Step-by-Step Guide To Evaluating Mathematical Expressions

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In the realm of mathematics, evaluating expressions is a fundamental skill. This article will guide you through a step-by-step process of evaluating complex mathematical expressions, ensuring clarity and accuracy in your calculations. Understanding the order of operations and applying them diligently is crucial for success in mathematics and related fields. We'll dissect three distinct problems, each requiring a unique approach and a firm grasp of mathematical principles. From mixed fractions to division and multiplication, we'll cover a wide array of concepts. By the end of this guide, you'll be well-equipped to tackle even the most daunting mathematical expressions with confidence. Remember, the key to mastering mathematics lies in consistent practice and a thorough understanding of the underlying principles. So, let's embark on this journey of mathematical exploration together, unraveling the intricacies of each expression and solidifying your understanding along the way. Whether you're a student, a professional, or simply someone with a passion for numbers, this guide will undoubtedly enhance your mathematical prowess and equip you with the tools necessary to excel in this fascinating domain.

1. Evaluating Expressions with Mixed Fractions and Order of Operations

The first expression we'll tackle involves mixed fractions, multiplication, and division. Mastering the order of operations (PEMDAS/BODMAS) is paramount here. First, we'll convert the mixed fractions into improper fractions. This simplifies the calculations and allows us to perform the arithmetic operations more easily. Then, we'll adhere to the order of operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This ensures that we perform the operations in the correct sequence, leading to an accurate result. Division can be tricky, but remember that dividing by a fraction is the same as multiplying by its reciprocal. This simple trick can save you a lot of headaches. Let's break down the expression step by step, highlighting each operation and the reasoning behind it. By understanding the logic behind each step, you'll be able to apply these principles to a wide range of mathematical problems. Remember, practice makes perfect, so don't hesitate to work through similar examples to solidify your understanding. With patience and persistence, you'll become a master of mathematical expressions in no time. So, let's dive into the first expression and unravel its secrets together!

Problem 1: 3 3/7 - 2 2/3 × 1 1/5 ÷ 24/25

To evaluate this expression, we follow the order of operations (PEMDAS/BODMAS).

  1. Convert mixed fractions to improper fractions:

    • 3 3/7 = (3 × 7 + 3) / 7 = 24/7
    • 2 2/3 = (2 × 3 + 2) / 3 = 8/3
    • 1 1/5 = (1 × 5 + 1) / 5 = 6/5

  2. Rewrite the expression: 24/7 - 8/3 × 6/5 ÷ 24/25

  3. Perform multiplication and division from left to right:

    • 8/3 × 6/5 = 48/15 = 16/5
    • 16/5 ÷ 24/25 = 16/5 × 25/24 = (16 × 25) / (5 × 24) = 400/120 = 10/3

  4. Perform subtraction: 24/7 - 10/3 = (24 × 3 - 10 × 7) / (7 × 3) = (72 - 70) / 21 = 2/21

Therefore, the value of the expression 3 3/7 - 2 2/3 × 1 1/5 ÷ 24/25 is 2/21.

2. Evaluating Expressions with Parentheses, Multiplication, and Division

The second problem brings parentheses into the mix, emphasizing their priority in the order of operations. Anything within parentheses must be calculated first, regardless of the operation inside. This could involve addition, subtraction, multiplication, or division. Once the expression inside the parentheses is simplified, we proceed with the remaining operations, again adhering to the PEMDAS/BODMAS rule. This problem also reinforces the concept of reciprocals in division, a crucial aspect of fraction manipulation. We'll break down the problem into manageable steps, focusing on clarity and precision. Each step will be explained in detail, ensuring that you understand not only the how but also the why behind each operation. This deeper understanding will empower you to tackle similar problems with confidence and ease. So, let's put on our mathematical hats and delve into this problem, unraveling its complexities one step at a time. Remember, the journey of a thousand miles begins with a single step, and the journey to mathematical mastery begins with understanding the fundamentals.

Problem 2: (3 3/7 - 2 2/3) × (1 1/5 ÷ 24/25)

  1. Convert mixed fractions to improper fractions:

    • 3 3/7 = 24/7
    • 2 2/3 = 8/3
    • 1 1/5 = 6/5

  2. Rewrite the expression: (24/7 - 8/3) × (6/5 ÷ 24/25)

  3. Evaluate the expression within the first parentheses:

    • Find a common denominator for 7 and 3, which is 21.
    • 24/7 - 8/3 = (24 × 3) / (7 × 3) - (8 × 7) / (3 × 7) = 72/21 - 56/21 = 16/21

  4. Evaluate the expression within the second parentheses:

    • Dividing by a fraction is the same as multiplying by its reciprocal.
    • 6/5 ÷ 24/25 = 6/5 × 25/24 = (6 × 25) / (5 × 24) = 150/120 = 5/4

  5. Multiply the results: 16/21 × 5/4 = (16 × 5) / (21 × 4) = 80/84

  6. Simplify the fraction: 80/84 = 20/21

Therefore, the value of the expression (3 3/7 - 2 2/3) × (1 1/5 ÷ 24/25) is 20/21.

3. Evaluating and Verifying Expressions with Multiple Operations

The third problem takes us a step further by incorporating a variety of operations: subtraction, division, addition, and mixed fractions. This requires a meticulous approach, ensuring that we follow the order of operations precisely. We'll also introduce the concept of verification, a crucial step in ensuring the accuracy of our calculations. Verification involves substituting the calculated value back into the original expression and checking if the equation holds true. This provides a valuable safeguard against errors and reinforces our understanding of the mathematical principles involved. We'll break down the problem into smaller, more manageable steps, explaining each operation in detail and highlighting the importance of careful calculation. By mastering this type of problem, you'll gain a deeper appreciation for the interconnectedness of mathematical concepts and develop a robust problem-solving skillset. So, let's embark on this final challenge, armed with our knowledge of order of operations and our commitment to accuracy. Together, we'll conquer this expression and emerge with a renewed sense of confidence in our mathematical abilities.

Problem 3: Evaluate and verify the expression: 1/2 - 1 1/3 ÷ 5/8 + 4 3/14 = ?

Stepwise solution:

  1. Original expression: 1/2 - 1 1/3 ÷ 5/8 + 4 3/14

  2. Convert mixed fractions to improper fractions:

    • 1 1/3 = (1 × 3 + 1) / 3 = 4/3
    • 4 3/14 = (4 × 14 + 3) / 14 = 59/14

  3. Rewrite the expression: 1/2 - 4/3 ÷ 5/8 + 59/14

  4. Perform division:

    • 4/3 ÷ 5/8 = 4/3 × 8/5 = 32/15

  5. Rewrite the expression: 1/2 - 32/15 + 59/14

  6. Find the least common denominator (LCD) for 2, 15, and 14:

    • The LCD is 210.

  7. Convert fractions to equivalent fractions with the LCD:

    • 1/2 = 105/210
    • 32/15 = 448/210
    • 59/14 = 885/210

  8. Rewrite the expression: 105/210 - 448/210 + 885/210

  9. Perform subtraction and addition from left to right:

    • 105/210 - 448/210 = -343/210
    • -343/210 + 885/210 = 542/210

  10. Simplify the fraction: 542/210 = 271/105

Therefore, the value of the expression 1/2 - 1 1/3 ÷ 5/8 + 4 3/14 is 271/105.

Verification:

To verify, we would substitute 271/105 back into the original equation. This process can be a bit cumbersome by hand, and it's often more practical to use a calculator or computer to verify such complex expressions. However, the key is to ensure that the left-hand side of the equation equals the right-hand side after the substitution.

In conclusion, evaluating mathematical expressions requires a systematic approach and a thorough understanding of the order of operations. By breaking down complex problems into smaller, more manageable steps, we can minimize errors and arrive at accurate solutions. Remember to practice regularly and don't hesitate to seek help when needed. With dedication and perseverance, you can master the art of mathematical expression evaluation and unlock a world of possibilities.

This article provides a comprehensive guide to evaluating mathematical expressions, covering a range of problem types and emphasizing the importance of accuracy and attention to detail. By mastering these skills, you'll be well-equipped to tackle a wide variety of mathematical challenges.