Calculating 8.5 [15.3 / (6.05 - 1.87) - (9.4 - 5.22) × 0.03] A Step-by-Step Guide

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This article delves into the step-by-step calculation of the mathematical expression: 8.5 [15.3 / (6.05 - 1.87) - (9.4 - 5.22) × 0.03]. We will break down each operation, providing a clear and detailed explanation to ensure a comprehensive understanding of the solution. This exercise not only reinforces fundamental arithmetic principles but also highlights the importance of order of operations in achieving accurate results. Whether you're a student looking to improve your math skills or simply curious about how to solve such expressions, this guide offers valuable insights and a methodical approach.

Understanding the Order of Operations

Before we dive into the calculation, it's crucial to understand the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which operations should be performed to arrive at the correct answer. Ignoring this order can lead to significantly different and incorrect results. In our expression, we will first address the operations within the parentheses, followed by division and multiplication, and finally, subtraction and multiplication outside the brackets.

Step 1: Solving the Parentheses

Our expression contains two sets of parentheses: (6.05 - 1.87) and (9.4 - 5.22). We'll start by solving each of these.

  • (6.05 - 1.87): This is a simple subtraction. Subtracting 1.87 from 6.05 gives us 4.18. This result will be used in the subsequent division step.
  • (9.4 - 5.22): Similarly, we subtract 5.22 from 9.4, which equals 4.18. This value will be used in the multiplication step.

Now, our expression looks like this: 8.5 [15.3 / 4.18 - 4.18 × 0.03]. We have successfully simplified the expressions within the parentheses.

Step 2: Division and Multiplication

Next, we tackle the division and multiplication operations within the brackets. According to PEMDAS, we perform these operations from left to right.

  • 15.3 / 4.18: This division yields approximately 3.6603. It's important to carry several decimal places to maintain accuracy in the final result. This value represents the quotient of the division.
  • 4.18 × 0.03: Multiplying 4.18 by 0.03 gives us 0.1254. This is the product of the multiplication operation. Now, our expression inside the brackets is simplified to 3.6603 - 0.1254.

At this stage, our overall expression is: 8.5 [3.6603 - 0.1254]. We have completed the division and multiplication steps, bringing us closer to the final solution.

Step 3: Subtraction within the Brackets

Now, we perform the subtraction within the brackets: 3. 6603 - 0.1254. This subtraction results in 3.5349. This step simplifies the expression inside the brackets to a single value.

Our expression now looks like this: 8.5 [3.5349]. The brackets now simply indicate multiplication, as there are no more operations to perform within them.

Step 4: Final Multiplication

Finally, we perform the last operation: multiplying 8.5 by 3.5349. This multiplication gives us the final result.

  • 8.5 × 3.5349: This multiplication equals approximately 30.04665. We can round this to a more manageable number, depending on the desired level of precision. For practical purposes, rounding to two decimal places gives us 30.05.

Therefore, the final answer to the expression 8.5 [15.3 / (6.05 - 1.87) - (9.4 - 5.22) × 0.03] is approximately 30.05. This completes the step-by-step calculation, demonstrating how to systematically solve a complex mathematical expression using the order of operations.

Detailed Breakdown of Each Step

To further clarify the process, let's revisit each step with a more granular focus. This will reinforce the methodology and help solidify your understanding of the calculation.

Parentheses in Depth

The initial step involves simplifying the expressions within the parentheses. This is paramount because parentheses dictate the priority of operations. Ignoring this can lead to a cascading effect of errors throughout the calculation.

  • Focusing on (6.05 - 1.87): The subtraction of 1.87 from 6.05 is a straightforward decimal subtraction. Aligning the decimal points and performing the subtraction yields 4.18. This seemingly simple step is the foundation for the subsequent division.
  • Analyzing (9.4 - 5.22): Similarly, subtracting 5.22 from 9.4 also requires careful alignment of decimal points. The result, 4.18, might seem coincidental but is a crucial element in the overall calculation. This value is then used in the multiplication operation, highlighting the interconnectedness of each step.

Diving into Division and Multiplication

With the parentheses resolved, we move onto division and multiplication, working from left to right as per PEMDAS. This section requires a keen eye for detail and precision.

  • Exploring 15.3 / 4.18: Dividing 15.3 by 4.18 is a decimal division that results in a non-terminating decimal. For accuracy, we carry several decimal places, resulting in approximately 3.6603. This value represents the quotient and plays a pivotal role in the next subtraction step.
  • Dissecting 4.18 × 0.03: Multiplying 4.18 by 0.03 involves multiplying a decimal by a small decimal, resulting in a smaller decimal value. The product, 0.1254, is the result of this multiplication and is crucial for the subsequent subtraction within the brackets.

Subtraction Unveiled

The subtraction within the brackets, 3.6603 - 0.1254, is a critical step that consolidates the results of the previous division and multiplication. This operation simplifies the expression within the brackets to a single value.

  • The Nuances of 3.6603 - 0.1254: This subtraction requires aligning the decimal points and carefully subtracting each digit. The result, 3.5349, is the simplified value within the brackets and is the penultimate step before the final multiplication.

Final Multiplication Elucidated

The final multiplication, 8.5 × 3.5349, is the culmination of all the previous steps. This operation yields the final answer to the expression.

  • The Grand Finale: 8.5 × 3.5349: Multiplying 8.5 by 3.5349 results in approximately 30.04665. Rounding this to two decimal places, we arrive at 30.05. This final result is the solution to the original expression, showcasing the power of methodical calculation and adherence to the order of operations.

Common Mistakes and How to Avoid Them

When tackling mathematical expressions, it's easy to make mistakes if you're not careful. Understanding common pitfalls can help you avoid them and improve your accuracy. Here are some common errors and how to prevent them:

  1. Ignoring the Order of Operations: This is the most frequent mistake. Always remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Solve operations in this order to avoid incorrect results. For instance, in our example, performing multiplication before dealing with parentheses would lead to a completely different answer.
  2. Incorrectly Handling Decimal Points: Decimal points can be tricky. Make sure to align them properly during addition and subtraction. In multiplication and division, keep track of the number of decimal places to ensure the correct placement in the final answer. A misplaced decimal point can drastically alter the value of the result.
  3. Rounding Errors: Rounding too early in the calculation can lead to inaccuracies in the final answer. It's best to carry several decimal places throughout the calculation and only round at the very end. In our example, rounding the result of the division 15.3 / 4.18 too early could affect the final result.
  4. Sign Errors: Pay close attention to negative signs, especially when subtracting negative numbers. A simple sign error can change the entire outcome of the calculation. Double-check each step to ensure you've correctly applied the signs.
  5. Misinterpreting the Expression: Sometimes, the way an expression is written can be confusing. Take a moment to understand the expression fully before you start calculating. Identify all the operations and their correct order. If necessary, rewrite the expression in a clearer format.

By being mindful of these common mistakes and taking the necessary precautions, you can significantly improve your accuracy and confidence in solving mathematical expressions. Practice and attention to detail are key to mastering these skills.

Practical Applications of Order of Operations

The order of operations isn't just a theoretical concept; it has numerous practical applications in everyday life and various fields. Understanding and applying PEMDAS (or BODMAS) is crucial for accurate calculations in diverse scenarios.

  1. Financial Calculations: In finance, the order of operations is vital for calculating interest, investments, and loans. For instance, determining the future value of an investment involves complex formulas that require strict adherence to the order of operations. Incorrect application can lead to significant errors in financial planning.
  2. Cooking and Baking: Recipes often involve measurements and ratios that need precise calculations. Scaling a recipe up or down requires applying the correct order of operations to maintain the proper proportions of ingredients. This ensures the final dish turns out as intended.
  3. Engineering and Construction: Engineers and construction workers use mathematical calculations extensively for designing structures, estimating materials, and managing projects. Complex calculations involving multiple operations are common, and adhering to the order of operations is essential for safety and accuracy.
  4. Computer Programming: In programming, mathematical expressions are used extensively for various purposes, from simple calculations to complex algorithms. Programming languages follow the order of operations, and understanding this concept is crucial for writing correct and efficient code.
  5. Data Analysis: Analyzing data often involves complex statistical calculations. Whether it's calculating averages, standard deviations, or performing regressions, the order of operations is necessary for accurate results. This is vital for making informed decisions based on data.
  6. Physics and Science: Scientific calculations, particularly in physics, chemistry, and biology, often involve complex formulas with multiple operations. Understanding the order of operations is crucial for solving problems related to motion, energy, chemical reactions, and biological processes.

These examples highlight the widespread relevance of the order of operations in various fields. By mastering this fundamental concept, you can enhance your problem-solving skills and perform accurate calculations in real-world scenarios.

Conclusion

In conclusion, the step-by-step calculation of the expression 8.5 [15.3 / (6.05 - 1.87) - (9.4 - 5.22) × 0.03] demonstrates the crucial role of the order of operations in mathematics. By meticulously following PEMDAS, we arrived at the final answer of approximately 30.05. This exercise not only reinforces basic arithmetic skills but also highlights the importance of precision and attention to detail in mathematical problem-solving. Whether you're a student, professional, or simply someone who enjoys mathematics, understanding and applying the order of operations is essential for achieving accurate results in a wide range of applications. Remember to break down complex expressions into smaller, manageable steps, and always double-check your work to avoid common errors. With practice and a methodical approach, you can confidently tackle any mathematical challenge.