Perform The Following Arithmetic Operations: 1. \( \frac{6}{9} - \frac{2}{9} + \frac{5}{9} \) 2. \( 12\frac{1}{2} + 14\frac{4}{10} \) 3. \( 5\frac{4}{6} + 3\frac{2}{3} + 6\frac{7}{12} - 7\frac{1}{2} \) 4. \( \frac{34}{6} + \frac{11}{3} + \frac{79}{12} \)

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In mathematics, mastering arithmetic operations with fractions and mixed numbers is crucial for various calculations and problem-solving scenarios. This article aims to provide a comprehensive guide on how to perform these operations efficiently and accurately. We will cover addition, subtraction, and a combination of these operations, ensuring a clear understanding through step-by-step explanations and examples. Whether you are a student looking to improve your math skills or someone needing a refresher, this guide will equip you with the necessary knowledge and techniques.

1. Subtracting and Adding Fractions with Common Denominators

When dealing with fractions that have the same denominator, the process of addition and subtraction becomes straightforward. Fractions with common denominators simplify the initial steps, allowing us to focus directly on the numerators. The key principle here is that you can only add or subtract fractions if they share a common denominator. If they do, you simply perform the operation (addition or subtraction) on the numerators and keep the denominator the same. This method ensures that we are combining equal parts of a whole, which is fundamental to understanding fraction arithmetic. This section will delve into the mechanics of this process, providing a clear, step-by-step approach to solving such problems. Let's explore how to tackle such arithmetic operations effectively.

Step-by-Step Guide

To perform a series of addition and subtraction operations on fractions with common denominators, follow these steps:

  1. Identify the Common Denominator: Ensure all fractions have the same denominator. This is crucial because it allows you to combine the fractions directly.
  2. Perform Operations on Numerators: Add or subtract the numerators as indicated in the problem. The denominator remains constant throughout the operation. For example, if you have the expression rac{a}{d} + rac{b}{d} - rac{c}{d}, you would compute a + b - c, and the result would be rac{a + b - c}{d}.
  3. Simplify the Resulting Fraction: After performing the operations on the numerators, check if the resulting fraction can be simplified. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). This ensures the fraction is in its simplest form.

Example 1

Let’s apply these steps to the first arithmetic operation: 69−29+59{ \frac{6}{9} - \frac{2}{9} + \frac{5}{9} }

  1. Identify the Common Denominator: All fractions have the same denominator, which is 9.
  2. Perform Operations on Numerators: Subtract and add the numerators: 6 - 2 + 5 = 9. Thus, the fraction becomes rac{9}{9}.
  3. Simplify the Resulting Fraction: Simplify rac{9}{9} by dividing both the numerator and the denominator by their GCD, which is 9. The simplified fraction is rac{9 ÷ 9}{9 ÷ 9} = rac{1}{1} = 1.

Therefore, the result of the operation rac{6}{9} - rac{2}{9} + rac{5}{9} is 1. This methodical approach ensures accuracy and clarity in dealing with fractions.

2. Adding Mixed Numbers

Adding mixed numbers involves combining whole numbers and fractions. A mixed number consists of a whole number and a proper fraction (where the numerator is less than the denominator). The process of adding mixed numbers can be approached in a few different ways, but the most common method involves converting the mixed numbers into improper fractions first. This method allows for easier manipulation of the numbers, especially when the fractional parts have different denominators. Understanding how to add mixed numbers is essential for real-world applications, such as measuring ingredients in a recipe or calculating distances.

Converting Mixed Numbers to Improper Fractions

Before adding mixed numbers, it's often easiest to convert them into improper fractions. An improper fraction is one where the numerator is greater than or equal to the denominator. The conversion process involves multiplying the whole number part by the denominator of the fractional part, adding the numerator, and then placing the result over the original denominator. This transformation allows us to work with the numbers more efficiently, particularly when dealing with addition and subtraction.

For example, to convert the mixed number 12 rac{1}{2} to an improper fraction:

  1. Multiply the whole number (12) by the denominator (2): 12 × 2 = 24.
  2. Add the numerator (1): 24 + 1 = 25.
  3. Place the result over the original denominator (2): rac{25}{2}.

So, 12 rac{1}{2} is equivalent to rac{25}{2} as an improper fraction. This conversion is a crucial step in simplifying the addition of mixed numbers.

Step-by-Step Guide

To add mixed numbers, follow these steps:

  1. Convert Mixed Numbers to Improper Fractions: If you have mixed numbers, convert them into improper fractions using the method described above. This step is essential for simplifying the addition process, especially when dealing with different denominators.
  2. Find a Common Denominator: If the fractions have different denominators, find the least common multiple (LCM) of the denominators. This LCM will be the new common denominator. Finding the common denominator is crucial for adding fractions accurately.
  3. Adjust the Fractions: Convert each fraction to an equivalent fraction with the common denominator. This involves multiplying both the numerator and the denominator of each fraction by the same factor, ensuring the value of the fraction remains unchanged.
  4. Add the Numerators: Once the fractions have the same denominator, add the numerators. The denominator remains the same. This step combines the fractional parts of the numbers.
  5. Simplify the Result: If the resulting fraction is improper, convert it back to a mixed number. Simplify the fraction if possible by dividing both the numerator and the denominator by their greatest common divisor (GCD). This ensures the answer is in its simplest form.

Example 2

Let’s apply these steps to the second arithmetic operation:

1212+14410{ 12\frac{1}{2} + 14\frac{4}{10} }

  1. Convert Mixed Numbers to Improper Fractions:
    • 12\frac{1}{2} = \frac{(12 × 2) + 1}{2} = \frac{25}{2}
    • 14\frac{4}{10} = \frac{(14 × 10) + 4}{10} = \frac{144}{10}
  2. Find a Common Denominator: The denominators are 2 and 10. The least common multiple (LCM) of 2 and 10 is 10.
  3. Adjust the Fractions: Convert each fraction to an equivalent fraction with the common denominator of 10.
    • rac{25}{2} = \frac{25 × 5}{2 × 5} = \frac{125}{10}
    • rac{144}{10} remains the same.
  4. Add the Numerators: Add the numerators: 125 + 144 = 269. The fraction is now rac{269}{10}.
  5. Simplify the Result: Convert the improper fraction back to a mixed number:
    • rac{269}{10} = 26\frac{9}{10}

Thus, 12\frac{1}{2} + 14\frac{4}{10} = 26\frac{9}{10}. This systematic approach ensures accuracy in adding mixed numbers.

3. Combining Addition, Subtraction, and Mixed Numbers

When tackling arithmetic problems involving a mix of addition, subtraction, and mixed numbers, a methodical approach is essential. Combining different operations requires careful attention to detail and a clear understanding of the order of operations and fraction manipulation. This section will guide you through the process of handling complex arithmetic expressions involving fractions and mixed numbers, ensuring you can solve them accurately and efficiently.

Step-by-Step Guide

To solve arithmetic problems that involve a combination of addition, subtraction, and mixed numbers, follow these steps:

  1. Convert Mixed Numbers to Improper Fractions: If the problem includes mixed numbers, convert them to improper fractions. This step simplifies the overall calculation process and makes it easier to perform addition and subtraction.
  2. Find a Common Denominator: Identify all the denominators in the fractions and find their least common multiple (LCM). The LCM will be the common denominator for all fractions. This is a crucial step as it allows you to combine the fractions easily.
  3. Adjust the Fractions: Convert each fraction to an equivalent fraction with the common denominator. Multiply both the numerator and the denominator of each fraction by the same factor to achieve the common denominator without changing the fraction's value.
  4. Perform Addition and Subtraction: Perform the addition and subtraction operations in the order they appear from left to right. Combine the numerators while keeping the common denominator. This step involves careful calculation to ensure accuracy.
  5. Simplify the Result: Simplify the resulting fraction. If the fraction is improper, convert it back to a mixed number. Reduce the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD). This ensures the final answer is presented in its most concise form.

Example 3

Let's apply these steps to the third arithmetic operation:

546+323+6712−712{ 5\frac{4}{6} + 3\frac{2}{3} + 6\frac{7}{12} - 7\frac{1}{2} }

  1. Convert Mixed Numbers to Improper Fractions:
    • 5\frac{4}{6} = \frac{(5 × 6) + 4}{6} = \frac{34}{6}
    • 3\frac{2}{3} = \frac{(3 × 3) + 2}{3} = \frac{11}{3}
    • 6\frac{7}{12} = \frac{(6 × 12) + 7}{12} = \frac{79}{12}
    • 7\frac{1}{2} = \frac{(7 × 2) + 1}{2} = \frac{15}{2}
  2. Find a Common Denominator: The denominators are 6, 3, 12, and 2. The least common multiple (LCM) of these numbers is 12.
  3. Adjust the Fractions: Convert each fraction to an equivalent fraction with the common denominator of 12.
    • rac{34}{6} = \frac{34 × 2}{6 × 2} = \frac{68}{12}
    • rac{11}{3} = \frac{11 × 4}{3 × 4} = \frac{44}{12}
    • rac{79}{12} remains the same.
    • rac{15}{2} = \frac{15 × 6}{2 × 6} = \frac{90}{12}
  4. Perform Addition and Subtraction: Add and subtract the numerators: 6812+4412+7912−9012=68+44+79−9012=10112{ \frac{68}{12} + \frac{44}{12} + \frac{79}{12} - \frac{90}{12} = \frac{68 + 44 + 79 - 90}{12} = \frac{101}{12} }
  5. Simplify the Result: Convert the improper fraction back to a mixed number: 10112=8512{ \frac{101}{12} = 8\frac{5}{12} }

Thus, 5\frac{4}{6} + 3\frac{2}{3} + 6\frac{7}{12} - 7\frac{1}{2} = 8\frac{5}{12}. This detailed process ensures accuracy when combining various arithmetic operations.

Example 4

Let’s apply these steps to the fourth arithmetic operation:

346+113+7912{ \frac{34}{6} + \frac{11}{3} + \frac{79}{12} }

  1. No Mixed Numbers: There are no mixed numbers in this problem, so we can skip the first step.
  2. Find a Common Denominator: The denominators are 6, 3, and 12. The least common multiple (LCM) of these numbers is 12.
  3. Adjust the Fractions: Convert each fraction to an equivalent fraction with the common denominator of 12.
    • rac{34}{6} = \frac{34 × 2}{6 × 2} = \frac{68}{12}
    • rac{11}{3} = \frac{11 × 4}{3 × 4} = \frac{44}{12}
    • rac{79}{12} remains the same.
  4. Perform Addition: Add the numerators: 6812+4412+7912=68+44+7912=19112{ \frac{68}{12} + \frac{44}{12} + \frac{79}{12} = \frac{68 + 44 + 79}{12} = \frac{191}{12} }
  5. Simplify the Result: Convert the improper fraction back to a mixed number: 19112=151112{ \frac{191}{12} = 15\frac{11}{12} }

Thus, 346+113+7912=151112{\frac{34}{6} + \frac{11}{3} + \frac{79}{12} = 15\frac{11}{12}}. This systematic approach ensures accuracy when performing combined arithmetic operations.

Conclusion

In conclusion, performing arithmetic operations with fractions and mixed numbers requires a systematic approach and a clear understanding of the underlying principles. By following the step-by-step guides outlined in this article, you can confidently tackle various arithmetic problems involving fractions and mixed numbers. Mastering these skills is essential for success in mathematics and has practical applications in everyday life. From simplifying fractions to adding mixed numbers and combining multiple operations, each technique discussed will enhance your mathematical proficiency. Remember to convert mixed numbers to improper fractions, find common denominators, and simplify your results to ensure accuracy. With consistent practice, you'll become adept at handling these operations, making mathematical calculations more manageable and precise.