Evaluating The Expression 500 - [(10 × 15) × (1/2 + 95)] A Step-by-Step Guide

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In this article, we will walk through the step-by-step evaluation of the mathematical expression: 500 - [(10 × 15) × (1/2 + 95)]. This problem involves the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). We will break down each step to ensure a clear and thorough understanding of how to arrive at the correct solution. This evaluation is crucial for anyone looking to sharpen their arithmetic skills, especially in contexts like algebra, calculus, or even everyday financial calculations. Let’s dive in and simplify this expression together!

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we begin, it's essential to grasp the order of operations, which dictates the sequence in which mathematical operations must be performed. The acronym PEMDAS (or BODMAS in some regions) provides a helpful mnemonic:

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

Following this order ensures that we evaluate expressions consistently and arrive at the correct answer. In our given expression, 500 - [(10 × 15) × (1/2 + 95)], we will first tackle the operations within the parentheses, then multiplication, and finally subtraction. Ignoring this order can lead to drastically different (and incorrect) results. So, keeping PEMDAS in mind is the cornerstone of accurate mathematical computations.

Step-by-Step Evaluation

Let's break down the evaluation of the expression 500 - [(10 × 15) × (1/2 + 95)] step by step.

1. Innermost Parentheses: (1/2 + 95)

We start with the innermost parentheses: (1/2 + 95). This requires us to add a fraction to a whole number. To do this, we can convert 95 into a fraction with a denominator of 2.

  • 95 = 95/1
  • To get a common denominator, multiply both the numerator and denominator by 2:
  • 95/1 = (95 × 2) / (1 × 2) = 190/2

Now, we can add the fractions:

  • 1/2 + 190/2 = 191/2

So, the expression inside the innermost parentheses simplifies to 191/2, which is also equal to 95.5.

2. Outer Parentheses: (10 × 15)

Next, we evaluate the other set of parentheses: (10 × 15). This is a straightforward multiplication:

  • 10 × 15 = 150

Thus, the second set of parentheses simplifies to 150.

3. Multiplication within the Brackets: (150 × 191/2)

Now we deal with the brackets: [(10 × 15) × (1/2 + 95)], which we have simplified to [150 × 191/2]. We need to multiply 150 by 191/2. We can rewrite this as:

  • 150 × (191/2) = (150 × 191) / 2

First, multiply 150 by 191:

  • 150 × 191 = 28650

Now, divide the result by 2:

  • 28650 / 2 = 14325

So, the expression within the brackets simplifies to 14325.

4. Final Subtraction: 500 - 14325

Finally, we perform the subtraction:

  • 500 - 14325 = -13825

Therefore, the final result of the expression 500 - [(10 × 15) × (1/2 + 95)] is -13825. This step is crucial as it gives the final value after considering all other operations in the correct order. Proper execution of subtraction ensures the accuracy of the entire evaluation.

Common Mistakes to Avoid

When evaluating mathematical expressions, several common mistakes can lead to incorrect answers. Recognizing and avoiding these pitfalls is key to mastering arithmetic and algebra. Here are some frequent errors to watch out for:

  1. Ignoring the Order of Operations: As discussed earlier, the order of operations (PEMDAS/BODMAS) is crucial. A common mistake is to perform operations in the order they appear from left to right, rather than following the correct hierarchy. For example, performing subtraction before multiplication can lead to significant errors.

  2. Incorrectly Handling Parentheses: Parentheses indicate which operations should be performed first. Failing to address operations within parentheses before moving on to other parts of the expression is a frequent mistake. Always start with the innermost set of parentheses and work your way outwards.

  3. Arithmetic Errors: Simple arithmetic mistakes such as incorrect addition, subtraction, multiplication, or division can derail the entire evaluation. It’s always a good idea to double-check your calculations, especially in complex expressions.

  4. Sign Errors: Dealing with negative numbers can be tricky. Forgetting to apply the correct sign during operations (especially in subtraction and multiplication) is a common source of errors. Pay close attention to the signs of numbers and how they interact during calculations.

  5. Fraction and Decimal Mistakes: Working with fractions and decimals requires careful attention to detail. Mistakes can occur during conversion, addition, subtraction, multiplication, or division of fractions and decimals. Ensure you have a solid understanding of these operations to avoid errors.

  6. Misinterpreting the Expression: Sometimes, the structure of the expression itself can be misinterpreted. For instance, students might overlook a set of parentheses or misread a mathematical symbol. Always take a moment to thoroughly understand the expression before beginning the evaluation.

  7. Rushing Through the Problem: Math problems, especially those involving multiple steps, require patience and attention to detail. Rushing through the problem increases the likelihood of making mistakes. Take your time, work methodically, and double-check each step.

By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy in evaluating mathematical expressions.

Practice Problems

To reinforce your understanding of evaluating expressions using the order of operations, let’s work through a few more practice problems. These examples will help you apply the concepts we’ve discussed and build confidence in your mathematical skills.

Practice Problem 1: 100 + (25 × 4) - 50 / 2

  1. Parentheses: 25 × 4 = 100
  2. Division: 50 / 2 = 25
  3. Addition: 100 + 100 = 200
  4. Subtraction: 200 - 25 = 175

So, the final answer is 175.

Practice Problem 2: (15 + 5) × 3 - 2 × (18 / 6)

  1. Parentheses (first set): 15 + 5 = 20
  2. Parentheses (second set): 18 / 6 = 3
  3. Multiplication (first): 20 × 3 = 60
  4. Multiplication (second): 2 × 3 = 6
  5. Subtraction: 60 - 6 = 54

The final answer is 54.

Practice Problem 3: 36 / (3 × 4) + 2^3 - 10

  1. Parentheses: 3 × 4 = 12
  2. Exponent: 2^3 = 8
  3. Division: 36 / 12 = 3
  4. Addition: 3 + 8 = 11
  5. Subtraction: 11 - 10 = 1

The final answer is 1.

By working through these problems, you can see how the order of operations is consistently applied to achieve the correct results. Practice is key to mastering these skills, so try tackling more examples on your own.

Conclusion

In summary, we have successfully evaluated the expression 500 - [(10 × 15) × (1/2 + 95)], arriving at the solution -13825. This process underscores the importance of adhering to the order of operations (PEMDAS/BODMAS) to ensure accurate results. By systematically addressing each part of the expression—starting with the innermost parentheses, moving through multiplication, and finally completing the subtraction—we were able to break down a complex problem into manageable steps.

We also highlighted common mistakes to avoid, such as neglecting the order of operations, making arithmetic errors, or mishandling signs and fractions. Awareness of these pitfalls can significantly improve your accuracy in mathematical calculations. The practice problems further illustrated how to apply these concepts, reinforcing the need for methodical and careful calculation.

Mastering the evaluation of mathematical expressions is a fundamental skill that extends beyond the classroom. It’s essential for various fields, including science, engineering, finance, and even everyday problem-solving. By consistently practicing and applying the principles discussed, you can build a strong foundation in mathematics and enhance your analytical abilities.