Adding And Subtracting Fractions Step-by-Step Solutions

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Welcome to a comprehensive guide on mastering the art of adding and subtracting fractions! Fractions are a fundamental concept in mathematics, and the ability to confidently perform operations with them is crucial for success in various mathematical domains. In this article, we will delve into the process of finding the sums and differences of fractions, providing step-by-step solutions and explanations to enhance your understanding.

This guide aims to equip you with the knowledge and skills necessary to tackle fraction-related problems effectively. Whether you are a student looking to improve your grades or an adult seeking to refresh your math skills, this resource will serve as your trusted companion. We will break down each problem into manageable steps, ensuring that you grasp the underlying concepts and techniques.

So, let's embark on this mathematical journey together and unlock the secrets of fraction operations! By the end of this guide, you will be able to confidently add and subtract fractions, paving the way for more advanced mathematical pursuits.

1. Adding Fractions with Unlike Denominators: rac{1}{4} + rac{5}{7} = N

When adding fractions with unlike denominators, the first crucial step involves finding a common denominator. This common denominator serves as the foundation for combining the fractions effectively. In this case, we aim to find the least common multiple (LCM) of 4 and 7, which will be our common denominator.

To determine the LCM of 4 and 7, we can list out the multiples of each number until we identify a common one. The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, and so on. Similarly, the multiples of 7 are 7, 14, 21, 28, and so on. The smallest multiple that appears in both lists is 28, making it the LCM of 4 and 7.

Now that we have our common denominator, 28, we need to convert each fraction to an equivalent fraction with this denominator. To do this, we multiply both the numerator and the denominator of each fraction by the factor that makes the denominator equal to 28.

For the fraction rac{1}{4}, we multiply both the numerator and denominator by 7, since 4 multiplied by 7 equals 28. This gives us the equivalent fraction rac{7}{28}.

Similarly, for the fraction rac{5}{7}, we multiply both the numerator and denominator by 4, since 7 multiplied by 4 equals 28. This gives us the equivalent fraction rac{20}{28}.

Now that both fractions have the same denominator, we can proceed with adding them. To add fractions with a common denominator, we simply add the numerators and keep the denominator the same.

Adding the numerators, we have 7 + 20 = 27. The denominator remains 28. Therefore, the sum of rac{1}{4} and rac{5}{7} is rac{27}{28}.

In summary, to add fractions with unlike denominators, we first find a common denominator, convert each fraction to an equivalent fraction with the common denominator, and then add the numerators while keeping the denominator the same. This systematic approach ensures accurate results and a deeper understanding of fraction addition.

Solution:

  • Find the least common denominator (LCD) of 4 and 7, which is 28.
  • Convert the fractions: rac{1}{4} = rac{7}{28} and rac{5}{7} = rac{20}{28}
  • Add the fractions: rac{7}{28} + rac{20}{28} = rac{27}{28}
  • Therefore, rac{1}{4} + rac{5}{7} = rac{27}{28}

2. Adding Mixed Numbers and Fractions: 3 rac{1}{5} + rac{3}{4} = N

Adding mixed numbers and fractions requires a slightly different approach compared to adding simple fractions. The first crucial step involves converting the mixed number into an improper fraction. This conversion allows us to work with a single fraction representation, simplifying the addition process.

To convert a mixed number to an improper fraction, we multiply the whole number part by the denominator of the fractional part and then add the numerator. This result becomes the new numerator, while the denominator remains the same.

In the case of 3 rac{1}{5}, we multiply 3 by 5, which gives us 15. Then, we add the numerator 1, resulting in 16. Therefore, the improper fraction equivalent of 3 rac{1}{5} is rac{16}{5}.

Now that we have converted the mixed number to an improper fraction, we can proceed with adding it to the other fraction, rac{3}{4}. As with adding fractions with unlike denominators, we need to find a common denominator before we can combine the fractions.

The least common multiple (LCM) of 5 and 4 is 20. This will be our common denominator.

Next, we convert each fraction to an equivalent fraction with a denominator of 20. For the fraction rac{16}{5}, we multiply both the numerator and denominator by 4, since 5 multiplied by 4 equals 20. This gives us the equivalent fraction rac{64}{20}.

Similarly, for the fraction rac{3}{4}, we multiply both the numerator and denominator by 5, since 4 multiplied by 5 equals 20. This gives us the equivalent fraction rac{15}{20}.

Now that both fractions have the same denominator, we can add them by adding the numerators and keeping the denominator the same.

Adding the numerators, we have 64 + 15 = 79. The denominator remains 20. Therefore, the sum of rac{16}{5} and rac{3}{4} is rac{79}{20}.

Finally, we can convert the improper fraction rac{79}{20} back to a mixed number for a more conventional representation. To do this, we divide the numerator by the denominator. The quotient becomes the whole number part of the mixed number, and the remainder becomes the numerator of the fractional part, while the denominator remains the same.

Dividing 79 by 20, we get a quotient of 3 and a remainder of 19. Therefore, the mixed number equivalent of rac{79}{20} is 3 rac{19}{20}.

In summary, adding mixed numbers and fractions involves converting the mixed number to an improper fraction, finding a common denominator, converting each fraction to an equivalent fraction with the common denominator, adding the numerators, and then converting the resulting improper fraction back to a mixed number if desired. This comprehensive approach ensures accurate calculations and a thorough understanding of the process.

Solution:

  • Convert the mixed number to an improper fraction: 3 rac{1}{5} = rac{16}{5}
  • Find the LCD of 5 and 4, which is 20.
  • Convert the fractions: rac{16}{5} = rac{64}{20} and rac{3}{4} = rac{15}{20}
  • Add the fractions: rac{64}{20} + rac{15}{20} = rac{79}{20}
  • Convert the improper fraction to a mixed number: rac{79}{20} = 3 rac{19}{20}
  • Therefore, 3 rac{1}{5} + rac{3}{4} = 3 rac{19}{20}

3. Subtracting Fractions with Unlike Denominators: rac{9}{10} - rac{3}{7} = N

When subtracting fractions with unlike denominators, the process closely mirrors that of addition, with a key distinction in the final step. The initial focus remains on finding a common denominator, which serves as the foundation for performing the subtraction effectively.

In this case, we need to determine the least common multiple (LCM) of 10 and 7. The multiples of 10 are 10, 20, 30, 40, 50, 60, 70, and so on. The multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, and so on. The smallest multiple that appears in both lists is 70, making it the LCM of 10 and 7.

With the common denominator established as 70, we proceed to convert each fraction to an equivalent fraction with this denominator. For the fraction rac{9}{10}, we multiply both the numerator and denominator by 7, since 10 multiplied by 7 equals 70. This yields the equivalent fraction rac{63}{70}.

Similarly, for the fraction rac{3}{7}, we multiply both the numerator and denominator by 10, since 7 multiplied by 10 equals 70. This gives us the equivalent fraction rac{30}{70}.

Now that both fractions share the same denominator, we can perform the subtraction. To subtract fractions with a common denominator, we subtract the numerators and keep the denominator the same.

Subtracting the numerators, we have 63 - 30 = 33. The denominator remains 70. Therefore, the difference between rac{9}{10} and rac{3}{7} is rac{33}{70}.

In essence, subtracting fractions with unlike denominators involves finding a common denominator, converting each fraction to an equivalent fraction with the common denominator, and then subtracting the numerators while maintaining the denominator. This methodical approach ensures accurate results and a solid understanding of fraction subtraction.

Solution:

  • Find the least common denominator (LCD) of 10 and 7, which is 70.
  • Convert the fractions: rac{9}{10} = rac{63}{70} and rac{3}{7} = rac{30}{70}
  • Subtract the fractions: rac{63}{70} - rac{30}{70} = rac{33}{70}
  • Therefore, rac{9}{10} - rac{3}{7} = rac{33}{70}

4. Subtracting Fractions from Mixed Numbers: 9 rac{4}{5} - rac{3}{10} = N

Subtracting fractions from mixed numbers follows a similar pattern to adding them, with the initial step focusing on converting the mixed number into an improper fraction. This conversion streamlines the subtraction process by allowing us to work with a single fractional representation.

To convert the mixed number 9 rac{4}{5} to an improper fraction, we multiply the whole number part (9) by the denominator of the fractional part (5), which gives us 45. Then, we add the numerator (4) to get 49. The improper fraction equivalent of 9 rac{4}{5} is therefore rac{49}{5}.

Now that we have converted the mixed number to an improper fraction, we can proceed with subtracting the fraction rac{3}{10}. As with previous fraction operations, we need to find a common denominator before we can perform the subtraction.

The least common multiple (LCM) of 5 and 10 is 10. This will serve as our common denominator.

Next, we convert each fraction to an equivalent fraction with a denominator of 10. For the fraction rac{49}{5}, we multiply both the numerator and denominator by 2, since 5 multiplied by 2 equals 10. This gives us the equivalent fraction rac{98}{10}.

The fraction rac{3}{10} already has the desired denominator, so no conversion is necessary.

Now that both fractions have the same denominator, we can subtract them by subtracting the numerators and keeping the denominator the same.

Subtracting the numerators, we have 98 - 3 = 95. The denominator remains 10. Therefore, the difference between rac{49}{5} and rac{3}{10} is rac{95}{10}.

Finally, we can simplify the improper fraction rac{95}{10} by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 5. This gives us the simplified fraction rac{19}{2}.

Alternatively, we can convert the improper fraction rac{95}{10} back to a mixed number. Dividing 95 by 10, we get a quotient of 9 and a remainder of 5. Therefore, the mixed number equivalent of rac{95}{10} is 9 rac{5}{10}, which can be further simplified to 9 rac{1}{2}.

In summary, subtracting fractions from mixed numbers involves converting the mixed number to an improper fraction, finding a common denominator, converting each fraction to an equivalent fraction with the common denominator, subtracting the numerators, and then simplifying the resulting fraction or converting it back to a mixed number if desired. This systematic approach ensures accurate calculations and a comprehensive understanding of the process.

Solution:

  • Convert the mixed number to an improper fraction: 9 rac{4}{5} = rac{49}{5}
  • Find the LCD of 5 and 10, which is 10.
  • Convert the fractions: rac{49}{5} = rac{98}{10}
  • Subtract the fractions: rac{98}{10} - rac{3}{10} = rac{95}{10}
  • Simplify the fraction: rac{95}{10} = rac{19}{2}
  • Convert the improper fraction to a mixed number: rac{19}{2} = 9 rac{1}{2}
  • Therefore, 9 rac{4}{5} - rac{3}{10} = 9 rac{1}{2}

5. Subtracting Mixed Numbers A Comprehensive Guide

Subtracting mixed numbers requires a careful approach, often involving converting them to improper fractions for easier calculation. This process ensures accuracy, especially when borrowing is necessary.

Understanding Mixed Numbers

A mixed number combines a whole number and a fraction, like 8 rac{1}{3}. To subtract mixed numbers, we can convert them into improper fractions. This means rewriting the mixed number as a single fraction where the numerator is greater than the denominator.

Converting Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction and add the numerator. This result becomes the new numerator, and the denominator stays the same. For example, to convert 8 rac{1}{3} to an improper fraction:

  • Multiply the whole number (8) by the denominator (3): 8 * 3 = 24
  • Add the numerator (1): 24 + 1 = 25
  • The improper fraction is rac{25}{3}

The Subtraction Process

Once the mixed numbers are converted to improper fractions, the subtraction process becomes straightforward, provided the fractions share a common denominator. If they don't, finding the least common denominator (LCD) is essential.

Finding the Least Common Denominator (LCD)

The LCD is the smallest multiple that the denominators of both fractions share. This involves listing the multiples of each denominator until a common one is found. For instance, to subtract fractions with denominators 3 and 4, the LCD would be 12.

Creating Equivalent Fractions

After finding the LCD, each fraction needs to be converted into an equivalent fraction with the LCD as the new denominator. This is done by multiplying both the numerator and the denominator by the same number, ensuring the value of the fraction remains unchanged.

Subtracting the Fractions

With equivalent fractions established, the numerators can be subtracted directly, while the denominator remains constant. This yields the difference as an improper fraction.

Simplifying the Result

The final step often involves simplifying the improper fraction. This might mean reducing it to its simplest form by dividing both numerator and denominator by their greatest common divisor (GCD), or converting it back into a mixed number for clarity.

Converting Back to a Mixed Number

To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient becomes the whole number, the remainder becomes the new numerator, and the denominator stays the same. For example, if the result is rac{29}{3}, dividing 29 by 3 gives a quotient of 9 and a remainder of 2, making the mixed number 9 rac{2}{3}.

Special Cases and Borrowing

Sometimes, subtracting mixed numbers requires borrowing from the whole number part, especially when the fraction being subtracted is larger. In these cases, borrowing involves reducing the whole number by 1 and adding the denominator to the numerator of the fraction.

Conclusion

Subtracting mixed numbers, while seemingly complex, can be mastered by breaking down the process into manageable steps. Converting to improper fractions, finding the LCD, creating equivalent fractions, subtracting, and simplifying are the key components. With practice, these steps become second nature, enabling accurate and confident mixed number subtraction.

Solution:

This part is incomplete and requires the specific problem to provide a detailed solution.

In conclusion, mastering the art of adding and subtracting fractions is a fundamental step in building a strong mathematical foundation. By understanding the concepts of common denominators, equivalent fractions, and improper fractions, you can confidently tackle a wide range of fraction-related problems.

This comprehensive guide has provided you with step-by-step solutions and explanations for various scenarios, including adding fractions with unlike denominators, adding mixed numbers and fractions, subtracting fractions with unlike denominators, subtracting fractions from mixed numbers, and subtracting mixed numbers. By practicing these techniques and applying them to real-world situations, you will solidify your understanding and enhance your problem-solving skills.

Remember, mathematics is a journey of continuous learning and exploration. Embrace the challenges, seek out new problems to solve, and never stop expanding your mathematical horizons. With dedication and perseverance, you can unlock the power of fractions and excel in your mathematical pursuits.

So, go forth and conquer the world of fractions! Your newfound knowledge and skills will serve you well in various academic and professional endeavors. Keep practicing, keep learning, and keep growing as a mathematician.