Step-by-Step Solution To 32 + [72 ÷ (7.5 - 13.2)]

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Navigating mathematical expressions often requires a clear understanding of the order of operations and a methodical approach. In this article, we will dissect the expression 32 + [72 ÷ (7.5 - 13.2)] step by step, ensuring clarity and accuracy in our calculations. Whether you're a student tackling algebra or simply brushing up on your math skills, this guide will provide a comprehensive breakdown of the process.

Understanding the Order of Operations

Before we dive into the specifics of this problem, 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 we perform mathematical operations:

  1. Parentheses: Operations within parentheses or brackets are performed first.
  2. Exponents: Next, we handle exponents and roots.
  3. Multiplication and Division: These operations are performed from left to right.
  4. Addition and Subtraction: Finally, addition and subtraction are carried out from left to right.

By adhering to this order, we ensure that we arrive at the correct solution. Now, let's apply this knowledge to our expression.

Step 1: Solve the Parenthetical Expression

The first step in solving 32 + [72 ÷ (7.5 - 13.2)] is to address the expression within the parentheses: (7.5 - 13.2). This is a straightforward subtraction problem. When subtracting a larger number from a smaller number, we can think of it as finding the difference and applying a negative sign.

So, we calculate 13.2 - 7.5, which equals 5.7. Since we are subtracting 13.2 from 7.5, the result is negative. Therefore,

(7.5 - 13.2) = -5.7

Now, we can substitute this result back into our original expression, which becomes:

32 + [72 ÷ (-5.7)]

Step 2: Perform the Division

With the parentheses addressed, our next step is to perform the division operation: 72 ÷ (-5.7). Dividing by a negative number will result in a negative quotient. To find the value, we divide 72 by 5.7. This may require a calculator or long division.

72 ÷ 5.7 ≈ 12.6316

Since we are dividing by -5.7, the result is negative:

72 ÷ (-5.7) ≈ -12.6316

Now, we substitute this result back into our expression:

32 + (-12.6316)

Step 3: Complete the Addition

The final step is to perform the addition: 32 + (-12.6316). Adding a negative number is the same as subtracting its positive counterpart. So, we can rewrite this as:

32 - 12.6316

Now, we perform the subtraction:

32 - 12.6316 ≈ 19.3684

Therefore, the final result of the expression 32 + [72 ÷ (7.5 - 13.2)] is approximately 19.3684.

Summarizing the Steps

Let's recap the steps we took to solve this expression:

  1. Parentheses: We first solved the expression within the parentheses: (7.5 - 13.2) = -5.7.
  2. Division: Next, we performed the division: 72 ÷ (-5.7) ≈ -12.6316.
  3. Addition: Finally, we completed the addition: 32 + (-12.6316) ≈ 19.3684.

By following the order of operations and breaking down the problem into smaller, manageable steps, we were able to arrive at the solution accurately.

The Importance of Precision

In mathematics, precision is key. When dealing with decimal numbers, as in our example, it's essential to maintain sufficient decimal places throughout the calculation to ensure an accurate final result. Rounding off too early can lead to significant errors, especially in more complex expressions.

In our case, we carried out the division to four decimal places, which provided a reasonably accurate answer. Depending on the context of the problem, you may need to carry out calculations to even more decimal places.

Common Mistakes to Avoid

When solving mathematical expressions, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

  • Ignoring the Order of Operations: This is the most common mistake. Always remember PEMDAS (or BODMAS) and follow the correct order.
  • Incorrectly Handling Negative Signs: Pay close attention to negative signs, especially when subtracting or dividing. A misplaced negative sign can completely change the result.
  • Rounding Off Too Early: As mentioned earlier, rounding off intermediate results can lead to inaccuracies. Try to keep as many decimal places as possible until the final step.
  • Misinterpreting the Expression: Make sure you understand exactly what the expression is asking. Read carefully and break it down into smaller parts if necessary.

By being aware of these common mistakes, you can minimize your chances of making errors and improve your accuracy in mathematical calculations.

Practical Applications

Understanding how to solve mathematical expressions is not just an academic exercise; it has practical applications in many real-world scenarios. From calculating finances to designing structures, mathematical operations are essential tools in various fields.

For instance, in finance, you might use similar calculations to determine the return on an investment or the cost of a loan. In engineering, these skills are crucial for calculating forces, stresses, and strains in structures. Even in everyday situations, such as cooking or home improvement, you often need to perform calculations involving multiple operations.

Conclusion

In conclusion, solving the expression 32 + [72 ÷ (7.5 - 13.2)] involves a methodical application of the order of operations. By breaking down the problem into smaller steps—parentheses, division, and addition—we can arrive at the correct solution: approximately 19.3684. Remember to pay attention to detail, especially when dealing with negative signs and decimal places, to avoid common mistakes. Mastering these skills is not only important for academic success but also for practical applications in various aspects of life. So, practice regularly, stay focused, and you'll become more confident in tackling mathematical challenges.

Understanding mathematical expressions and their solutions is a fundamental skill that extends beyond the classroom. Whether you're managing your personal finances, planning a project, or pursuing a career in a STEM field, the ability to accurately perform calculations is invaluable. By mastering the order of operations and practicing regularly, you can build confidence in your mathematical abilities and unlock new opportunities for problem-solving and critical thinking.

The key to success in mathematics lies in consistent effort and a willingness to learn from mistakes. Don't be discouraged by challenges; instead, view them as opportunities to grow and improve. With each problem you solve, you'll strengthen your understanding and develop a deeper appreciation for the power and elegance of mathematics.

So, embrace the challenge, stay curious, and continue to explore the fascinating world of numbers and equations. The more you practice, the more proficient you'll become, and the more you'll realize the countless ways in which mathematics enriches our lives.