Solving 3/4 ÷ 3/6 × 1/2 - 1/6 + 1/2 A Step-by-Step Guide

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Navigating the world of fractions can often feel like traversing a complex maze, especially when faced with a seemingly intricate expression like 3/4 ÷ 3/6 × 1/2 - 1/6 + 1/2. However, with a clear understanding of the order of operations and the fundamental principles of fraction arithmetic, even the most daunting problems can be broken down into manageable steps. In this comprehensive guide, we will embark on a journey to unravel this particular expression, equipping you with the knowledge and confidence to tackle similar challenges in the future. By meticulously dissecting each operation and employing the correct sequence, we will arrive at the accurate solution, demystifying the process and solidifying your grasp of fraction manipulation. So, let's delve into the realm of fractions and conquer this mathematical puzzle together.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we dive into the specific problem at hand, it's crucial to revisit the bedrock of mathematical calculations: the order of operations. Often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), this principle dictates the sequence in which mathematical operations must be performed to ensure a consistent and accurate result. In essence, it provides a roadmap for navigating complex expressions, preventing ambiguity and ensuring that everyone arrives at the same answer. Ignoring this fundamental rule can lead to drastically different outcomes, highlighting its importance in the world of mathematics. Therefore, understanding and adhering to PEMDAS/BODMAS is paramount to successfully solving any mathematical problem, particularly those involving multiple operations.

Let's break down the acronym:

  • P/B: Parentheses/Brackets: Operations within parentheses or brackets are always performed first. This is the highest priority in the order of operations. They act as containers, isolating specific parts of the expression that need to be evaluated before interacting with the rest of the equation. Ignoring parentheses can completely alter the outcome of a calculation, emphasizing their crucial role in mathematical expressions.
  • E/O: Exponents/Orders: Next, we address exponents or orders (powers and square roots, etc.). Exponents represent repeated multiplication and contribute significantly to the magnitude of a number. Handling them correctly is vital for accurate calculations. Understanding exponents is not just about following the order of operations; it's about grasping the concept of exponential growth, which has applications far beyond basic arithmetic, including fields like finance and computer science.
  • MD: Multiplication and Division: Multiplication and division are performed from left to right. These operations are inverse to each other and hold equal precedence in the order of operations. This left-to-right approach ensures consistency when an expression contains both multiplication and division. It's a subtle rule but essential for maintaining accuracy in mathematical calculations.
  • AS: Addition and Subtraction: Finally, addition and subtraction are performed from left to right, similar to multiplication and division. These are the fundamental operations of arithmetic, and their correct execution is the final step in simplifying an expression. Just like multiplication and division, addition and subtraction are inverse operations and are performed from left to right to ensure consistency.

By consistently applying PEMDAS/BODMAS, we can systematically simplify complex expressions and arrive at the correct solution. This order of operations serves as a universal language in mathematics, ensuring that regardless of who is solving a problem, the approach and the answer will be the same.

Step-by-Step Solution of 3/4 ÷ 3/6 × 1/2 - 1/6 + 1/2

Now that we have a firm grasp on the order of operations, let's apply it to the expression 3/4 ÷ 3/6 × 1/2 - 1/6 + 1/2. We will meticulously break down each step, explaining the reasoning behind each operation and highlighting the importance of adhering to PEMDAS/BODMAS.

Step 1: Division (3/4 ÷ 3/6)

According to the order of operations, division and multiplication take precedence over addition and subtraction. When dividing fractions, we invert the second fraction and multiply. So, 3/4 ÷ 3/6 becomes 3/4 × 6/3. This transformation is a fundamental rule of fraction division, turning a division problem into a multiplication problem. It's a critical step in simplifying expressions involving fractions and requires careful attention to detail.

Multiplying the numerators (3 × 6) gives us 18, and multiplying the denominators (4 × 3) gives us 12. Therefore, 3/4 × 6/3 = 18/12. This result is an improper fraction, meaning the numerator is larger than the denominator. While technically correct, it's often beneficial to simplify improper fractions into mixed numbers or their simplest form for easier interpretation and further calculations.

Now, we simplify 18/12 by finding the greatest common divisor (GCD) of 18 and 12, which is 6. Dividing both the numerator and the denominator by 6, we get 18 ÷ 6 = 3 and 12 ÷ 6 = 2. So, 18/12 simplifies to 3/2. This simplification process is crucial for obtaining the most concise and manageable form of the fraction, making it easier to work with in subsequent steps. Simplified fractions also provide a clearer understanding of the fraction's value and its relationship to other numbers.

Step 2: Multiplication (3/2 × 1/2)

Next, we perform the multiplication operation. Multiplying fractions involves multiplying the numerators together and the denominators together. So, 3/2 × 1/2 becomes (3 × 1) / (2 × 2). This straightforward process of multiplying numerators and denominators is a core concept in fraction arithmetic and is essential for accurately combining fractions.

This gives us 3/4. The result of this multiplication is a proper fraction, meaning the numerator is smaller than the denominator. This fraction is already in its simplest form, as 3 and 4 share no common factors other than 1. This makes it ready to be used in the next steps of the calculation without further simplification.

Step 3: Subtraction (3/4 - 1/6)

Now we move on to subtraction. To subtract fractions, they must have a common denominator. The least common multiple (LCM) of 4 and 6 is 12. Finding the LCM is a crucial step in adding or subtracting fractions with different denominators. It allows us to express the fractions with a common base, making the addition or subtraction operation possible. If we use a common denominator that is not the LCM, we can still arrive at the correct answer, but the fraction will need to be simplified at the end.

We convert 3/4 to an equivalent fraction with a denominator of 12 by multiplying both the numerator and denominator by 3: (3 × 3) / (4 × 3) = 9/12. This process ensures that we are not changing the value of the fraction, only its representation. Multiplying both the numerator and denominator by the same number maintains the fraction's proportion while allowing us to achieve the desired common denominator.

Similarly, we convert 1/6 to an equivalent fraction with a denominator of 12 by multiplying both the numerator and denominator by 2: (1 × 2) / (6 × 2) = 2/12. Just like with the previous fraction, this conversion maintains the fraction's value while providing the necessary common denominator for subtraction.

Now we can subtract: 9/12 - 2/12 = (9 - 2) / 12 = 7/12. Subtracting fractions with a common denominator involves subtracting the numerators and keeping the denominator the same. This is a fundamental rule of fraction arithmetic and is essential for accurately performing subtraction operations.

Step 4: Addition (7/12 + 1/2)

Finally, we perform the addition. Again, we need a common denominator. The least common multiple (LCM) of 12 and 2 is 12. This step reinforces the importance of finding the LCM when adding or subtracting fractions with different denominators. It ensures that the fractions can be combined accurately and efficiently.

We convert 1/2 to an equivalent fraction with a denominator of 12 by multiplying both the numerator and denominator by 6: (1 × 6) / (2 × 6) = 6/12. This conversion process is consistent with the previous steps, ensuring that we maintain the value of the fraction while achieving the necessary common denominator.

Now we can add: 7/12 + 6/12 = (7 + 6) / 12 = 13/12. Adding fractions with a common denominator involves adding the numerators and keeping the denominator the same. This is a straightforward process that allows us to combine fractions and arrive at the final result.

Step 5: Simplify the Result

The result, 13/12, is an improper fraction. We can convert it to a mixed number by dividing 13 by 12. 13 divided by 12 is 1 with a remainder of 1. Therefore, 13/12 is equal to 1 1/12. Converting improper fractions to mixed numbers often makes the result easier to understand and interpret, especially in practical applications.

Therefore, 3/4 ÷ 3/6 × 1/2 - 1/6 + 1/2 = 13/12 or 1 1/12.

Common Mistakes to Avoid When Working with Fractions

Working with fractions can be tricky, and it's easy to make mistakes if you're not careful. Recognizing common pitfalls can help you avoid errors and build a stronger foundation in fraction arithmetic. Let's explore some frequent mistakes and how to steer clear of them.

  • Forgetting the Order of Operations: As we've emphasized, the order of operations (PEMDAS/BODMAS) is paramount. Neglecting this order can lead to incorrect results. For example, performing addition before multiplication or division can drastically change the outcome. Always double-check the order of operations before starting a problem and make sure you're following it consistently throughout your calculations.
  • Incorrectly Dividing Fractions: A common mistake is dividing fractions directly without inverting and multiplying. Remember, dividing by a fraction is the same as multiplying by its reciprocal. Failing to invert the second fraction before multiplying is a fundamental error that will lead to the wrong answer. Always take the extra moment to ensure you've correctly applied this rule.
  • Adding or Subtracting Fractions Without a Common Denominator: Fractions must have a common denominator before they can be added or subtracted. Attempting to add or subtract fractions with different denominators without finding a common denominator is a frequent error. Remember to find the least common multiple (LCM) of the denominators and convert the fractions accordingly before performing the addition or subtraction.
  • Simplifying Fractions Incorrectly: Simplifying fractions is crucial, but it's essential to do it correctly. Make sure you're dividing both the numerator and denominator by their greatest common divisor (GCD). Dividing by a common factor that is not the GCD will result in a simplified fraction, but it won't be in its simplest form, which can lead to complications in later calculations. Always aim for the most simplified form of the fraction.
  • Misinterpreting Mixed Numbers and Improper Fractions: Confusing mixed numbers and improper fractions can lead to errors. Remember that a mixed number combines a whole number and a proper fraction, while an improper fraction has a numerator larger than or equal to the denominator. Be comfortable converting between these forms as needed, and understand when each form is most appropriate for a given calculation.
  • Careless Arithmetic: Simple arithmetic errors, such as incorrect multiplication or addition, can derail your entire calculation. It's essential to be meticulous and double-check your work at each step. Even a small mistake can propagate through the problem and lead to a wrong final answer. Practice and careful attention to detail are key to minimizing these errors.

By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy and confidence when working with fractions. Fraction arithmetic is a foundational skill in mathematics, and mastering it will benefit you in various areas of study and real-life applications.

Practice Problems to Sharpen Your Skills

To solidify your understanding of fraction operations and master the techniques we've discussed, practice is key. Working through a variety of problems will not only reinforce your knowledge but also help you develop problem-solving skills and build confidence in your abilities. Here are some practice problems to challenge yourself:

  1. (2/5 + 1/4) × 2/3: This problem combines addition and multiplication, requiring you to apply the order of operations correctly. Remember to find a common denominator before adding the fractions within the parentheses.
  2. 1/2 ÷ (3/4 - 1/8): This problem involves division and subtraction. Pay close attention to the parentheses and perform the subtraction first. Then, remember to invert and multiply when dividing fractions.
  3. 5/6 × 2/3 + 1/4: This problem combines multiplication and addition. Perform the multiplication before the addition, and remember to find a common denominator before adding the fractions.
  4. (3/4 ÷ 1/2) - 1/3 × 2/5: This problem includes division, subtraction, and multiplication. Follow the order of operations carefully, performing division and multiplication from left to right before subtraction.
  5. 1 1/2 + 2/3 ÷ 4/5: This problem involves a mixed number, addition, and division. Convert the mixed number to an improper fraction before proceeding with the calculations. Remember to invert and multiply when dividing fractions.

Tips for Practicing:

  • Show Your Work: Write down each step of your solution. This will help you track your progress, identify any errors, and understand the reasoning behind each operation. Showing your work is a valuable habit that promotes clarity and accuracy.
  • Check Your Answers: Once you've solved a problem, check your answer using a calculator or by working through the problem again. This will help you catch any mistakes and ensure you've arrived at the correct solution. Verification is a crucial step in the problem-solving process.
  • Vary the Problems: Work through a variety of problems with different levels of difficulty. This will challenge you to apply your knowledge in different contexts and help you develop a deeper understanding of fraction operations. Variety is the spice of learning!
  • Seek Help When Needed: If you're struggling with a particular problem or concept, don't hesitate to seek help from a teacher, tutor, or online resources. There are many resources available to support your learning, and asking for help is a sign of strength, not weakness.

By consistently practicing and applying the techniques you've learned, you'll become more proficient and confident in your ability to solve fraction problems. Fraction arithmetic is a foundational skill that will serve you well in future math courses and in many real-life applications.

Conclusion

In this comprehensive guide, we have meticulously dissected the expression 3/4 ÷ 3/6 × 1/2 - 1/6 + 1/2, demonstrating the step-by-step process of solving it using the order of operations and the principles of fraction arithmetic. By understanding PEMDAS/BODMAS, mastering fraction operations, and avoiding common mistakes, you can confidently tackle complex expressions and build a strong foundation in mathematics. Remember, practice is key to success. Work through the provided practice problems and continue to challenge yourself to expand your knowledge and skills. With dedication and perseverance, you can conquer the world of fractions and unlock new mathematical horizons.