Solving 30-5.6:7+2.10.40-30+5 A Step-by-Step Guide

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In this comprehensive guide, we will meticulously walk through the process of solving the mathematical expression 30 - 5 . 6 : 7 + 2 . 10 . 40 - 30 + 5. This seemingly complex equation requires a careful application of the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). By understanding and following these rules, we can break down the problem into manageable steps and arrive at the correct solution. This guide will not only provide the answer but also explain the rationale behind each step, ensuring a clear understanding of the mathematical principles involved. Whether you're a student looking to brush up on your skills, or simply someone who enjoys the challenge of solving mathematical puzzles, this guide will provide a detailed and insightful approach to tackling this equation.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we dive into the solution, it's crucial to understand the order of operations, often referred to as PEMDAS or BODMAS. This acronym dictates the sequence in which mathematical operations should be performed to ensure accuracy. PEMDAS stands for:

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

Alternatively, BODMAS stands for:

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

Both acronyms represent the same hierarchy of operations. The key takeaway is that we must address parentheses (or brackets) and exponents (or orders) first, followed by multiplication and division (performed from left to right), and finally, addition and subtraction (also performed from left to right). Ignoring this order will lead to incorrect results. Applying this order systematically is the cornerstone of accurately solving mathematical expressions. Mastering PEMDAS/BODMAS is not just about getting the right answer; it's about developing a logical and structured approach to problem-solving, a skill that transcends mathematics and applies to various aspects of life. By consistently applying these rules, you build a solid foundation for more advanced mathematical concepts and problem-solving techniques. Remember, practice makes perfect, so working through various examples while keeping PEMDAS/BODMAS in mind will solidify your understanding and build your confidence in tackling even the most complex equations.

Step-by-Step Solution of 30 - 5 . 6 : 7 + 2 . 10 . 40 - 30 + 5

Now, let's apply the order of operations to solve the expression 30 - 5 . 6 : 7 + 2 . 10 . 40 - 30 + 5 step by step. This meticulous approach will help us avoid common errors and ensure an accurate final result. Each step will be clearly explained, highlighting the specific operation being performed and why it's being done in that order. By breaking down the problem into smaller, more manageable chunks, we can minimize the chances of making mistakes and gain a deeper understanding of the mathematical process.

Step 1: Multiplication and Division (from left to right)

According to PEMDAS, we perform multiplication and division before addition and subtraction. It's crucial to remember that these operations are performed from left to right. This means we tackle them in the order they appear in the expression.

  • First, we have 5 . 6, which equals 30. Our expression now becomes:

    30 - 30 : 7 + 2 . 10 . 40 - 30 + 5

  • Next, we perform the division 30 : 7, which is approximately 4.2857. Let's keep this value for now. The expression is now:

    30 - 4.2857 + 2 . 10 . 40 - 30 + 5

  • Now, we continue with the multiplication. We have 2 . 10, which equals 20. Our expression becomes:

    30 - 4.2857 + 20 . 40 - 30 + 5

  • Finally, we multiply 20 . 40, which equals 800. The expression is now:

    30 - 4.2857 + 800 - 30 + 5

By systematically addressing the multiplication and division operations from left to right, we've simplified the expression considerably. This methodical approach is key to avoiding errors and ensuring an accurate final answer. Each step builds upon the previous one, progressively reducing the complexity of the equation. This process illustrates the power of PEMDAS in breaking down complex mathematical problems into manageable components. The careful attention to detail in this step-by-step approach not only leads to the correct solution but also reinforces the understanding of the fundamental principles of arithmetic.

Step 2: Addition and Subtraction (from left to right)

Now that we've completed the multiplication and division, we move on to addition and subtraction. Just like with multiplication and division, we perform these operations from left to right. This consistent approach ensures that we maintain the correct order and arrive at the accurate solution. It's important to resist the temptation to jump around or prioritize certain additions or subtractions over others; adhering to the left-to-right rule is crucial for maintaining mathematical integrity.

  • First, we have 30 - 4.2857, which is approximately 25.7143. The expression becomes:

    25.7143 + 800 - 30 + 5

  • Next, we add 25.7143 + 800, which is approximately 825.7143. The expression is now:

    825.7143 - 30 + 5

  • Now, we subtract 825.7143 - 30, which is approximately 795.7143. The expression becomes:

    795.7143 + 5

  • Finally, we add 795.7143 + 5, which gives us 800.7143.

Therefore, the final answer is approximately 800.7143. By meticulously performing the addition and subtraction operations from left to right, we've successfully solved the equation. This step-by-step approach not only provides the correct answer but also demonstrates the importance of following the order of operations. Each calculation builds upon the previous one, ensuring that we maintain accuracy throughout the process. This methodical approach reinforces the understanding that mathematics is a sequential process, where each step logically follows the previous one, ultimately leading to the solution.

Final Answer and Conclusion

After carefully following the order of operations (PEMDAS), we have determined that the solution to the expression 30 - 5 . 6 : 7 + 2 . 10 . 40 - 30 + 5 is approximately 800.7143. This result underscores the importance of adhering to mathematical rules and conventions to arrive at the correct answer. The step-by-step approach we employed not only led us to the solution but also provided a clear and transparent pathway, making the process easy to understand and replicate. By breaking down the complex expression into smaller, more manageable parts, we were able to minimize the risk of errors and maintain accuracy throughout the calculation.

In conclusion, solving complex mathematical expressions requires a systematic approach and a thorough understanding of the order of operations. PEMDAS (or BODMAS) serves as a valuable tool for guiding us through the process, ensuring that we perform operations in the correct sequence. This methodical approach is not only essential for mathematical accuracy but also fosters a logical and structured mindset that can be applied to various problem-solving scenarios. By mastering these principles, you can confidently tackle even the most challenging mathematical problems and develop a deeper appreciation for the elegance and precision of mathematics.

Practice Problems

To solidify your understanding of the order of operations, try solving these practice problems:

  1. 10 + 5 . 3 - 15 : 5
  2. (2 + 3) . 4 - 12 : 2
  3. 20 - 4 . (6 - 2) + 8

Solving these problems will reinforce your skills and help you become more comfortable with applying PEMDAS/BODMAS. Remember to break down each problem into steps, carefully following the order of operations. Good luck!