Simplifying Complex Mathematical Expressions A Step-by-Step Guide

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Introduction

In this comprehensive guide, we will embark on a detailed exploration of the mathematical expressions provided. Our objective is to dissect each component, apply the correct order of operations, and simplify the expressions to their most fundamental forms. By delving into the intricacies of these equations, we aim to enhance your understanding of mathematical principles and problem-solving techniques. This exploration will not only provide step-by-step solutions but also illuminate the underlying concepts that govern these calculations. Whether you are a student seeking to improve your mathematical skills or simply a curious mind eager to unravel mathematical puzzles, this guide is designed to equip you with the knowledge and confidence to tackle similar challenges.

Analyzing the First Expression

The first expression we will tackle is: (2 + 3/4) (1 - 6/11) ⋅ (2 - 3/4) (1 + 1/5) ⋅ 1/7 ⋅ 1/5. This expression involves a series of multiplications and additions/subtractions within parentheses, followed by additional multiplications. To solve this, we will adhere to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). The key is to break down the expression into manageable parts, addressing each set of parentheses individually before moving on to the multiplications. By systematically simplifying each component, we can avoid errors and arrive at the correct solution. Our approach will be methodical, ensuring that each step is clearly explained and justified, providing a robust understanding of the solution process.

Step-by-Step Solution

  1. Simplify within the parentheses:

    • (2 + 3/4): To add these terms, we need a common denominator. Converting 2 to a fraction with a denominator of 4, we get 8/4. Thus, 8/4 + 3/4 = 11/4.
    • (1 - 6/11): Similarly, we convert 1 to 11/11. Hence, 11/11 - 6/11 = 5/11.
    • (2 - 3/4): Again, converting 2 to 8/4, we have 8/4 - 3/4 = 5/4.
    • (1 + 1/5): Converting 1 to 5/5, we get 5/5 + 1/5 = 6/5.

  2. Rewrite the expression with simplified parentheses:

    • The expression now becomes: (11/4) (5/11) ⋅ (5/4) (6/5) ⋅ 1/7 ⋅ 1/5.

  3. Perform the multiplications:

    • Multiply the fractions from left to right. To multiply fractions, we multiply the numerators (top numbers) together and the denominators (bottom numbers) together. We can also simplify by canceling out common factors between numerators and denominators before multiplying.
    • (11/4) (5/11) = (11 * 5) / (4 * 11) = 55/44. We can simplify this by dividing both numerator and denominator by 11, resulting in 5/4.
    • (5/4) (5/4) = (5 * 5) / (4 * 4) = 25/16.
    • (25/16) (6/5) = (25 * 6) / (16 * 5) = 150/80. Simplifying by dividing both by 10, we get 15/8. Further simplification by recognizing common factors is not possible at this stage.
    • (15/8) (1/7) = (15 * 1) / (8 * 7) = 15/56.
    • (15/56) (1/5) = (15 * 1) / (56 * 5) = 15/280. Simplifying by dividing both by 5, we get 3/56.

  4. Final Result:

    • The simplified form of the expression is 3/56.

By meticulously following each step, we have successfully simplified the initial expression. This process highlights the importance of adhering to the order of operations and employing simplification techniques to arrive at the final answer. The step-by-step breakdown not only provides the solution but also reinforces the underlying mathematical principles.

Deconstructing the Second Expression

The second expression we will dissect is: (1 + 1/2 - 3/4) (1 - (1/2 ÷ 3/4)). This expression combines addition, subtraction, division, and parentheses, necessitating a careful application of the order of operations. The presence of division within the second set of parentheses adds an extra layer of complexity, requiring us to address this operation before proceeding with the subtraction. Our strategy will involve simplifying each set of parentheses independently, then performing the final multiplication. This methodical approach ensures accuracy and clarity in the solution process. We will pay close attention to the rules governing fraction arithmetic, such as finding common denominators for addition and subtraction, and inverting and multiplying for division. By meticulously executing each step, we will arrive at the simplified form of the expression.

Step-by-Step Solution

  1. Simplify the first set of parentheses:

    • (1 + 1/2 - 3/4): To add and subtract these fractions, we need a common denominator. The least common denominator for 2 and 4 is 4. Convert 1 and 1/2 to fractions with a denominator of 4.
    • 1 = 4/4
    • 1/2 = 2/4
    • Now the expression within the first parentheses becomes: 4/4 + 2/4 - 3/4
    • Adding and subtracting the numerators: (4 + 2 - 3) / 4 = 3/4

  2. Simplify the second set of parentheses, starting with the division:

    • (1/2 ÷ 3/4): To divide fractions, we multiply by the reciprocal of the divisor. The reciprocal of 3/4 is 4/3.
    • (1/2) * (4/3) = (1 * 4) / (2 * 3) = 4/6. This fraction can be simplified by dividing both numerator and denominator by 2, resulting in 2/3.
    • Now the expression within the second parentheses becomes: (1 - 2/3)
    • To subtract, convert 1 to a fraction with a denominator of 3: 1 = 3/3
    • 3/3 - 2/3 = 1/3

  3. Rewrite the expression with simplified parentheses:

    • The expression now becomes: (3/4) (1/3)

  4. Perform the multiplication:

    • (3/4) * (1/3) = (3 * 1) / (4 * 3) = 3/12

  5. Simplify the result:

    • 3/12 can be simplified by dividing both numerator and denominator by 3.
    • 3/12 = 1/4

  6. Final Result:

    • The simplified form of the expression is 1/4.

By systematically simplifying each component, we have successfully navigated the complexities of the second expression. The step-by-step solution underscores the importance of adhering to the order of operations and employing fraction arithmetic techniques to arrive at the correct answer. This detailed breakdown provides a clear understanding of the problem-solving process, enhancing your ability to tackle similar mathematical challenges.

Conclusion

In conclusion, we have meticulously analyzed and simplified two complex mathematical expressions. By adhering to the order of operations and employing fundamental arithmetic principles, we have successfully broken down each expression into manageable steps, leading to accurate solutions. The first expression, (2 + 3/4) (1 - 6/11) ⋅ (2 - 3/4) (1 + 1/5) ⋅ 1/7 ⋅ 1/5, simplified to 3/56, while the second expression, (1 + 1/2 - 3/4) (1 - (1/2 ÷ 3/4)), simplified to 1/4. These step-by-step solutions not only provide the answers but also serve as a comprehensive guide to problem-solving strategies in mathematics. The process of dissecting each expression, simplifying its components, and systematically performing operations reinforces the importance of precision and methodical thinking. By mastering these techniques, you can confidently approach a wide range of mathematical challenges and enhance your overall understanding of mathematical concepts. This guide serves as a valuable resource for students, educators, and anyone seeking to improve their mathematical skills and problem-solving abilities.