How to Evaluate Math Expressions & Equations Using Order of Operations? (Examples) - BYJUS

# Evaluating Order of Operations

A math expression is a phrase containing numbers, variables, and operators. We can interpret different meanings from a math expression, just like how we can interpret different meanings from words. So, it is important to approach math expressions from a certain angle by following the order of operations to arrive at the right result....Read MoreRead Less

## The Order of Operations

The rules that state the order in which the multiple operations in an expression should be solved are known as the order of operations in mathematics.

## What is the meaning of a Numerical Expression?

A numerical expression is a mathematical statement comprised solely of numbers and one or more operation symbols. Adding, subtracting, multiplying, and dividing are examples of operation symbols. It can also be written as a radical (square root) or absolute value symbol.

## Definition of the Order of Operations

Definition: The order of operations is a set of basic precedence rules that we use to solve any mathematical expression that involves multiple operations. When a sub expression appears between two operators, the operator that appears first in the following list should be applied first.

The following are the rules and the order of operations:

1. Brackets (), {}, []
2. Exponents
3. Division $$(\div)$$ and Multiplication $$(\times)$$
4. Addition (+) and Subtraction (-)

The preceding set of rules is always different depending on the given mathematical expressions.

## Rules for the Order of Operations

The order of operations is a rule that specifies the correct steps to be applied while evaluating a mathematical expression.

We will follow the given basic rules in a particular sequence when performing any sort of operation on the respective numbers present in expressions.

1. Perform symbol grouping operations.
2. Analyze numbers with exponents.
3. From the left to the right, multiply and divide.
4. From left to right, add and subtract.

## Solved Order of Operations Examples

Question 1:

Solve $$8+5\div~5+4\times~3-7$$

Solution:
The problem given is $$8+5\div~5+4\times~3-7$$.

The division operation is performed first.

$$5\div~5=1$$

So, the expression reduces to $$8+1+4\times~3-7$$

The multiplication operation is taken next.

$$4\times~3=12$$

So, the expression reduces to $$8+1+12-7$$

The addition operation is performed next.

$$8+1+12=21$$

Lastly, subtraction is performed.

21 – 7 = 14.

Hence, 8 + 9 ÷ 9 + 4 × 3 – 7 = 14

Question 2:

Simplify the expression [25 – 3(6 + 1)] ÷ 4 + 9.

Solution:
Let us solve this sequentially [25 – 3(6 + 1)] ÷ 4 + 9

The round bracket is (6 + 1) = 7.

The next bracket is 3(7) = 21

Take [25 – 21] ÷ 4 + 9

(25 – 21) = 4

Then, the division operation is performed.

4 ÷ 4 = 1

Then 1 + 9 = 10

Question 3:

Solve 9 + 10 ÷ 5 + 4 × 3 – 3

Solution:
The problem given is 9 + 10 ÷ 5 + 4 × 3 – 3.

The division operation is performed first.

$$10\div~5=2$$

So, the expression reduces to 9 + 2 + 4 × 3 – 3

The multiplication operation is taken next.

$$4\times~3=12$$

So, the expression reduces to 9 + 2 + 12 – 3

The addition operation is performed next.

9 + 2 + 12 = 23

So, the expression reduces to 23 – 3

Lastly, the subtraction operation is performed.

23 – 3 = 20

Hence, 9 + 10 ÷ 5 + 4 × 3 – 3 = 20.

Frequently Asked Questions on Order of Operations

In math, the order of operation is a set of rules centered on four major operators. There is a specific sequence that we must follow for each operator while solving the given mathematical expression based on the order of operations.

When it comes to simplifying numerical expressions, there are four basic operations to consider. Addition, subtraction, multiplication, and division are the four operations that include whole numbers, fractional numbers, and decimals.

To solve mathematical expressions, we must follow the rules of the order of operations so that everyone gets the same answer.