Fraction Division Examples And Solutions

Fraction division can seem tricky at first, but with a clear understanding of the underlying principles, it becomes a straightforward process. This article aims to provide a comprehensive guide on how to evaluate various fraction division expressions. We will break down each expression step by step, ensuring clarity and accuracy in our calculations. Understanding how to divide fractions is a fundamental skill in mathematics, crucial for various applications in both academic and real-world scenarios. Let's delve into the mechanics of fraction division and equip you with the tools to tackle these problems with confidence.

The core concept in dividing fractions is to invert and multiply. This means that when we divide one fraction by another, we actually multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping its numerator and denominator. For instance, the reciprocal of ${\frac{a}{b}}$ is ${\frac{b}{a}}$. This simple yet powerful rule is the key to unlocking the world of fraction division. We will apply this rule to each of the expressions provided, demonstrating how to arrive at the correct answer methodically. The ability to perform these calculations accurately is essential for anyone studying mathematics or working in fields that require mathematical proficiency.

Throughout this guide, we will emphasize the importance of simplifying fractions whenever possible. Simplifying fractions involves reducing them to their lowest terms, making them easier to work with and understand. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. Simplifying fractions not only makes calculations more manageable but also provides a clearer representation of the fraction's value. Before we dive into the examples, let's recap the key steps involved in dividing fractions: 1) Identify the fractions to be divided. 2) Find the reciprocal of the divisor (the fraction we are dividing by). 3) Multiply the first fraction by the reciprocal of the second fraction. 4) Simplify the resulting fraction if necessary. With these steps in mind, let's begin our journey into the world of fraction division.

1.1. ${\frac{1}{5} \div 9}$

To evaluate ${\frac{1}{5} \div 9}$, we first need to express the whole number 9 as a fraction. We can do this by writing it as ${\frac{9}{1}}$. Now, we have the expression ${\frac{1}{5} \div \frac{9}{1}}$. The next step is to apply the invert and multiply rule. We find the reciprocal of ${\frac{9}{1}}$, which is ${\frac{1}{9}}$. Then, we multiply ${\frac{1}{5}}$ by ${\frac{1}{9}}$. This gives us ${\frac{1}{5} \times \frac{1}{9} = \frac{1 \times 1}{5 \times 9} = \frac{1}{45}}$. The resulting fraction, ${\frac{1}{45}}$, is already in its simplest form, as 1 and 45 have no common factors other than 1. Therefore, the final answer is ${\frac{1}{45}}$. This example highlights how whole numbers can be easily incorporated into fraction division problems by expressing them as fractions with a denominator of 1. Mastering this technique is crucial for tackling more complex fraction division problems. Remember, the key to success in mathematics lies in understanding the fundamental principles and applying them consistently.

1.2. ${\frac{4}{3} \div 4}$

In this case, we need to evaluate ${\frac{4}{3} \div 4}$. Similar to the previous example, we express the whole number 4 as a fraction, which is ${\frac{4}{1}}$. Our expression now becomes ${\frac{4}{3} \div \frac{4}{1}}$. To divide, we invert and multiply. The reciprocal of ${\frac{4}{1}}$ is ${\frac{1}{4}}$. We then multiply ${\frac{4}{3}}$ by ${\frac{1}{4}}$, which gives us ${\frac{4}{3} \times \frac{1}{4} = \frac{4 \times 1}{3 \times 4} = \frac{4}{12}}$. Now, we need to simplify the fraction ${\frac{4}{12}}$. Both the numerator and the denominator are divisible by 4. Dividing both by 4, we get ${\frac{4 \div 4}{12 \div 4} = \frac{1}{3}}$. Therefore, the simplified answer is ${\frac{1}{3}}$. This example emphasizes the importance of simplifying fractions after performing the division. Simplifying not only provides the answer in its most concise form but also makes it easier to compare with other fractions. Always remember to look for common factors between the numerator and denominator to simplify your answer.

1.3. ${\frac{3}{7} \div 9}$

To evaluate ${\frac{3}{7} \div 9}$, we start by expressing the whole number 9 as a fraction, writing it as ${\frac{9}{1}}$. Our expression is now ${\frac{3}{7} \div \frac{9}{1}}$. We apply the invert and multiply rule. The reciprocal of ${\frac{9}{1}}$ is ${\frac{1}{9}}$. Multiplying ${\frac{3}{7}}$ by ${\frac{1}{9}}$ gives us ${\frac{3}{7} \times \frac{1}{9} = \frac{3 \times 1}{7 \times 9} = \frac{3}{63}}$. To simplify ${\frac{3}{63}}$, we notice that both 3 and 63 are divisible by 3. Dividing both by 3, we get ${\frac{3 \div 3}{63 \div 3} = \frac{1}{21}}$. The simplified fraction is ${\frac{1}{21}}$, which is our final answer. This example reinforces the need to simplify fractions to obtain the most accurate and concise answer. Always look for common factors and divide both the numerator and denominator by their greatest common factor.

1.4. ${\frac{14}{4} \div 2}$

Evaluating ${\frac{14}{4} \div 2}$ begins with expressing the whole number 2 as a fraction, which is ${\frac{2}{1}}$. The expression becomes ${\frac{14}{4} \div \frac{2}{1}}$. Applying the invert and multiply rule, we find the reciprocal of ${\frac{2}{1}}$, which is ${\frac{1}{2}}$. Multiplying ${\frac{14}{4}}$ by ${\frac{1}{2}}$ gives us ${\frac{14}{4} \times \frac{1}{2} = \frac{14 \times 1}{4 \times 2} = \frac{14}{8}}$. Now, we simplify ${\frac{14}{8}}$. Both 14 and 8 are divisible by 2. Dividing both by 2, we get ${\frac{14 \div 2}{8 \div 2} = \frac{7}{4}}$. The simplified fraction is ${\frac{7}{4}}$. This fraction can also be expressed as a mixed number. To convert ${\frac{7}{4}}$ to a mixed number, we divide 7 by 4, which gives us 1 with a remainder of 3. So, ${\frac{7}{4}}$ is equal to 1${\frac{3}{4}}$. Therefore, the final answer is ${\frac{7}{4}}$ or 1${\frac{3}{4}}$. This example demonstrates the process of simplifying fractions and converting them to mixed numbers when appropriate.

1.5. ${\frac{7}{1} \div 4}$

To evaluate ${\frac{7}{1} \div 4}$, we express the whole number 4 as a fraction, which is ${\frac{4}{1}}$. Our expression now is ${\frac{7}{1} \div \frac{4}{1}}$. Applying the invert and multiply rule, we find the reciprocal of ${\frac{4}{1}}$, which is ${\frac{1}{4}}$. Multiplying ${\frac{7}{1}}$ by ${\frac{1}{4}}$ gives us ${\frac{7}{1} \times \frac{1}{4} = \frac{7 \times 1}{1 \times 4} = \frac{7}{4}}$. The fraction ${\frac{7}{4}}$ is already in its simplest form, but we can convert it to a mixed number for a different representation. Dividing 7 by 4, we get 1 with a remainder of 3. Therefore, ${\frac{7}{4}}$ is equal to 1${\frac{3}{4}}$. The final answer is ${\frac{7}{4}}$ or 1${\frac{3}{4}}$. This example shows how a fraction can be expressed in both improper and mixed number forms, depending on the context and preference.

1.6. ${\frac{1}{11} \div 6}$

To evaluate ${\frac{1}{11} \div 6}$, we express the whole number 6 as a fraction, writing it as ${\frac{6}{1}}$. The expression becomes ${\frac{1}{11} \div \frac{6}{1}}$. Applying the invert and multiply rule, the reciprocal of ${\frac{6}{1}}$ is ${\frac{1}{6}}$. Multiplying ${\frac{1}{11}}$ by ${\frac{1}{6}}$ gives us ${\frac{1}{11} \times \frac{1}{6} = \frac{1 \times 1}{11 \times 6} = \frac{1}{66}}$. The fraction ${\frac{1}{66}}$ is already in its simplest form, as 1 and 66 have no common factors other than 1. Thus, the final answer is ${\frac{1}{66}}$. This example is a straightforward application of the invert and multiply rule, resulting in a simplified fraction.

1.7. ${\frac{2}{15} \div 8}$

Evaluating ${\frac{2}{15} \div 8}$ involves expressing the whole number 8 as a fraction, which is ${\frac{8}{1}}$. Our expression is now ${\frac{2}{15} \div \frac{8}{1}}$. Applying the invert and multiply rule, we find the reciprocal of ${\frac{8}{1}}$, which is ${\frac{1}{8}}$. Multiplying ${\frac{2}{15}}$ by ${\frac{1}{8}}$ gives us ${\frac{2}{15} \times \frac{1}{8} = \frac{2 \times 1}{15 \times 8} = \frac{2}{120}}$. To simplify ${\frac{2}{120}}$, we notice that both 2 and 120 are divisible by 2. Dividing both by 2, we get ${\frac{2 \div 2}{120 \div 2} = \frac{1}{60}}$. Therefore, the simplified answer is ${\frac{1}{60}}$. This example highlights the importance of simplifying fractions to obtain the most concise answer.

1.8. ${\frac{2}{3} \div 3}$

To evaluate ${\frac{2}{3} \div 3}$, we express the whole number 3 as a fraction, which is ${\frac{3}{1}}$. The expression becomes ${\frac{2}{3} \div \frac{3}{1}}$. Applying the invert and multiply rule, the reciprocal of ${\frac{3}{1}}$ is ${\frac{1}{3}}$. Multiplying ${\frac{2}{3}}$ by ${\frac{1}{3}}$ gives us ${\frac{2}{3} \times \frac{1}{3} = \frac{2 \times 1}{3 \times 3} = \frac{2}{9}}$. The fraction ${\frac{2}{9}}$ is already in its simplest form, as 2 and 9 have no common factors other than 1. Thus, the final answer is ${\frac{2}{9}}$. This example is a straightforward application of the invert and multiply rule, resulting in a simplified fraction.

1.9. ${\frac{3}{1} \div 8}$

Evaluating ${\frac{3}{1} \div 8}$ begins with expressing the whole number 8 as a fraction, which is ${\frac{8}{1}}$. Our expression now is ${\frac{3}{1} \div \frac{8}{1}}$. Applying the invert and multiply rule, we find the reciprocal of ${\frac{8}{1}}$, which is ${\frac{1}{8}}$. Multiplying ${\frac{3}{1}}$ by ${\frac{1}{8}}$ gives us ${\frac{3}{1} \times \frac{1}{8} = \frac{3 \times 1}{1 \times 8} = \frac{3}{8}}$. The fraction ${\frac{3}{8}}$ is already in its simplest form, as 3 and 8 have no common factors other than 1. Therefore, the final answer is ${\frac{3}{8}}$. This example is another clear demonstration of the invert and multiply rule in action.

1.10. ${\frac{14}{6} \div 3}$

To evaluate ${\frac{14}{6} \div 3}$, we express the whole number 3 as a fraction, which is ${\frac{3}{1}}$. The expression becomes ${\frac{14}{6} \div \frac{3}{1}}$. Applying the invert and multiply rule, the reciprocal of ${\frac{3}{1}}$ is ${\frac{1}{3}}$. Multiplying ${\frac{14}{6}}$ by ${\frac{1}{3}}$ gives us ${\frac{14}{6} \times \frac{1}{3} = \frac{14 \times 1}{6 \times 3} = \frac{14}{18}}$. To simplify ${\frac{14}{18}}$, we notice that both 14 and 18 are divisible by 2. Dividing both by 2, we get ${\frac{14 \div 2}{18 \div 2} = \frac{7}{9}}$. The simplified fraction is ${\frac{7}{9}}$, which is our final answer. This example reinforces the importance of simplifying fractions to obtain the most accurate and concise answer.

The discussion category for the problems presented is mathematics, specifically focusing on the subtopic of fraction division. Fraction division is a fundamental concept in arithmetic and algebra, forming the basis for more advanced mathematical operations. A strong grasp of fraction division is essential for success in higher-level math courses and various real-world applications. The problems discussed in this article provide a comprehensive overview of the techniques and strategies involved in dividing fractions, including the crucial step of simplifying fractions to their lowest terms. Understanding the concept of reciprocals and their role in division is also paramount. The examples provided cover a range of scenarios, from dividing fractions by whole numbers to dividing fractions by other fractions, ensuring a well-rounded understanding of the topic. Furthermore, the ability to convert between improper fractions and mixed numbers is a valuable skill that is highlighted in several examples. By mastering these concepts, students can build a solid foundation for future mathematical endeavors.

In conclusion, this article has provided a detailed guide on how to evaluate fraction division expressions. By understanding the invert and multiply rule and practicing simplification techniques, you can confidently tackle any fraction division problem. Remember, mathematics is a subject that builds upon itself, so mastering these fundamental concepts is crucial for future success. Keep practicing, and you'll become a fraction division expert in no time!