Mastering Order Of Operations PEMDAS BODMAS With Examples

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In mathematics, the order of operations is a crucial concept for solving complex expressions accurately. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), this set of rules dictates the sequence in which mathematical operations should be performed. Understanding and applying PEMDAS ensures consistency and correctness in mathematical calculations. This article provides a detailed explanation of the order of operations, accompanied by step-by-step solutions to various examples. By mastering this fundamental principle, you can confidently tackle more advanced mathematical problems.

Understanding the Order of Operations (PEMDAS/BODMAS)

The order of operations, commonly known as PEMDAS in the United States and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) in other parts of the world, is a standard convention used in mathematics to clarify the sequence in which operations should be performed in an expression. This ensures that everyone arrives at the same correct answer. Let's break down each component of PEMDAS/BODMAS in detail:

1. Parentheses/Brackets

Parentheses, or brackets, are always the first priority in any mathematical expression. This is because they group terms together, indicating that the operations within the parentheses should be performed before any operations outside of them. Parentheses can include various types of brackets such as round brackets (), square brackets [], and curly braces {}. When dealing with nested parentheses (one set of parentheses inside another), it is crucial to start with the innermost set and work your way outwards. This systematic approach ensures that each operation is performed in the correct sequence. For example, in the expression 2 x (3 + (4 - 1)), we would first solve (4 - 1), then (3 + 3), and finally multiply by 2. Ignoring the parentheses and performing operations in a left-to-right manner would lead to an incorrect result. Therefore, understanding the precedence of parentheses is essential for accurate mathematical calculations.

2. Exponents/Orders

Exponents, also known as orders, represent the power to which a number is raised. After addressing parentheses, the next step in the order of operations is to evaluate exponents. Exponents indicate repeated multiplication of a base number by itself. For instance, in the expression 5^3, the base is 5, and the exponent is 3, meaning 5 is multiplied by itself three times (5 x 5 x 5). Evaluating exponents involves calculating the result of this repeated multiplication. Consider the expression 2 + 3^2. According to PEMDAS/BODMAS, we must first compute the exponent 3^2, which equals 9, and then add 2 to get the final result of 11. Neglecting exponents and performing addition before exponentiation would lead to a wrong answer. The correct handling of exponents is vital in various mathematical contexts, including algebraic equations, scientific notation, and compound interest calculations. Therefore, understanding the placement and effect of exponents within an expression is key to ensuring accuracy in mathematical problem-solving.

3. Multiplication and Division

Multiplication and Division hold equal precedence in the order of operations. After addressing parentheses and exponents, these operations are performed from left to right within the expression. This means that if multiplication appears before division, it is carried out first, and vice versa. For instance, in the expression 10 ÷ 2 x 3, division comes before multiplication, so we first divide 10 by 2 to get 5, and then multiply 5 by 3 to obtain the result of 15. However, if the expression were 10 x 3 ÷ 2, we would first multiply 10 by 3 to get 30, and then divide 30 by 2 to get 15. The left-to-right rule is crucial when both multiplication and division are present to avoid ambiguity and ensure consistent results. This aspect of PEMDAS/BODMAS is particularly important in more complex calculations where the order can significantly impact the outcome. Understanding and applying this rule correctly is fundamental to mastering arithmetic and algebraic problem-solving.

4. Addition and Subtraction

Addition and Subtraction are the final operations to be performed in the order of operations, and they also hold equal precedence. Similar to multiplication and division, addition and subtraction are carried out from left to right within the expression. This means that if addition appears before subtraction, it is performed first, and vice versa. For instance, in the expression 8 - 3 + 5, subtraction comes before addition, so we first subtract 3 from 8 to get 5, and then add 5 to obtain the result of 10. Conversely, if the expression were 8 + 5 - 3, we would first add 8 and 5 to get 13, and then subtract 3 to get 10. The left-to-right rule ensures consistency when both addition and subtraction are present in a calculation. This principle is essential for maintaining accuracy in arithmetic and algebraic expressions. Properly applying the order of operations for addition and subtraction ensures that mathematical problems are solved correctly, regardless of their complexity.

Solved Examples

Let's apply the order of operations (PEMDAS/BODMAS) to solve the following mathematical expressions:

1. 36 + (6 + 6) x 5

  • Step 1: Parentheses First, we solve the operation inside the parentheses: (6 + 6) = 12.

  • Step 2: Multiplication Next, we perform the multiplication: 12 x 5 = 60.

  • Step 3: Addition Finally, we add the result to 36: 36 + 60 = 96.

    Therefore, the solution to the expression 36 + (6 + 6) x 5 is 96.

2. 8 x 9 + (48 + 6)

  • Step 1: Parentheses First, we solve the operation inside the parentheses: (48 + 6) = 54.

  • Step 2: Multiplication Next, we perform the multiplication: 8 x 9 = 72.

  • Step 3: Addition Finally, we add the results: 72 + 54 = 126.

    Therefore, the solution to the expression 8 x 9 + (48 + 6) is 126.

3. (72 + 15) x 4 - (625 + 125)

  • Step 1: Parentheses Solve the operations inside the parentheses:

    • (72 + 15) = 87
    • (625 + 125) = 750

  • Step 2: Multiplication Perform the multiplication: 87 x 4 = 348.

  • Step 3: Subtraction Finally, subtract the results: 348 - 750 = -402.

    Therefore, the solution to the expression (72 + 15) x 4 - (625 + 125) is -402.

4. 5 x 6 + 6 + 6 - 12 x 2

  • Step 1: Multiplication Perform the multiplications from left to right:

    • 5 x 6 = 30
    • 12 x 2 = 24

  • Step 2: Addition and Subtraction Perform addition and subtraction from left to right: 30 + 6 + 6 - 24

    • 30 + 6 = 36
    • 36 + 6 = 42
    • 42 - 24 = 18

    Therefore, the solution to the expression 5 x 6 + 6 + 6 - 12 x 2 is 18.

5. 81 - 86 ÷ 2 + (9 x 2) - 50

  • Step 1: Parentheses Solve the operation inside the parentheses: (9 x 2) = 18.

  • Step 2: Division Perform the division: 86 ÷ 2 = 43.

  • Step 3: Addition and Subtraction Perform addition and subtraction from left to right: 81 - 43 + 18 - 50

    • 81 - 43 = 38
    • 38 + 18 = 56
    • 56 - 50 = 6

    Therefore, the solution to the expression 81 - 86 ÷ 2 + (9 x 2) - 50 is 6.

6. 40 + 2 x 4 + 6

  • Step 1: Multiplication Perform the multiplication: 2 x 4 = 8.

  • Step 2: Addition Perform addition from left to right: 40 + 8 + 6

    • 40 + 8 = 48
    • 48 + 6 = 54

    Therefore, the solution to the expression 40 + 2 x 4 + 6 is 54.

7. (15 - 6) + (4 - 1) x 24

  • Step 1: Parentheses Solve the operations inside the parentheses:

    • (15 - 6) = 9
    • (4 - 1) = 3

  • Step 2: Multiplication Perform the multiplication: 3 x 24 = 72.

  • Step 3: Addition Finally, add the results: 9 + 72 = 81.

    Therefore, the solution to the expression (15 - 6) + (4 - 1) x 24 is 81.

8. 3 x [3 + 2 x (10 - 4)]

  • Step 1: Innermost Parentheses Solve the operation inside the innermost parentheses: (10 - 4) = 6.

  • Step 2: Inner Parentheses Perform the multiplication inside the square brackets: 2 x 6 = 12.

  • Step 3: Remaining Parentheses Add within the square brackets: 3 + 12 = 15.

  • Step 4: Multiplication Finally, multiply: 3 x 15 = 45.

    Therefore, the solution to the expression 3 x [3 + 2 x (10 - 4)] is 45.

Conclusion

Mastering the order of operations is fundamental to success in mathematics. By consistently applying the rules of PEMDAS/BODMAS, you can accurately solve complex expressions and avoid common errors. The examples provided illustrate the step-by-step process of solving various types of mathematical problems. With practice and a solid understanding of these principles, you will be well-equipped to tackle more advanced mathematical concepts and challenges. Remember, mathematics builds upon itself, and a strong foundation in basic operations is essential for future learning and problem-solving.