Order of Operations

Concept

The order of operations is a set of rules that is to be followed in a particular sequence while solving an expression. In mathematics with the word operations we mean, the process of evaluating any mathematical expression, involving arithmetic operations such as division, multiplication, addition, and subtraction.

The order of Operations is the rule in math that states we evaluate the parentheses/brackets first, the exponents/the orders second, division or multiplication third (from left to right, whichever comes first), and the addition or subtraction at the last (from left to right, whichever comes first). In math, there might be several operations to be done while evaluating an expression, and simplification at the end yields different results. However, we can only have one correct answer for any sort of expression. To identify the correct answer we simplify any given mathematical expression using a certain set of rules. These rules revolve around all the basic operators used in maths. Operators such as addition (+), subtraction (-), division (÷), and multiplication (×).

Rules

While performing any sort of an operation on the respective numbers present in the expression we will follow the given basic rules in the particular sequence.

1. Observe the expression. The first rule is to solve the numbers present inside the parentheses or brackets. We solve inside to out grouping operations. Note the pattern of brackets present in the expression, there is a particular order to solve the parentheses, i.e., [ { ( ) } ]. First, solve the round brackets ( ) → curly brackets { } → box brackets [ ]. Inside the parentheses the order of operations are to be followed.
2. After solving the numbers in the parentheses, look for any term present in the form of exponents and solve it.
3. Now we are left with the basic four operators. Look for the numbers with the operation of multiplication or division, solve them from left to right.
4. Lastly, look for the terms with addition or subtraction and solve them from left to right.

These rules have a specific acronym name. We call them PEMDAS or BODMAS.

Order of Operations PEMDAS
P stands for Parentheses ( ), { }, [ ]
E stands for Exponents (a²) (For example, here, a is a number with exponent 2)
M stands for Multiplication (×)
D stands for Division (÷)
A stands for Addition (+)
S stands for Subtraction (-)

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

Example

Evaluate.
(3 + 7)² – 4 × (2 + 3)

Solution

1. Start by solving problems in parentheses.
(3 + 7) = 10
(2 + 3) = 5

This would result in: 10² – 4 x 5

2. Next solve any exponents. 10² = 100
100 – 4 x 5

3. Now look for any multiplication or division to be solved. 4 x 5 = 20
100 – 20

4. Last do any addition or subtraction: 100 – 20 = 80

5. (3 + 7)² – 4 x (2 + 3) = 80

Practice Order of Operations

Practice Problem 1

Which operation should be performed second?
20 ÷ 4 + 10 – 2

Practice Problem 2

Evaluate.
(50 – 10) ÷ 4 × 5

Practice Problem 3

Evaluate.
(51 – 41)² ÷ 9 × 3

Practice Problem 4

Zoey pays 18 dollars for materials to make bracelets. She makes 14 bracelets and sells 9 for 5 dollars and 5 for 3 dollars. How much profit does Zoey make?