 2 = 10

So, the result is 10

Solved Examples

Example 1: Solve expression: 8 + (5 × 3) − 2² using PEMDAS.

Solution:

Step 1: Parentheses: 8 + (5 × 3) − 2² = 8 + (15) − 2²
Step 2: Exponents: 8 + (15) − 2² = 8 + 15 − 4
Step 3: Addition: 8 + 15 − 4 = 23 − 4
Step 4: Subtraction: 23 − 4 = 19

Therefore, the solution is 19

Example 2: Solve expression: 12 − 4 × (6 ÷ 2) + 5 using BODMAS.

Solution:

Step 1: Brackets: 12 − 4 × (6 ÷ 2) + 5 = 12 − 4 × 3 + 5
Step 2: Multiplication: 12 − 4 × 3 + 5 = 12 − 12 + 5
Step 3: Addition: 12 − 12 + 5 = 12 - 17
Step 4: Subtraction: 12 - 17 = 5

Therefore, the solution is 5

Example 3: Solve expression: 3 × (4 + 2)² − 10 using Order of operation.

Solution:

Step 1: Parentheses: 3 × (4 + 2)² − 10 = 3 × (6)² − 10
Step 2: Exponents: 3 × (6)² − 10 = 3 × 36− 10
Step 3: Multiplication: 108− 10
Step 4: Subtraction: 98

Therefore, the solution is 98