Order of Operations

If a math expression has only one type of operation, such as only addition or multiplication, you can solve it from left to right. But when expressions mix operations like addition, multiplication, subtraction, or division, it often confuses. Thus, in mathematics, a set of rules known as the order of operations is followed. The order of operations tells us how to solve equations when they contain more than one operation. The common operations used in math are: 
 

• Addition (+)
• Subtraction (-)
• Multiplication (×)
• Division (÷)
• Brackets 
• Exponents 

As these operations do not have the same priority, we follow a standard sequence. A simple way to remember this sequence is by memorizing the acronym PEMDAS, where each letter stands for a mathematical operation. 

Order of Operation Definition
 

The order of operations is a set of mathematical rules that determines the correct sequence for solving expressions that contain more than one operation. It ensures the order by evaluating parentheses or brackets first, exponents next, followed by multiplication or division, and finally addition or subtraction.
 

We follow the order of operations, when mathematical expressions contain more than one operation. These rules ensure that everyone simplifies an expression the same way and arrives at the correct answer. The rules involved in order of operations are: 
 

Order of operations rule 1: Solve inside parentheses or brackets first.
 

Observe the expression and identify any grouping symbols like ( ), { } or [ ].  The first rule is to solve the numbers present inside the parenthesis or brackets. These must be always solved from inside to outside in this order. 

• () - round brackets
• {} - Curly brackets
• [ ] - square brackets

Inside the parentheses, you must still follow the overall order of operations. Grouping symbols come first because they show operations that must be completed before anything else. 

Order of operations rule 2: Evaluate exponents/orders.
 

After the brackets are cleared, look for any powers, roots, or exponents.
These operations come next and must be completed before moving to multiplication or division.

Order of operations rule 3: Perform Multiplication and Division (left to right).
 

Now, focus on the operators, multiplication and division. These two operations have equal priority, so you solve whichever appears first from left to right in the expression.

Order of operations rule 4: Perform addition and subtraction (left to right).
 

Finally, complete any addition or subtraction. Just as with multiplication and division, these operations also have equal priority and should be solved from left to right.

These steps together form the sequence commonly remembered as PEMDAS or BODMAS, which helps students recall the correct order:
 

• P/B – Parentheses / Brackets
• E/O – Exponents / Orders
• M/D – Multiplication or Division
• A/S – Addition or Subtraction.

PEMDAS and BODMAS are acronyms that help people remember the correct order of operations in math.

PEMDAS
 

PEMDAS is used in the U.S.A. It tells us the sequence to follow so we solve expressions correctly and consistently. PEMDAS specifies that we solve parentheses first, then exponents, then multiplication or division, and finally addition or subtraction. While calculating PEMDAS, do them from left to right.

• P stands for parentheses (brackets) (), [], {}
• E stands for exponents (x2, x3. . .)
• M stands for multiplication
• D stands for division
• A stands for addition
• S stands for subtraction

​
Example: Evaluate Order of Operation 12 ÷ 3 × 2.

​You have a question about whether to do multiplication or division first. The solution is simple, just go from left to right.
 

Step 1: First, perform division from left to right: 
12 ÷ 3 = 4. 

Step 2: Next, multiply the result by 2:

4 × 2 = 8. 

Therefore, the answer is 8. 

BODMAS 
 

BODMAS is used in countries like the UK and India. It helps to avoid confusion while solving the expression. In BODMAS is a rule that tells us the order to solve math problems: Brackets first, then Order (like powers), then Division and Multiplication (left to right), and finally Addition and Subtraction (left to right). In PEMDAS or BODMAS, multiplication and division are at the same level. You just solve them from left to right, in the order they appear.

• B stands for Brackets (), [], {}
• O stands for Order (x2, x3. … )
• D stands for Division 
• M stands for Multiplication
• A stands for Addition
• S stands for Subtraction.
   

Example: Evaluate order of operations 6 + 2 (3 + 1).
 

Step 1: According to the order of operations (PEMDAS), evaluate parentheses first:

(3 + 1) = 4.
 

Step 2: Next, multiply the result by 2.

2 × 4 = 8. 

Step 3: Finally, add the result to 6.

6 + 8 = 14. 

Therefore, the answer is 14. 

Let’s try a different example to understand the order of operations. 

How to use PEMDAS step-by-step:
 

• Start with the innermost parentheses and simplify inside out.
   
• Calculate any exponents.
   
• Perform multiplication and division as they appear from left to right.

• Perform addition and subtraction as they appear from left to right.

Example: Evaluate the order of operations 3 + 4  (23 - 5)

Step 1: Parentheses first (Brackets).

\(2^3 - 5 = 2 × 2 × 2 - 5 =  8 -5\)

\(8 - 5 =3\)    

Step 2: Exponents.

There are no additional exponents to evaluate, as 2³ was solved in Step 1.                 

Step 3: Expression becomes,

\( 3 + 4 × 3\)

Step 4: Multiplication. 

\(4 × 3 = 12\)

The expression becomes \(3 + 12\)

Step 5: Addition.  

\(3 + 12 = 15\)

The answer is 15.
 

How to use BODMAS step by step: 

•  Solve everything inside brackets first.
   

•  Calculate Orders (powers and roots).
   

• Do multiplication and division in order, starting from the left.
  
   
• Perform Addition and Subtraction from left to right.

Example using BODMAS.

Simplify:

\(5 + 2  (32 - 1)\)

Step 1: Solve the brackets:

\(3^2 -  1 =  3 × 3 -1 = 9 - 1\)

\(9 - 1 = 8\)

The expression becomes \(5 + 2 × 8\).

Step 2: There are no additional exponents to evaluate, as 3² was solved in Step 1.

Step 3: Multiplication.

\(2 × 8 = 16\)

The expression becomes \(5 + 16 \)

Step 4: Addition.

\(5 + 16 = 21\)

The value of the expression is 21.

Use these tips and tricks to follow the order of operations correctly and solve problems with confidence.

• Remember the acronyms (PEMDAS and BODMAS). It will help keep the order clear while solving the expression.

• When solving a math problem, do the brackets or parentheses first.

• Calculate the power and roots before other operations.

• Multiply and divide from left to right.

• Add and subtract from left to right.

• Review the steps and avoid simple mistakes.
  
   
• Use real-life examples, such as calculating shopping totals or measuring ingredients, to show students why the order of operations matters.
  
   
• Parents and teachers can use various visual aids, such as color-coded steps, flowcharts, or PEMDAS posters, to help students memorize the sequence. 
  
   
• Please encourage students to write step-by-step and ask them to show their work clearly. This helps them track the order and catch errors. 
  
   
• Parents and teachers can use the trick of making students practice with mistakes intentionally. Give them an expression and make them solve it in correct and incorrect order to strengthen their reasoning skills. 
  
   
• Use interactive activities, such as games or puzzles, and enable efficient learning with online tools like an order of operations calculator or order of operations worksheets.

The order of operations is important in real-life situations where correct calculations are needed to ensure the correct result.

• Financial calculation: The order of operations helps to calculate monthly interest or loan payments. The compound of interest formula includes exponents, multiplication, and subtraction. Expressions like \(A = P ×(1 + r)^t \).

• Construction and engineering: In construction and engineering, they calculate the measurements for materials. Using the order of operations gives a correct result that helps to identify the dimension, quantities. When calculating the volume of concrete, volume = length × (width × thickness).  

• Shopping and budgeting: Buying items at the store that offers discounts and adding the tax by applying the order of operations to identify the final price. Final price = (Price - discount) + tax.

• Cooking and recipes: Adjusting ingredient amounts often involves operations like multiplication and addition, where the order affects the final quantity.

•  Science experiments: In science experiments, we use formulas to calculate energy, pressure, or chemical reactions. The order of operations helps us solve these formulas the right way, so our results are accurate and make sense.