Fraction Division Examples And Step-by-Step Solutions

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Delving into the world of fraction division can initially seem daunting, but with a clear understanding of the underlying principles, it becomes a manageable and even enjoyable mathematical exercise. This section focuses on solving the problem 34รท25{\frac{3}{4} \div \frac{2}{5} }, providing a step-by-step explanation to ensure clarity and comprehension. Fraction division, at its core, involves multiplying the first fraction by the reciprocal of the second fraction. This seemingly simple rule is the key to unlocking the solution. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. In this case, the reciprocal of 25{\frac{2}{5}} is 52{\frac{5}{2}}. Therefore, the division problem transforms into a multiplication problem: 34ร—52{\frac{3}{4} \times \frac{5}{2}}. Now, the process becomes straightforward: multiply the numerators (3 and 5) to get the new numerator, and multiply the denominators (4 and 2) to get the new denominator. This yields 3ร—54ร—2=158{\frac{3 \times 5}{4 \times 2} = \frac{15}{8}}. The resulting fraction, 158{\frac{15}{8}}, is an improper fraction, meaning the numerator is greater than the denominator. While it is a correct answer, it's often preferable to express it as a mixed number for better understanding of its magnitude. To convert an improper fraction to a mixed number, divide the numerator by the denominator. 15 divided by 8 gives a quotient of 1 and a remainder of 7. This means 158{\frac{15}{8}} is equal to 1 whole and 78{\frac{7}{8}} of another whole. Thus, the mixed number representation is 1 78{\frac{7}{8}}. Understanding this process thoroughly equips you to tackle similar problems with confidence. Mastering the concept of reciprocals and the conversion between improper fractions and mixed numbers are crucial steps in building a strong foundation in fraction division. This detailed explanation aims to make the process clear and accessible, ensuring you can confidently divide fractions. Practice is key to solidifying this understanding, so working through various examples will further enhance your skills.

Moving on to the next problem, 38รท212{\frac{3}{8} \div \frac{2}{12} }, we continue to reinforce the principles of fraction division. The cornerstone of fraction division, as previously discussed, is the transformation of the division problem into a multiplication problem by using the reciprocal of the second fraction. Here, we will meticulously walk through the steps, ensuring a comprehensive understanding of the process. Identifying the reciprocal is the first crucial step. The reciprocal of 212{\frac{2}{12}} is obtained by swapping the numerator and the denominator, resulting in 122{\frac{12}{2}}. Now, the original division problem, 38รท212{\frac{3}{8} \div \frac{2}{12} }, is transformed into a multiplication problem: 38ร—122{\frac{3}{8} \times \frac{12}{2}}. Multiplying fractions involves multiplying the numerators together and the denominators together. Thus, we have 3ร—128ร—2=3616{\frac{3 \times 12}{8 \times 2} = \frac{36}{16}}. The resulting fraction, 3616{\frac{36}{16}}, is an improper fraction, indicating that the numerator is larger than the denominator. To simplify this fraction, we can first look for common factors between the numerator and the denominator. Both 36 and 16 are divisible by 4. Dividing both by 4 gives us 36รท416รท4=94{\frac{36 \div 4}{16 \div 4} = \frac{9}{4}}. This simplified improper fraction, 94{\frac{9}{4}}, can further be converted into a mixed number. Dividing 9 by 4 gives a quotient of 2 and a remainder of 1. Therefore, 94{\frac{9}{4}} is equivalent to 2 14{\frac{1}{4}}. This process of simplifying fractions, both before and after multiplication, is essential for arriving at the most concise and easily understandable answer. Simplifying fractions not only makes the numbers smaller and easier to work with, but also provides a clearer representation of the quantity. Understanding the relationship between improper fractions and mixed numbers is also crucial for practical applications of fractions. This problem highlights the importance of these skills in the context of fraction division.

Let's tackle the third problem, 410รท12{\frac{4}{10} \div \frac{1}{2} }, which provides another opportunity to solidify our understanding of fraction division. As with the previous examples, the fundamental principle remains the same: we convert the division problem into a multiplication problem by multiplying the first fraction by the reciprocal of the second fraction. This core concept is the key to successfully navigating fraction division. The first step is to identify the reciprocal of the second fraction. The reciprocal of 12{\frac{1}{2}} is 21{\frac{2}{1}}, which is simply 2. Now, we transform the division problem 410รท12{\frac{4}{10} \div \frac{1}{2}} into a multiplication problem: 410ร—21{\frac{4}{10} \times \frac{2}{1}}. Multiplying the numerators and the denominators, we get 4ร—210ร—1=810{\frac{4 \times 2}{10 \times 1} = \frac{8}{10}}. The resulting fraction, 810{\frac{8}{10}}, can be simplified. Both 8 and 10 are divisible by 2. Dividing both the numerator and the denominator by 2 gives us 8รท210รท2=45{\frac{8 \div 2}{10 \div 2} = \frac{4}{5}}. Therefore, the simplified fraction is 45{\frac{4}{5}}. This example also illustrates the importance of simplification. While 810{\frac{8}{10}} is a correct answer, 45{\frac{4}{5}} is a more concise and standard representation. Simplifying fractions is a crucial skill in mathematics, as it allows for easier comparison and manipulation of fractional values. It is always best practice to present fractions in their simplest form. Furthermore, this problem highlights how a fraction can be divided by a simple whole number (in this case, 2), as 21{\frac{2}{1}} is equivalent to the whole number 2. Understanding this equivalence is beneficial in various mathematical contexts. Mastering fraction division requires not only understanding the core principle of multiplying by the reciprocal but also the ability to simplify fractions effectively. This combination of skills is essential for success in more advanced mathematical concepts.

Now, letโ€™s address the fourth problem: 69รท85{\frac{6}{9} \div \frac{8}{5} }. This problem, similar to the others, reinforces the central concept of fraction division โ€“ transforming division into multiplication by using the reciprocal. However, it also presents a valuable opportunity to practice simplification both before and after multiplication, a skill that can greatly streamline the calculation process. The first step, as always, is to identify the reciprocal of the second fraction. The reciprocal of 85{\frac{8}{5}} is 58{\frac{5}{8}}. Therefore, the division problem 69รท85{\frac{6}{9} \div \frac{8}{5}} becomes the multiplication problem 69ร—58{\frac{6}{9} \times \frac{5}{8}}. Before we proceed with multiplication, let's consider simplifying the fraction 69{\frac{6}{9}}. Both 6 and 9 are divisible by 3. Dividing both by 3 gives us 6รท39รท3=23{\frac{6 \div 3}{9 \div 3} = \frac{2}{3}}. Now, our multiplication problem is 23ร—58{\frac{2}{3} \times \frac{5}{8}}. Multiplying the numerators and the denominators, we get 2ร—53ร—8=1024{\frac{2 \times 5}{3 \times 8} = \frac{10}{24}}. The resulting fraction, 1024{\frac{10}{24}}, can be further simplified. Both 10 and 24 are divisible by 2. Dividing both by 2 gives us 10รท224รท2=512{\frac{10 \div 2}{24 \div 2} = \frac{5}{12}}. Therefore, the simplified answer is 512{\frac{5}{12}}. This example beautifully illustrates the power of simplifying fractions before multiplying. By simplifying 69{\frac{6}{9}} to 23{\frac{2}{3}} early in the process, we worked with smaller numbers, making the multiplication and subsequent simplification easier. Simplifying fractions at any stage of the calculation is a valuable strategy to minimize computational complexity. Recognizing common factors and simplifying before multiplying can significantly reduce the chances of errors and make the overall process more efficient. This problem reinforces the importance of this skill in mastering fraction division.

Moving on to the fifth problem, 1012รท78{\frac{10}{12} \div \frac{7}{8} }, we continue to practice and refine our skills in fraction division. As with the previous examples, the core principle of converting division to multiplication using the reciprocal remains paramount. This problem also presents a good opportunity to practice simplification techniques, which can make the calculations more manageable. The first step is to find the reciprocal of the second fraction. The reciprocal of 78{\frac{7}{8}} is 87{\frac{8}{7}}. So, the division problem 1012รท78{\frac{10}{12} \div \frac{7}{8}} transforms into the multiplication problem 1012ร—87{\frac{10}{12} \times \frac{8}{7}}. Before multiplying, let's explore simplification possibilities. The fraction 1012{\frac{10}{12}} can be simplified. Both 10 and 12 are divisible by 2. Dividing both by 2, we get 10รท212รท2=56{\frac{10 \div 2}{12 \div 2} = \frac{5}{6}}. Now our multiplication problem is 56ร—87{\frac{5}{6} \times \frac{8}{7}}. We can also notice that the 6 in the denominator of the first fraction and the 8 in the numerator of the second fraction share a common factor of 2. Dividing 6 by 2 gives 3, and dividing 8 by 2 gives 4. So, our problem further simplifies to 53ร—47{\frac{5}{3} \times \frac{4}{7}}. Multiplying the numerators and the denominators, we get 5ร—43ร—7=2021{\frac{5 \times 4}{3 \times 7} = \frac{20}{21}}. The fraction 2021{\frac{20}{21}} is in its simplest form, as 20 and 21 share no common factors other than 1. Therefore, the final answer is 2021{\frac{20}{21}}. This example effectively demonstrates the benefits of simplifying fractions at multiple stages of the calculation. By simplifying both the initial fraction and cross-simplifying before multiplying, we were able to work with smaller numbers and arrive at the solution more efficiently. The ability to identify common factors and simplify fractions is a crucial skill in fraction manipulation. Mastering these techniques not only simplifies calculations but also enhances understanding of fractional relationships.

Finally, let's address the sixth problem: 915รท14{\frac{9}{15} \div \frac{1}{4} }. This problem provides a final opportunity to solidify our understanding of fraction division and the importance of simplification. The fundamental principle of converting division into multiplication using the reciprocal remains the key to solving this problem. We will also focus on simplifying fractions to make the calculation process as efficient as possible. The first step is to identify the reciprocal of the second fraction. The reciprocal of 14{\frac{1}{4}} is 41{\frac{4}{1}}, which is simply 4. Therefore, the division problem 915รท14{\frac{9}{15} \div \frac{1}{4}} becomes the multiplication problem 915ร—41{\frac{9}{15} \times \frac{4}{1}}. Before proceeding with multiplication, let's consider simplifying the fraction 915{\frac{9}{15}}. Both 9 and 15 are divisible by 3. Dividing both by 3 gives us 9รท315รท3=35{\frac{9 \div 3}{15 \div 3} = \frac{3}{5}}. Now our multiplication problem is 35ร—41{\frac{3}{5} \times \frac{4}{1}}. Multiplying the numerators and the denominators, we get 3ร—45ร—1=125{\frac{3 \times 4}{5 \times 1} = \frac{12}{5}}. The resulting fraction, 125{\frac{12}{5}}, is an improper fraction. To express it as a mixed number, we divide 12 by 5. 12 divided by 5 gives a quotient of 2 and a remainder of 2. Therefore, 125{\frac{12}{5}} is equivalent to 2 25{\frac{2}{5}}. This example reinforces the importance of simplifying fractions early in the process. By simplifying 915{\frac{9}{15}} to 35{\frac{3}{5}}, we worked with smaller numbers, making the multiplication straightforward. Recognizing and simplifying fractions is a crucial skill for efficient and accurate fraction manipulation. Converting improper fractions to mixed numbers is also important for a clear understanding of the magnitude of the fractional value. This final problem serves as a comprehensive review of the key concepts and techniques involved in fraction division.

This article falls under the discussion category of mathematics. The content focuses on the fundamental principles and practical application of fraction division. Understanding fractions and their operations is a cornerstone of mathematics, and this article aims to provide a clear and comprehensive guide to mastering fraction division. The examples provided and the step-by-step explanations are designed to make the topic accessible and understandable for a wide range of learners. From basic arithmetic to more advanced mathematical concepts, fractions play a vital role. This discussion on fraction division contributes to a broader understanding of mathematical principles and problem-solving strategies. The article is structured to promote a deeper understanding of the underlying concepts and encourages practice to solidify these skills. Mathematics is a vast and interconnected field, and mastering foundational concepts like fraction division is essential for building a strong mathematical foundation. This article serves as a valuable resource for students, educators, and anyone seeking to enhance their understanding of this important mathematical topic. Further exploration of related mathematical concepts can build upon the knowledge gained here, leading to a more comprehensive understanding of mathematics as a whole. The ability to confidently work with fractions is a key skill that extends far beyond the classroom, impacting various aspects of daily life and professional endeavors. This article aims to empower readers with this skill, fostering a greater appreciation for the beauty and practicality of mathematics.