Solving Arithmetic Expressions Step-by-Step Guide

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In the realm of mathematics, arithmetic expressions form the bedrock of more complex calculations. Understanding how to solve these expressions accurately is crucial for success in various mathematical disciplines. This comprehensive guide will walk you through the process of solving arithmetic expressions, providing step-by-step solutions and explanations for each example. We will delve into the order of operations, often remembered by the acronym PEMDAS or BODMAS, and demonstrate its application in solving a variety of expressions. Let's embark on this mathematical journey together and master the art of solving arithmetic expressions.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we dive into solving specific expressions, it's essential to grasp the fundamental principle governing arithmetic calculations: the order of operations. This principle dictates the sequence in which operations should be performed to arrive at the correct answer. The acronyms PEMDAS and BODMAS serve as handy mnemonics to remember this order:

  • PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  • BODMAS: Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Both acronyms essentially convey the same order, with slight variations in terminology. Parentheses and brackets serve the same function, as do exponents and orders. The key takeaway is that operations within parentheses or brackets are performed first, followed by exponents or orders, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). Mastering this order is the cornerstone of accurately solving arithmetic expressions. Without adhering to this principle, the results will invariably be incorrect.

The Importance of Order

Why is the order of operations so critical? Consider a simple example: 2 + 3 × 4. If we perform the addition first, we get 5 × 4 = 20. However, if we follow PEMDAS/BODMAS and perform the multiplication first, we get 2 + 12 = 14. The correct answer is 14, highlighting the significant impact of order. Ignoring the order of operations can lead to drastically different and incorrect results, underscoring the need for meticulous adherence to the established rules. The order of operations is not just a convention; it's a mathematical necessity for ensuring consistency and accuracy in calculations.

Breaking Down PEMDAS/BODMAS

To further clarify the order of operations, let's break down each component:

  1. Parentheses/Brackets: Operations within parentheses or brackets are always performed first. This includes any arithmetic operations, exponents, or even nested parentheses. Parentheses act as grouping symbols, indicating that the enclosed expression should be treated as a single entity. Think of them as creating a mini-problem within the larger expression that must be solved first.
  2. Exponents/Orders: Exponents or orders represent repeated multiplication. For example, 2³ (2 to the power of 3) means 2 × 2 × 2 = 8. Exponents are evaluated after parentheses, as they represent a higher level of operation than multiplication or division. They signify the power or magnitude of a number.
  3. Multiplication and Division: Multiplication and division are performed from left to right. This means that if an expression contains both multiplication and division, you perform the operation that appears first as you read from left to right. They are inverse operations, and their order of execution is determined by their position within the expression.
  4. Addition and Subtraction: Addition and subtraction are performed from left to right, similar to multiplication and division. Again, the order is determined by their position within the expression. These are also inverse operations, and their placement dictates the sequence of their execution.

Understanding each component of PEMDAS/BODMAS and their relative priority is key to navigating complex arithmetic expressions with confidence. Practice applying this order consistently, and you'll find that solving these expressions becomes much more straightforward.

Solving the Arithmetic Expressions

Now, let's apply the order of operations to solve the given arithmetic expressions step-by-step.

1) 81 - 6 × 3 + 7

  • Step 1: Multiplication
    • Following PEMDAS/BODMAS, we perform the multiplication first: 6 × 3 = 18.
    • The expression now becomes: 81 - 18 + 7.
  • Step 2: Subtraction
    • Next, we perform subtraction from left to right: 81 - 18 = 63.
    • The expression is now: 63 + 7.
  • Step 3: Addition
    • Finally, we perform the addition: 63 + 7 = 70.
  • Solution: Therefore, 81 - 6 × 3 + 7 = 70.

This first example clearly illustrates the importance of prioritizing multiplication before addition and subtraction. By adhering to PEMDAS/BODMAS, we arrived at the correct answer. Let's move on to the next expression and continue honing our skills.

2) 51 - 49 = 7 × 4 + 3

This expression presents a slight twist, as it involves an equation with operations on both sides. We need to solve each side separately, following the order of operations, and then verify if the equation holds true.

  • Step 1: Solve the Left-Hand Side (LHS)
    • The LHS is: 51 - 49.
    • Performing the subtraction: 51 - 49 = 2.
  • Step 2: Solve the Right-Hand Side (RHS)
    • The RHS is: 7 × 4 + 3.
    • Following PEMDAS/BODMAS, we perform the multiplication first: 7 × 4 = 28.
    • The RHS now becomes: 28 + 3.
    • Performing the addition: 28 + 3 = 31.
  • Step 3: Compare LHS and RHS
    • We found that LHS = 2 and RHS = 31.
    • Since 2 ≠ 31, the equation is not true.
  • Solution: Therefore, 51 - 49 ≠ 7 × 4 + 3. The equation is false. It's crucial to remember that when dealing with equations, we must solve each side independently and then compare the results. This example emphasizes that not all mathematical statements are true, and careful evaluation is essential.

3) 81 ÷ (9 × 3)

This expression includes parentheses, which, according to PEMDAS/BODMAS, take precedence.

  • Step 1: Solve the Parentheses
    • Inside the parentheses, we have: 9 × 3 = 27.
    • The expression now becomes: 81 ÷ 27.
  • Step 2: Division
    • Performing the division: 81 ÷ 27 = 3.
  • Solution: Therefore, 81 ÷ (9 × 3) = 3. This example highlights the importance of parentheses in dictating the order of operations. By solving the expression within the parentheses first, we obtained the correct result. Ignoring the parentheses would lead to a different, incorrect answer.

4) 45 - 6 - 32 - 8

This expression involves only subtraction. According to PEMDAS/BODMAS, we perform subtraction from left to right.

  • Step 1: Subtraction (Left to Right)
    • First, 45 - 6 = 39.
    • The expression becomes: 39 - 32 - 8.
    • Next, 39 - 32 = 7.
    • The expression is now: 7 - 8.
  • Step 2: Final Subtraction
    • Finally, 7 - 8 = -1.
  • Solution: Therefore, 45 - 6 - 32 - 8 = -1. This example reinforces the rule of performing addition and subtraction (or multiplication and division) from left to right when they are the only operations present. Failing to do so can result in an incorrect answer. The concept of negative numbers also comes into play here, demonstrating the broader application of arithmetic principles.

5) 36 ÷ (6 + 6) × 5

This expression combines parentheses, division, and multiplication. Let's tackle it step-by-step using PEMDAS/BODMAS.

  • Step 1: Solve the Parentheses
    • Inside the parentheses, we have: 6 + 6 = 12.
    • The expression now becomes: 36 ÷ 12 × 5.
  • Step 2: Division and Multiplication (Left to Right)
    • We perform division and multiplication from left to right. First, 36 ÷ 12 = 3.
    • The expression becomes: 3 × 5.
  • Step 3: Multiplication
    • Finally, we perform the multiplication: 3 × 5 = 15.
  • Solution: Therefore, 36 ÷ (6 + 6) × 5 = 15. This example showcases the combined application of parentheses and the left-to-right rule for division and multiplication. It's a comprehensive demonstration of PEMDAS/BODMAS in action. By meticulously following the order of operations, we arrive at the accurate solution.

Conclusion

Solving arithmetic expressions accurately requires a firm grasp of the order of operations (PEMDAS/BODMAS). By consistently applying this principle, you can confidently tackle a wide range of mathematical problems. Remember to prioritize parentheses, exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right). Practice is key to mastering this skill, so continue working through examples and challenging yourself with more complex expressions. With dedication and a clear understanding of the order of operations, you'll become proficient in solving arithmetic expressions and build a strong foundation for further mathematical endeavors. The ability to solve these expressions accurately is not just a mathematical skill; it's a fundamental life skill that enhances logical thinking and problem-solving abilities in various contexts.