Solving 7 * 2 - 5 * 3 A Guide To Order Of Operations

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In the realm of mathematics, order of operations is a fundamental concept that dictates the sequence in which mathematical operations should be performed. This ensures consistency and accuracy in calculations, preventing ambiguity and leading to a universally accepted solution. Without a defined order, the same expression could yield different results, making mathematical communication and problem-solving impossible. This article will provide an in-depth exploration of the order of operations, using the expression 7 * 2 - 5 * 3 as a practical example. By understanding the rules and applying them systematically, you can confidently tackle more complex mathematical expressions and equations.

Understanding the Order of Operations (PEMDAS/BODMAS)

The order of operations is often remembered by the acronyms PEMDAS or BODMAS. Both acronyms represent the same hierarchy of operations:

  • Parentheses / Brackets
  • Exponents / Orders
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

This hierarchy means that expressions within parentheses or brackets are evaluated first, followed by exponents or orders (such as square roots and powers). Multiplication and division are performed next, working from left to right, and finally, addition and subtraction are carried out, also from left to right. By adhering to this order, we ensure that mathematical expressions are evaluated consistently, leading to accurate and reliable results. For instance, consider the expression 7 * 2 - 5 * 3. Without the order of operations, one might mistakenly subtract 5 from 2 first, leading to an incorrect answer. However, following PEMDAS/BODMAS, we prioritize multiplication before subtraction, yielding the correct result.

Applying PEMDAS/BODMAS to 7 * 2 - 5 * 3

Let's apply the order of operations to the expression 7 * 2 - 5 * 3. According to PEMDAS/BODMAS, multiplication should be performed before subtraction. Therefore, we begin by multiplying 7 by 2 and 5 by 3:

  1. Multiply:
    • 7 * 2 = 14
    • 5 * 3 = 15

Now, the expression simplifies to 14 - 15. Next, we perform the subtraction:

  1. Subtract:
    • 14 - 15 = -1

Therefore, the solution to the expression 7 * 2 - 5 * 3 is -1. By following the order of operations, we've arrived at a precise and unambiguous result. This methodical approach is crucial for handling more complex expressions involving multiple operations and nested parentheses. Remember, the key is to break down the expression into smaller, manageable steps, applying the rules of PEMDAS/BODMAS at each stage. This ensures that you maintain accuracy and avoid common pitfalls in mathematical calculations.

Step-by-Step Breakdown of the Calculation

To further illustrate the application of the order of operations, let's break down the calculation of 7 * 2 - 5 * 3 step by step:

  1. Identify the operations: The expression contains multiplication and subtraction.
  2. Apply PEMDAS/BODMAS: Multiplication comes before subtraction.
  3. Perform the first multiplication: 7 * 2 = 14. The expression now becomes 14 - 5 * 3.
  4. Perform the second multiplication: 5 * 3 = 15. The expression now becomes 14 - 15.
  5. Perform the subtraction: 14 - 15 = -1.

Thus, the final answer is -1. This step-by-step approach highlights the importance of methodical calculation. By focusing on one operation at a time and adhering to the order of operations, you minimize the risk of errors and gain confidence in your mathematical abilities. This systematic approach is particularly valuable when dealing with complex expressions involving numerous operations and nested parentheses. In such cases, breaking down the problem into smaller, manageable steps becomes essential for accuracy and clarity.

Common Mistakes to Avoid

When evaluating mathematical expressions, it's easy to make mistakes if the order of operations is not followed correctly. One common mistake is performing addition or subtraction before multiplication or division. For example, in the expression 7 * 2 - 5 * 3, someone might incorrectly subtract 5 from 2 first, leading to an incorrect result. Another mistake is failing to work from left to right when performing multiplication and division or addition and subtraction. Remember that these operations have equal precedence and should be performed in the order they appear in the expression.

To avoid these errors, it's crucial to always remember and apply PEMDAS/BODMAS. Before starting a calculation, take a moment to identify the operations and their order of precedence. Break down the expression into smaller steps, and perform each operation according to the rules. Double-check your work at each step to ensure accuracy. Practice regularly to reinforce your understanding of the order of operations and build confidence in your problem-solving skills. By being mindful of these common pitfalls and adopting a systematic approach, you can minimize errors and achieve accurate results consistently.

Practice Problems and Solutions

To solidify your understanding of the order of operations, let's work through a few practice problems:

Problem 1: Evaluate 10 + 2 * 3 - 4

Solution:

  1. Multiply: 2 * 3 = 6. The expression becomes 10 + 6 - 4.
  2. Add: 10 + 6 = 16. The expression becomes 16 - 4.
  3. Subtract: 16 - 4 = 12.

Therefore, the solution is 12.

Problem 2: Evaluate (8 - 2) * 4 + 6 / 2

Solution:

  1. Parentheses: 8 - 2 = 6. The expression becomes 6 * 4 + 6 / 2.
  2. Multiply: 6 * 4 = 24. The expression becomes 24 + 6 / 2.
  3. Divide: 6 / 2 = 3. The expression becomes 24 + 3.
  4. Add: 24 + 3 = 27.

Therefore, the solution is 27.

Problem 3: Evaluate 15 / 3 + 4 * (2 + 1)

Solution:

  1. Parentheses: 2 + 1 = 3. The expression becomes 15 / 3 + 4 * 3.
  2. Divide: 15 / 3 = 5. The expression becomes 5 + 4 * 3.
  3. Multiply: 4 * 3 = 12. The expression becomes 5 + 12.
  4. Add: 5 + 12 = 17.

Therefore, the solution is 17.

By working through these practice problems, you can reinforce your understanding of the order of operations and develop your problem-solving skills. Remember to break down each expression into smaller steps, applying PEMDAS/BODMAS systematically. Regular practice is key to mastering this fundamental concept and building confidence in your mathematical abilities.

Conclusion: The Importance of Order of Operations

The order of operations is a cornerstone of mathematics, ensuring that expressions are evaluated consistently and accurately. By following the rules of PEMDAS/BODMAS, we can avoid ambiguity and arrive at the correct solution every time. This article has provided a comprehensive guide to understanding and applying the order of operations, using the expression 7 * 2 - 5 * 3 as a practical example. We've explored the hierarchy of operations, broken down the calculation step by step, discussed common mistakes to avoid, and worked through practice problems to solidify your understanding.

Mastering the order of operations is essential for success in mathematics and related fields. It forms the basis for more advanced concepts and problem-solving techniques. By consistently applying the rules and practicing regularly, you can build a strong foundation in mathematics and confidently tackle complex expressions and equations. Remember, the key is to be methodical, break down problems into smaller steps, and always double-check your work. With a solid understanding of the order of operations, you'll be well-equipped to excel in your mathematical endeavors.