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Order of Operations: Definition and Example | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Order of OperationsOrder of Operations: Definition and ExampleTable of ContentsDefinition of Order of Operations

The order of operations is a fundamental mathematical rule that defines the sequence in which we should solve expressions containing multiple operations. When faced with a mathematical expression that includes various operations such as addition, subtraction, multiplication, and division, following a standardized order ensures everyone arrives at the same answer. This rule eliminates confusion and provides consistency in mathematical calculations across the world.

PEMDAS is the acronym commonly used to remember the correct order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), followed by Addition and Subtraction (from left to right). If an expression contains only operations of the same precedence (for example, only addition or only multiplication), the correct approach is to solve from left to right. However, for expressions with multiple operations, following PEMDAS is essential to arrive at the correct solution.

Examples of Order of Operations Example 1: Solving an Expression with Parentheses, Multiplication, Division, Addition, and Subtraction Problem:

Solve 2+6×(4+5)÷3−52 + 6 \times (4 + 5) \div 3 - 52+6×(4+5)÷3−5 using the order of operations.

Step-by-step solution:

Step 1 - Parentheses:

Begin by calculating the expression inside the parentheses: (4+5)=9(4 + 5) = 9(4+5)=9 So our expression becomes: 2+6×9÷3−52 + 6 \times 9 \div 3 - 52+6×9÷3−5

Step 2 - Exponents:

There are no exponents in this expression, so we move to the next step.

Step 3 - Multiplication and Division:

Working from left to right for multiplication and division: 6×9=546 \times 9 = 546×9=54, our expression is now: 2+54÷3−52 + 54 \div 3 - 52+54÷3−5 Next division: 54÷3=1854 \div 3 = 1854÷3=18, our expression becomes: 2+18−52 + 18 - 52+18−5

Step 4 - Addition and Subtraction:

First addition: 2+18=202 + 18 = 202+18=20 Then subtraction: 20−5=1520 - 5 = 1520−5=15

Therefore, 2+6×(4+5)÷3−5=152 + 6 \times (4 + 5) \div 3 - 5 = 152+6×(4+5)÷3−5=15

Example 2: Evaluating an Expression with Division Inside Parentheses Problem:

Solve 4−5÷(8−3)×2+54 - 5 \div (8 - 3) \times 2 + 54−5÷(8−3)×2+5 using the order of operations.

Step-by-step solution:

Step 1 - Parentheses:

Calculate what's inside the parentheses first. (8−3)=5(8 - 3) = 5(8−3)=5 So our expression becomes: 4−5÷5×2+54 - 5 \div 5 \times 2 + 54−5÷5×2+5

Step 2 - Exponents:

No exponents in this expression.

Step 3 - Multiplication and Division:

First division since it comes first when reading from left to right: 5÷5=15 \div 5 = 15÷5=1, our expression is now: 4−1×2+54 - 1 \times 2 + 54−1×2+5 Next multiplication: 1×2=21 \times 2 = 21×2=2, our expression becomes: 4−2+54 - 2 + 54−2+5

Step 4 - Addition and Subtraction:

First subtraction since it comes first from left to right: 4−2=24 - 2 = 24−2=2 Then addition: 2+5=72 + 5 = 72+5=7

Therefore, 4−5÷(8−3)×2+5=74 - 5 \div (8 - 3) \times 2 + 5 = 74−5÷(8−3)×2+5=7

Example 3: Working with Nested Operations Inside Parentheses Problem:

Solve 100÷(6+7×2)−5100 \div (6 + 7 \times 2) - 5100÷(6+7×2)−5 using the order of operations.

Step-by-step solution:

Step 1 - Parentheses:

When working with operations inside parentheses, we still apply PEMDAS within those parentheses. First multiplication inside parentheses: 7×2=147 \times 2 = 147×2=14 Then addition inside parentheses: 6+14=206 + 14 = 206+14=20 So our expression becomes: 100÷20−5100 \div 20 - 5100÷20−5

Step 2 - Exponents:

No exponents present.

Step 3 - Multiplication and Division:

Division: 100÷20=5100 \div 20 = 5100÷20=5, our expression is now: 5−55 - 55−5

Step 4 - Addition and Subtraction:

Subtraction: 5−5=05 - 5 = 05−5=0

Therefore, 100÷(6+7×2)−5=0100 \div (6 + 7 \times 2) - 5 = 0100÷(6+7×2)−5=0

Comments(7)TTeacherAmyNovember 4, 2025I've been struggling to explain order of operations to my students. This glossary page made it so much easier! Thanks for the clear examples.

NNatureLover87September 17, 2025This explanation of the order of operations was so clear! I used the examples to help my son with his math homework, and it really clicked for him. Thanks for breaking it down step by step!

MCMs. CarterSeptember 10, 2025This site explained the order of operations so clearly! I used the examples to help my kids with their math homework, and they finally got PEMDAS. Such a helpful resource for parents!

MCMs. CarterAugust 26, 2025I’ve been using this site to help my kids understand PEMDAS, and it’s been a game-changer! The clear examples really helped them see how to solve problems step by step. Great resource!

MCMs. CarterAugust 20, 2025This definition of the order of operations was super clear! I used the examples to help my son with his math homework, and it finally clicked for him. Great resource!

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