Solving (0.6 ÷ 0.3) - 0.5 - 0.2 + 0.3 A Step-by-Step Guide

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When faced with a mathematical expression like (0.6 ÷ 0.3) - 0.5 - 0.2 + 0.3, it's essential to understand the fundamental principles of the order of operations. This set of rules ensures that we arrive at a consistent and accurate answer, regardless of who is solving the problem. The universally accepted acronym for remembering the order of operations is PEMDAS, which stands for:

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

In our specific equation, we'll meticulously apply PEMDAS to break down each step and arrive at the correct solution. This involves first addressing the division within the parentheses, followed by sequential subtraction and addition. Understanding and applying the order of operations is not just crucial for this particular problem but forms the bedrock of more complex mathematical calculations. Mastering this principle allows for the accurate and consistent solving of equations in various mathematical and real-world scenarios. In this context, we will use the order of operations as a tool, helping us to not only find the correct solution but also to understand the logical sequence that mathematics demands. This approach to problem-solving cultivates both mathematical accuracy and a structured mindset applicable in diverse aspects of logical thinking and decision-making.

Step-by-Step Solution

To accurately solve the expression (0.6 ÷ 0.3) - 0.5 - 0.2 + 0.3, we will follow the order of operations (PEMDAS) meticulously. Our focus here is not just on arriving at the correct numerical answer but also on understanding the procedural steps involved in mathematical problem-solving. This step-by-step approach is crucial as it lays a foundation for tackling more complex mathematical problems later on. By breaking down the problem into manageable steps, we can minimize errors and enhance comprehension. Moreover, this systematic method of problem-solving is transferable to other areas, fostering analytical thinking and a methodical approach to challenges in various domains.

  1. Division: The first operation we tackle is the division within the parentheses: 0.6 ÷ 0.3. Performing this division, we find that 0.6 divided by 0.3 equals 2. This step is crucial as it simplifies the expression and sets the stage for the subsequent operations. Accurate execution of this step is paramount, as any error here will propagate through the rest of the calculation. The process of division, in this context, can be visualized as splitting 0.6 into portions of 0.3, which results in two equal parts.
  2. Rewriting the Expression: Now that we've performed the division, we can rewrite the expression as: 2 - 0.5 - 0.2 + 0.3. This simplified form makes it easier to proceed with the remaining operations. Rewriting the expression is not merely a cosmetic step; it’s a crucial part of maintaining clarity and preventing errors. It allows us to focus on the remaining operations without the distraction of the already resolved division.
  3. Subtraction (Left to Right): Following the order of operations, we perform subtraction and addition from left to right. First, we subtract 0.5 from 2: 2 - 0.5 = 1.5. This subtraction is a straightforward arithmetic operation, but it is crucial to perform it accurately. The result, 1.5, becomes the new value upon which the next operation will be performed. In practical terms, this can be thought of as reducing a quantity of 2 by half a unit, resulting in one and a half units.
  4. Continuing Subtraction: Next, we subtract 0.2 from 1.5: 1.5 - 0.2 = 1.3. This step further reduces the value, bringing us closer to the final answer. The continuous subtraction illustrates the concept of progressively decreasing a quantity. Accuracy in this step is essential as it builds on the previous calculation, and any error here will affect the final result.
  5. Final Addition: Finally, we add 0.3 to 1.3: 1.3 + 0.3 = 1.6. This addition completes the sequence of operations, leading us to the final answer. The addition of 0.3 can be seen as an increment to the existing quantity, bringing the final value to 1.6. This concluding step highlights the additive nature of mathematical operations, balancing out the subtractions performed earlier.

Identifying the Correct Answer

After carefully executing each step in the order of operations, we have arrived at the solution: 1.6. Now, let's compare this result with the options provided:

  • A. 2.4
  • B. 2.0
  • C. 1.6
  • D. 3.0
  • E. None of these

By comparing our calculated solution with the provided options, it becomes clear that option C. 1.6 matches our result. Therefore, option C is the correct answer.

The process of verifying our solution against the provided options is a critical step in mathematical problem-solving. It not only confirms the accuracy of our calculations but also reinforces our understanding of the problem-solving process. This validation step is a safeguard against errors and provides confidence in the correctness of the solution. In educational contexts, this practice encourages students to develop a habit of self-assessment, enhancing their critical thinking skills and mathematical proficiency. Moreover, it exemplifies the importance of precision and attention to detail in mathematical endeavors.

Why Other Options Are Incorrect

Understanding why the other options are incorrect is just as important as finding the correct answer. It deepens our understanding of the mathematical principles involved and helps us avoid making similar mistakes in the future. Let's analyze why options A, B, and D are not the correct answers:

  • A. 2.4: This answer likely results from an incorrect application of the order of operations. A common mistake might be adding all the numbers together before performing the division, or miscalculating one of the subtraction steps. For instance, if someone were to add 0.3 to 0.6 and then divide by 0.3, or if they mishandled the subtraction of 0.5 or 0.2, they might arrive at this incorrect answer. Understanding why this is wrong reinforces the importance of following the order of operations meticulously. Recognizing this common error helps in developing a more robust mathematical approach.
  • B. 2.0: This answer could arise from correctly performing the division but then making an error in the subsequent subtractions and additions. Perhaps the individual missed a subtraction or addition step, or made a mistake in the arithmetic. This highlights the necessity of carefully executing each step of the problem and double-checking the calculations. Analyzing this error teaches us the value of precision and thoroughness in mathematical problem-solving, emphasizing the need for a systematic approach to avoid overlooking any part of the equation.
  • D. 3.0: This result may stem from misinterpreting the operations or overlooking the impact of the decimal points. It could also be a consequence of a fundamental misunderstanding of how addition, subtraction, and division interact within the expression. This particular error underscores the importance of having a solid grasp of basic mathematical concepts and their application. Understanding the conceptual underpinnings of each operation, such as division as the inverse of multiplication, is crucial for avoiding such mistakes.
  • E. None of these: Option E is only correct if all the other options are incorrect and our calculation is also flawed. Since we have confirmed that option C is the correct answer, option E is not applicable in this scenario. This choice highlights the importance of methodically working through a problem and verifying the answer before resorting to a 'none of these' conclusion. It emphasizes the need for a thorough and confident approach to problem-solving, trusting in the process and the calculations.

Conclusion: Mastering Order of Operations

In conclusion, the correct answer to the expression (0.6 ÷ 0.3) - 0.5 - 0.2 + 0.3 is 1.6, which corresponds to option C. This solution was obtained by meticulously following the order of operations (PEMDAS), a fundamental principle in mathematics. This exercise underscores the critical importance of adhering to these rules to ensure accuracy in calculations. By correctly applying PEMDAS, we first addressed the division within the parentheses, followed by sequential subtraction and addition from left to right. This methodical approach not only led us to the correct answer but also illustrated the structured and logical nature of mathematical problem-solving.

Furthermore, the analysis of why other options were incorrect provided valuable insights into common errors and misconceptions. It highlighted the importance of precision, a thorough understanding of mathematical principles, and the necessity of a systematic approach. Avoiding these errors in future calculations will enhance mathematical proficiency and confidence. Mastery of the order of operations is not just essential for solving mathematical expressions; it is a foundational skill that applies to various scientific, engineering, and everyday problem-solving contexts. This proficiency enables individuals to approach complex problems with a structured mindset, breaking them down into manageable steps and arriving at accurate solutions. In essence, understanding and applying the order of operations is a cornerstone of mathematical literacy.