Simplify 4/9 + [1/2 - {3/4 - (3/5 + 3/7 - 1/5)}] A Step-by-Step Guide

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In mathematics, simplifying fractions and expressions is a fundamental skill. This comprehensive guide will walk you through the process of simplifying the complex expression: 4/9 + [1/2 - {3/4 - (3/5 + 3/7 - 1/5)}]. We will break down each step, providing clear explanations and examples to ensure a thorough understanding. By mastering these techniques, you'll be well-equipped to tackle a wide range of mathematical problems. Simplifying fractions involves reducing them to their simplest form, while simplifying expressions means combining like terms and performing operations to arrive at a more concise result. This article provides a detailed, step-by-step approach, enhancing understanding and skill in these crucial mathematical areas. Let's dive into the world of mathematical simplification!

Understanding the Order of Operations

Before we begin, it's crucial to understand the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which we perform mathematical operations. First, we handle parentheses, then exponents, followed by multiplication and division (from left to right), and finally, addition and subtraction (from left to right). Correctly applying PEMDAS is crucial for accurate simplification of any mathematical expression. It ensures consistency and avoids ambiguity in the results. For instance, if we were to perform addition before multiplication, the outcome would be incorrect. Therefore, understanding and adhering to the order of operations is paramount in mathematical calculations and will help in simplifying complex equations. Mastery of PEMDAS is not just a mathematical skill; it is a fundamental principle that underpins all algebraic and arithmetic operations, making it an essential tool for problem-solving in mathematics.

Step 1: Simplify the Innermost Parentheses

Our first step is to simplify the expression within the innermost parentheses: (3/5 + 3/7 - 1/5). To add or subtract fractions, we need a common denominator. In this case, the least common multiple (LCM) of 5 and 7 is 35. Therefore, we convert each fraction to an equivalent fraction with a denominator of 35. This involves multiplying both the numerator and the denominator of each fraction by the appropriate factor. By finding the LCM and converting the fractions, we ensure that they have a common base, which allows for straightforward addition and subtraction. This process is a cornerstone of fraction manipulation and is essential for simplifying complex expressions. Without a common denominator, adding or subtracting fractions would be like trying to add apples and oranges – it simply wouldn't work. The common denominator acts as a unifying unit, allowing us to perform the arithmetic accurately.

Converting to a Common Denominator

  • 3/5 becomes (3 * 7) / (5 * 7) = 21/35
  • 3/7 becomes (3 * 5) / (7 * 5) = 15/35
  • 1/5 becomes (1 * 7) / (5 * 7) = 7/35

Performing the Operations

Now we can rewrite the expression within the parentheses as: 21/35 + 15/35 - 7/35. Adding and subtracting the numerators, we get (21 + 15 - 7) / 35 = 29/35. This step demonstrates the practical application of finding a common denominator. By converting each fraction to an equivalent form with the same denominator, we create a scenario where we can simply add or subtract the numerators. The result, 29/35, is the simplified form of the expression within the innermost parentheses. This process showcases the fundamental principle of fraction arithmetic, which is the ability to combine fractions with the same denominator. It's a critical skill in simplifying more complex expressions and solving equations involving fractions.

Step 2: Simplify the Curly Braces

Next, we move to the curly braces: {3/4 - (29/35)}. To subtract these fractions, we again need a common denominator. The LCM of 4 and 35 is 140. So, we convert each fraction to an equivalent fraction with a denominator of 140. This step mirrors the process in Step 1, reinforcing the importance of finding a common denominator when adding or subtracting fractions. The LCM of 4 and 35 provides the smallest common multiple, ensuring that the resulting fractions are in their simplest form. This efficient method prevents the creation of unnecessarily large numbers, which could complicate the subsequent steps. The conversion to a common denominator is a crucial skill for simplifying fractions and ensuring accuracy in mathematical calculations.

Converting to a Common Denominator

  • 3/4 becomes (3 * 35) / (4 * 35) = 105/140
  • 29/35 becomes (29 * 4) / (35 * 4) = 116/140

Performing the Subtraction

Now we have: 105/140 - 116/140. Subtracting the numerators, we get (105 - 116) / 140 = -11/140. This subtraction results in a negative fraction, which is perfectly valid. It's important to remember that when subtracting fractions, the sign of the result depends on the relative sizes of the numerators. The negative sign here indicates that the fraction being subtracted (116/140) is larger than the fraction being subtracted from (105/140). This result, -11/140, is the simplified form of the expression within the curly braces. It highlights the importance of paying attention to signs when performing arithmetic operations. Furthermore, it underscores the flexibility of fractions, which can be positive, negative, or even zero.

Step 3: Simplify the Square Brackets

Now we tackle the square brackets: [1/2 - (-11/140)]. Subtracting a negative number is the same as adding its positive counterpart. So, we have 1/2 + 11/140. To add these fractions, we need a common denominator. The LCM of 2 and 140 is 140. This step continues the theme of finding a common denominator to add or subtract fractions. The LCM of 2 and 140 is straightforward, as 140 is a multiple of 2. This simplifies the conversion process, making it easier to add the fractions. Understanding LCMs is crucial for efficient fraction arithmetic, as it allows us to find the smallest common multiple, which leads to simpler calculations and results.

Converting to a Common Denominator

  • 1/2 becomes (1 * 70) / (2 * 70) = 70/140
  • 11/140 remains as 11/140

Performing the Addition

We now have: 70/140 + 11/140. Adding the numerators, we get (70 + 11) / 140 = 81/140. This addition combines the two fractions into a single fraction, 81/140. This result is the simplified form of the expression within the square brackets. It demonstrates the cumulative effect of the previous simplification steps, building upon each other to gradually reduce the complexity of the expression. The fraction 81/140 is in its simplest form, as 81 and 140 share no common factors other than 1. This outcome underscores the importance of simplification in mathematics, as it allows us to express results in their most concise and manageable form.

Step 4: Final Addition

Finally, we add the remaining fraction: 4/9 + 81/140. To add these fractions, we need a common denominator. The LCM of 9 and 140 is 1260. This final step brings us to the culmination of the simplification process. Finding the LCM of 9 and 140 is crucial for adding these fractions. The LCM of 1260 ensures that the resulting fractions have a common base, allowing for accurate addition. This methodical approach to fraction arithmetic is essential for solving complex mathematical problems. Without a common denominator, we cannot directly add the fractions. Therefore, finding the LCM is a necessary step for obtaining the correct result.

Converting to a Common Denominator

  • 4/9 becomes (4 * 140) / (9 * 140) = 560/1260
  • 81/140 becomes (81 * 9) / (140 * 9) = 729/1260

Performing the Addition

We now have: 560/1260 + 729/1260. Adding the numerators, we get (560 + 729) / 1260 = 1289/1260. This final addition combines the two fractions into a single fraction, 1289/1260. This result is the simplified form of the entire expression. The fraction 1289/1260 is an improper fraction, where the numerator is larger than the denominator. It can be left in this form or converted to a mixed number, depending on the context. The result demonstrates the power of step-by-step simplification, breaking down a complex expression into manageable parts and arriving at a final answer. This process underscores the importance of accuracy and attention to detail in mathematical calculations.

Final Answer

Therefore, the simplified form of the expression 4/9 + [1/2 - {3/4 - (3/5 + 3/7 - 1/5)}] is 1289/1260. This comprehensive step-by-step guide has demonstrated the process of simplifying a complex expression involving fractions. We have meticulously broken down each step, explaining the rationale behind each operation. From finding common denominators to adhering to the order of operations, we have covered the essential techniques for simplifying fractions and expressions. This knowledge empowers you to tackle similar problems with confidence and accuracy. Mastery of these skills is crucial for success in mathematics, as it provides a solid foundation for more advanced concepts. The final answer, 1289/1260, represents the culmination of this simplification process, showcasing the power of systematic problem-solving.

By following these steps, you can confidently simplify similar mathematical expressions. Remember, practice makes perfect! Continue to challenge yourself with new problems to further develop your skills. The key to success in mathematics is consistent effort and a willingness to learn from mistakes. As you practice, you will become more proficient in simplifying fractions and expressions, which will serve you well in your mathematical journey. So, keep practicing and keep learning!