Solving 7/2 × 1/4 - 7/2 + 9/2 × 1/2 A Step-by-Step Guide

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Navigating the world of mathematical expressions often requires a firm grasp of the order of operations. This fundamental principle ensures that we arrive at the correct solution, regardless of the complexity of the equation. In this comprehensive guide, we will meticulously break down the problem 7/2 × 1/4 - 7/2 + 9/2 × 1/2, employing the order of operations to arrive at the accurate answer. By the end of this exploration, you'll not only have the solution but also a deepened understanding of the mathematical principles at play.

Understanding the Order of Operations (PEMDAS/BODMAS)

The cornerstone of solving mathematical expressions lies in adhering to the order of operations, commonly remembered by the acronyms PEMDAS or BODMAS. These acronyms represent the sequence in which operations must be performed:

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

The PEMDAS/BODMAS rule is not merely a convention; it's a critical framework that ensures consistency and accuracy in mathematical calculations. Without it, the same expression could yield multiple different answers depending on the order in which operations are performed. This would lead to confusion and invalidate the very purpose of mathematics as a precise and universal language.

The order of operations acts as a roadmap, guiding us through the intricate landscape of mathematical expressions. By following this roadmap, we ensure that each operation is performed in its designated sequence, ultimately leading us to the correct destination – the solution. Ignoring this order is akin to navigating without a map, increasing the likelihood of getting lost or arriving at the wrong place.

Before we delve into the specific problem at hand, let's underscore the significance of PEMDAS/BODMAS. It's not just a mnemonic device; it's the bedrock of mathematical consistency. Imagine a world where addition took precedence over multiplication, or where exponents were calculated before parentheses were resolved. The resulting chaos would render mathematical communication impossible. The order of operations, therefore, is more than a rule; it's the foundation upon which the entire edifice of mathematics is built.

Step-by-Step Solution of 7/2 × 1/4 - 7/2 + 9/2 × 1/2

Now, let's apply our understanding of the order of operations to the problem 7/2 × 1/4 - 7/2 + 9/2 × 1/2. We will proceed systematically, adhering to the PEMDAS/BODMAS rule at each stage. This methodical approach will not only lead us to the correct answer but also provide a clear and understandable solution process.

Step 1: Multiplication

According to PEMDAS/BODMAS, multiplication takes precedence over addition and subtraction. Therefore, we begin by performing the multiplication operations in the expression. We have two multiplication operations to address: 7/2 × 1/4 and 9/2 × 1/2. Let's tackle them one by one.

7/2 × 1/4

To multiply fractions, we simply multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, 7/2 multiplied by 1/4 is calculated as follows:

(7 × 1) / (2 × 4) = 7/8

Thus, the result of the first multiplication operation is 7/8.

9/2 × 1/2

Similarly, we multiply the numerators and denominators for the second multiplication operation:

(9 × 1) / (2 × 2) = 9/4

Therefore, the result of the second multiplication operation is 9/4.

After completing the multiplication operations, our expression now looks like this:

7/8 - 7/2 + 9/4

We have successfully simplified the expression by addressing the multiplication components, bringing us closer to the final solution.

Step 2: Addition and Subtraction (from left to right)

With the multiplication operations completed, we now move on to addition and subtraction. It's crucial to remember that addition and subtraction are performed from left to right, as they hold equal precedence in the order of operations. This left-to-right approach ensures that we maintain the correct sequence of calculations and avoid any potential errors.

7/8 - 7/2

First, we perform the subtraction operation 7/8 - 7/2. To subtract fractions, they must have a common denominator. The least common multiple (LCM) of 8 and 2 is 8. So, we need to convert 7/2 to an equivalent fraction with a denominator of 8. To do this, we multiply both the numerator and the denominator of 7/2 by 4:

(7 × 4) / (2 × 4) = 28/8

Now we can perform the subtraction:

7/8 - 28/8 = -21/8

The result of the first subtraction is -21/8.

-21/8 + 9/4

Next, we perform the addition operation -21/8 + 9/4. Again, we need a common denominator. The LCM of 8 and 4 is 8. So, we convert 9/4 to an equivalent fraction with a denominator of 8 by multiplying both the numerator and the denominator by 2:

(9 × 2) / (4 × 2) = 18/8

Now we can perform the addition:

-21/8 + 18/8 = -3/8

Therefore, the final result of the expression is -3/8.

Final Answer: -3/8

By meticulously following the order of operations, we have successfully solved the expression 7/2 × 1/4 - 7/2 + 9/2 × 1/2. The final answer is -3/8. This solution not only provides the numerical result but also demonstrates the importance of adhering to the established rules of mathematical operations.

The journey through this problem has underscored the critical role of PEMDAS/BODMAS in mathematical problem-solving. The order of operations is not merely a set of guidelines; it's the very language of mathematics, ensuring that expressions are interpreted and solved consistently across the globe. Without this shared understanding, mathematical communication would be impossible.

The significance of PEMDAS/BODMAS extends far beyond the realm of textbooks and classrooms. It's a fundamental principle that underpins countless applications in science, engineering, finance, and many other fields. From calculating the trajectory of a spacecraft to determining the optimal interest rate on a loan, the order of operations is an indispensable tool for accurate and reliable calculations.

As you continue your mathematical journey, remember that the order of operations is your steadfast companion. It will guide you through complex expressions, ensuring that you arrive at the correct solutions every time. Embrace PEMDAS/BODMAS, and you'll unlock a world of mathematical possibilities.

Practice Problems

To solidify your understanding of the order of operations, try solving these practice problems:

  1. (1/2 + 1/4) × 2/3 - 1/6
  2. 3/4 ÷ 1/2 + 5/8 × 4/5
  3. 2/5 × (3/4 - 1/2) + 7/10

By working through these problems, you'll reinforce your grasp of PEMDAS/BODMAS and develop your problem-solving skills. Remember, practice makes perfect! The more you apply the order of operations, the more confident and proficient you'll become in tackling mathematical challenges.

Conclusion

Mastering the order of operations is a crucial step in your mathematical journey. By understanding and applying PEMDAS/BODMAS, you can confidently solve complex expressions and unlock the power of mathematics. Remember to approach each problem systematically, breaking it down into smaller, manageable steps. And most importantly, practice regularly to hone your skills and deepen your understanding. With dedication and perseverance, you can conquer any mathematical challenge that comes your way.