Solving 30-5.6:7+2.10.40-30+5 A Step-by-Step Guide
Introduction
In this comprehensive article, we will delve into the step-by-step solution of the mathematical expression 30-5.6:7+2.10.40-30+5. This expression involves a combination of arithmetic operations, including subtraction, multiplication, division, and addition. To solve it accurately, we need to adhere to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Understanding and applying PEMDAS is crucial for simplifying complex mathematical expressions and arriving at the correct result. This guide will not only provide the final answer but also break down each step in detail, ensuring a clear understanding of the process. Mastering the order of operations is a fundamental skill in mathematics, essential for various applications ranging from basic arithmetic to more advanced mathematical problems. Through this detailed explanation, readers will gain confidence in tackling similar expressions and enhance their problem-solving abilities in mathematics.
Breaking Down the Expression
To effectively solve the expression 30-5.6:7+2.10.40-30+5, we must first understand the significance of the order of operations. The acronym PEMDAS (or BODMAS, depending on the region) dictates the sequence in which mathematical operations should be performed: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Ignoring this order can lead to incorrect results. In our expression, there are no parentheses or exponents, so we focus on multiplication and division first, followed by addition and subtraction. This methodical approach ensures accuracy and clarity in each step. The breakdown involves identifying each operation and its position within the expression, setting the stage for a systematic simplification. Understanding this preliminary step is vital for preventing errors and ensuring a smooth progression toward the final solution. Recognizing the structure of the expression and the operations involved is the foundation of mathematical problem-solving.
Step 1: Multiplication and Division
According to the order of operations (PEMDAS), multiplication and division take precedence over addition and subtraction. In the expression 30-5.6:7+2.10.40-30+5, we first encounter the division operation 5.6:7. Performing this operation, we divide 5.6 by 7, which results in 0.8. Next, we identify the multiplication operations. We have 2.10, which equals 20, and then 20.40, which equals 800. These multiplication and division operations must be completed before we can proceed with addition and subtraction. This step is critical because altering the order would lead to a different, incorrect answer. By prioritizing multiplication and division, we maintain the mathematical integrity of the expression and ensure accurate simplification. This adherence to the rules of PEMDAS is what allows us to systematically reduce the complexity of the problem and move closer to the final solution. The careful execution of these initial operations is crucial for the subsequent steps.
Step 2: Substituting the Results
After performing the multiplication and division operations, we substitute the results back into the original expression. The original expression was 30-5.6:7+2.10.40-30+5. We found that 5.6:7 equals 0.8 and 2.10.40 equals 800. Substituting these values, the expression now becomes 30 - 0.8 + 800 - 30 + 5. This substitution simplifies the expression, making it easier to manage and solve. It transforms the problem into a series of addition and subtraction operations, which can be performed sequentially. This step is a crucial transition, bridging the initial complex form of the expression to a more manageable one. By accurately replacing the multiplication and division segments with their calculated values, we maintain the expression's mathematical equivalence while significantly reducing its visual and computational complexity. This careful substitution is key to preventing errors and streamlining the remaining steps of the solution process.
Step 3: Addition and Subtraction
With the expression simplified to 30 - 0.8 + 800 - 30 + 5, we now proceed with addition and subtraction. According to PEMDAS, addition and subtraction are performed from left to right. First, we subtract 0.8 from 30, resulting in 29.2. The expression then becomes 29.2 + 800 - 30 + 5. Next, we add 800 to 29.2, which gives us 829.2. The expression is further reduced to 829.2 - 30 + 5. Subtracting 30 from 829.2 yields 799.2. Finally, we add 5 to 799.2, resulting in 804.2. This step-by-step approach ensures that we accurately account for each operation, maintaining the integrity of the mathematical process. By performing addition and subtraction sequentially, we avoid potential confusion and errors, leading us to the correct final answer. This meticulous execution of the remaining operations is vital for the overall accuracy of the solution.
Final Answer
After carefully following the order of operations and performing each step methodically, we arrive at the final answer. Starting with the expression 30-5.6:7+2.10.40-30+5, we first addressed the multiplication and division, followed by addition and subtraction. Through these steps, we simplified the expression and calculated the result. The final answer to the expression 30-5.6:7+2.10.40-30+5 is 804.2. This result is the culmination of a series of precise calculations, each building upon the previous one. The accuracy of this answer highlights the importance of adhering to the order of operations and executing each step with diligence. This final value represents the solution to the initial mathematical problem and demonstrates the power of methodical problem-solving in mathematics. Achieving this answer reinforces the understanding of fundamental arithmetic principles and their application in complex expressions.
Conclusion
In conclusion, solving the mathematical expression 30-5.6:7+2.10.40-30+5 requires a clear understanding and application of the order of operations (PEMDAS). By breaking down the expression into manageable steps, we can accurately perform each operation and arrive at the correct answer. This process involves prioritizing multiplication and division, followed by addition and subtraction, all executed from left to right. The final result, 804.2, demonstrates the importance of precision and attention to detail in mathematical calculations. Mastering the order of operations is a crucial skill that extends beyond simple arithmetic, forming the foundation for more advanced mathematical concepts. This comprehensive guide provides a step-by-step approach to solving complex expressions, enhancing problem-solving abilities and fostering confidence in tackling mathematical challenges. The ability to systematically solve such expressions is invaluable in various fields, highlighting the practical significance of mathematical proficiency. Therefore, understanding and applying these principles is essential for both academic and real-world applications.
