Order of Operations

The order of operations is a set of rules in mathematics that tells us the correct order to follow when solving equations with more than one operation, like addition, subtraction, multiplication & division.

These are the common operations used in math:

• Addition(+)

• Subtraction(-)

• Multiplication()

• Division() 

• And Brackets (), [], {}

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

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:

\(4 × 2 = 8\)

The answer is 8.

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

\(6 + 8 = 14\)

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.

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.