Dividing Fractions Solving 6 2/3 ÷ 2/7

Introduction

In the realm of mathematics, dividing fractions is a fundamental skill that builds the foundation for more complex algebraic and arithmetic operations. Understanding how to divide fractions, simplify the result, and express it in different forms, such as a mixed number, is crucial. This article aims to provide a comprehensive guide on solving the problem $6 \frac{2}{3} \div \frac{2}{7}$, while also explaining the underlying concepts and principles. We will walk through each step, ensuring clarity and understanding for readers of all levels. Mastering this skill not only helps in academic settings but also in everyday situations where proportional reasoning and fractions are involved. So, let's delve into the world of fractions and unravel the process of division with ease and confidence.

Understanding Fractions and Mixed Numbers

Before we dive into the division problem, it's essential to have a solid understanding of what fractions and mixed numbers are. A fraction represents a part of a whole and is written in the form $\frac{a}{b}$, where 'a' is the numerator (the number of parts we have) and 'b' is the denominator (the total number of parts the whole is divided into). For example, $\frac{2}{7}$ means we have 2 parts out of a total of 7. A mixed number, on the other hand, is a combination of a whole number and a fraction, such as $6 \frac{2}{3}$. The whole number (6 in this case) represents the number of complete units, and the fraction ($\frac{2}{3}$) represents the remaining part of a unit. Understanding these concepts is crucial because dividing fractions often involves converting mixed numbers into improper fractions to simplify the calculations. An improper fraction is one where the numerator is greater than or equal to the denominator, making it easier to perform multiplication and division operations. The ability to fluently convert between mixed numbers and improper fractions is a cornerstone of fraction manipulation and is vital for solving more complex mathematical problems. In the next section, we will explore the process of converting mixed numbers to improper fractions, setting the stage for our division problem.

Converting Mixed Numbers to Improper Fractions

To effectively divide fractions, especially when dealing with mixed numbers, converting them into improper fractions is a crucial first step. This conversion simplifies the division process and allows us to apply the rules of fraction division more easily. A mixed number, like $6 \frac{2}{3}$, combines a whole number and a fraction. To convert it into an improper fraction, we follow a simple process: multiply the whole number by the denominator of the fraction, and then add the numerator. This result becomes the new numerator of the improper fraction, while the denominator remains the same. Let's apply this to our mixed number, $6 \frac{2}{3}$. We multiply the whole number 6 by the denominator 3, which gives us 18. Then, we add the numerator 2 to this product, resulting in 20. So, the new numerator is 20, and the denominator remains 3. Therefore, the improper fraction equivalent of $6 \frac{2}{3}$ is $\frac{20}{3}$. This conversion is not just a mechanical process; it reflects a deeper understanding of what mixed numbers and fractions represent. The improper fraction $\frac{20}{3}$ signifies that we have twenty thirds, which is the same as six whole units and two-thirds of another unit. Mastering this conversion is essential because it streamlines the subsequent steps in dividing fractions. Now that we've converted the mixed number to an improper fraction, we are ready to tackle the division operation itself.

The Process of Dividing Fractions

Now that we have a solid grasp of fractions, mixed numbers, and the conversion process, let's dive into the heart of our problem: dividing fractions. The rule for dividing fractions is often remembered by the simple phrase "keep, change, flip." This mnemonic helps us remember the three steps involved: keep the first fraction, change the division sign to multiplication, and flip the second fraction (i.e., take its reciprocal). The reciprocal of a fraction is obtained by swapping its numerator and denominator. For example, the reciprocal of $\frac{2}{7}$ is $\frac{7}{2}$. Applying this rule to our problem, $6 \frac{2}{3} \div \frac{2}{7}$, we first convert the mixed number to an improper fraction, as we did in the previous section, resulting in $\frac{20}{3}$. Now, we have the division problem $\frac{20}{3} \div \frac{2}{7}$. Following the "keep, change, flip" rule, we keep the first fraction $\frac{20}{3}$, change the division to multiplication, and flip the second fraction $\frac{2}{7}$ to its reciprocal $\frac{7}{2}$. This transforms our division problem into a multiplication problem: $\frac{20}{3} \times \frac{7}{2}$. This transformation is crucial because multiplying fractions is generally simpler than dividing them. The logic behind this method lies in the concept of inverse operations; dividing by a number is the same as multiplying by its reciprocal. Understanding this principle provides a deeper insight into why the "keep, change, flip" rule works. With our division problem now converted into a multiplication problem, we are ready to proceed with the multiplication process.

Multiplying Fractions

Having transformed our division problem into a multiplication problem, the next step is to multiply the fractions. Multiplying fractions is a straightforward process: we multiply the numerators together to get the new numerator, and we multiply the denominators together to get the new denominator. In our case, we have $\frac{20}{3} \times \frac{7}{2}$. Multiplying the numerators, 20 and 7, gives us 140. Multiplying the denominators, 3 and 2, gives us 6. Thus, the result of the multiplication is $\frac{140}{6}$. This fraction represents the solution to our division problem, but it's not yet in its simplest form. The fraction $\frac{140}{6}$ is an improper fraction, meaning the numerator is larger than the denominator. While improper fractions are perfectly valid, it's often preferable to simplify them, either by reducing them to their lowest terms or by converting them to mixed numbers. Simplifying fractions makes them easier to understand and compare. Before converting to a mixed number, it's often helpful to reduce the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor. This step ensures that the fraction is expressed in its most concise form. In the following section, we will explore the process of simplifying fractions and converting them to mixed numbers.

Simplifying Improper Fractions and Converting to Mixed Numbers

After performing the multiplication, we arrived at the improper fraction $\frac{140}{6}$. To fully solve the problem, we need to simplify this fraction and, if possible, convert it into a mixed number. Simplifying a fraction involves reducing it to its lowest terms. This is achieved by finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by it. In our case, the GCD of 140 and 6 is 2. Dividing both the numerator and the denominator by 2, we get $\frac{140 \div 2}{6 \div 2} = \frac{70}{3}$. Now, we have a simplified improper fraction. To convert this improper fraction to a mixed number, we divide the numerator (70) by the denominator (3). The quotient becomes the whole number part of the mixed number, the remainder becomes the numerator of the fractional part, and the denominator remains the same. When we divide 70 by 3, we get a quotient of 23 and a remainder of 1. Therefore, the mixed number equivalent of $\frac{70}{3}$ is $23 \frac{1}{3}$. This mixed number represents the final simplified answer to our division problem. It tells us that $6 \frac{2}{3}$ divided by $\frac{2}{7}$ equals 23 whole units and one-third of another unit. The process of simplifying and converting fractions is a crucial step in ensuring that our answer is presented in the most understandable and usable form. With this final step completed, we have successfully solved the division problem and can confidently interpret the result.

Final Solution and Conclusion

In conclusion, we have successfully navigated the process of dividing fractions, starting with the problem $6 \frac{2}{3} \div \frac{2}{7}$ and arriving at the simplified answer of $23 \frac{1}{3}$. We began by understanding the basics of fractions and mixed numbers, emphasizing the importance of recognizing and converting between the two. We then focused on converting the mixed number $6 \frac{2}{3}$ into an improper fraction, which is a crucial step in simplifying the division process. Following this, we applied the rule of "keep, change, flip" to transform the division problem into a multiplication problem. We multiplied the fractions, simplified the resulting improper fraction, and finally converted it into a mixed number. The final answer, $23 \frac{1}{3}$, represents the quotient of the original division problem in its simplest form. This journey through the division of fractions highlights the importance of understanding each step and the underlying principles. By mastering these fundamental concepts, you can confidently tackle more complex mathematical problems involving fractions. The ability to divide fractions, simplify results, and convert between different forms is not only essential for academic success but also for practical applications in everyday life. With this knowledge, you are well-equipped to handle fraction-related challenges with ease and precision.