Dividing Fractions Step By Step Solution For -16/21 Divided By (-8/3)

Introduction

In the realm of mathematics, dividing fractions can often seem daunting, but with the right approach, it becomes a manageable task. This comprehensive guide aims to demystify the process of dividing fractions, focusing specifically on the example $-\frac{16}{21} \div(-\frac{8}{3})$. We will break down the steps involved, provide clear explanations, and offer additional insights to ensure a thorough understanding. Mastering the division of fractions is crucial for various mathematical applications, from basic arithmetic to advanced algebra and calculus. This article will not only help you solve the given problem but also equip you with the skills to tackle similar challenges with confidence.

Understanding fractions is the foundational step in grasping the concept of dividing them. A fraction represents a part of a whole, expressed as a ratio between two numbers: the numerator (the top number) and the denominator (the bottom number). In our example, we are dealing with two fractions: $-\frac{16}{21}$ and $-\frac{8}{3}$. The negative signs indicate that these fractions represent negative quantities. When dividing fractions, we are essentially asking how many times the second fraction (the divisor) fits into the first fraction (the dividend). This concept is crucial for visualizing the operation and understanding the logic behind the steps involved. Before diving into the specific steps, let's clarify some essential terminology and principles that will guide our approach. Understanding the reciprocal of a fraction, the rules for multiplying fractions, and the handling of negative signs are all vital components of successfully dividing fractions. So, let's embark on this journey to master the art of dividing fractions.

Understanding Fractions and Division

To effectively divide fractions, a solid grasp of what fractions represent and how division works is essential. A fraction is a numerical quantity that is not a whole number. It represents a part of a whole and is written in the form ${\frac{a}{b}}$, where a is the numerator and b is the denominator. The numerator indicates the number of parts we have, while the denominator indicates the total number of parts the whole is divided into. For instance, the fraction ${\frac{16}{21}}$ means we have 16 parts out of a total of 21. Understanding this fundamental concept is crucial before we delve into the process of dividing fractions. The concept of division, in general, involves splitting a quantity into equal parts or determining how many times one quantity is contained within another. When we apply this to fractions, we are essentially asking how many times one fraction fits into another. This might sound complex, but the process becomes clearer when we understand the mathematical operation involved.

Division of fractions can be understood as the inverse operation of multiplication. When we divide one fraction by another, we are essentially finding out how many times the second fraction (the divisor) is contained within the first fraction (the dividend). To perform this operation, we use a simple yet powerful rule: we 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 example, the reciprocal of ${\frac{8}{3}}$ is ${\frac{3}{8}}$. This seemingly simple step is the key to unlocking the division of fractions. But why does this work? The reason lies in the fundamental relationship between multiplication and division. Dividing by a number is the same as multiplying by its inverse. In the context of fractions, the reciprocal serves as the multiplicative inverse. By multiplying by the reciprocal, we are effectively undoing the division, making the calculation straightforward. This principle is not just a mathematical trick; it’s a reflection of the underlying structure of numbers and operations. Understanding this rationale can greatly enhance your confidence and accuracy when dealing with fraction division problems.

Step-by-Step Solution for -rac{16}{21} ext{divided by} (-rac{8}{3})

Now, let's tackle the specific problem: $-\frac{16}{21} \div(-\frac{8}{3})$. We will break down the solution into manageable steps to ensure clarity and understanding. The first step in dividing fractions is to rewrite the division as a multiplication problem. This involves taking the reciprocal of the second fraction (the divisor) and changing the division sign to a multiplication sign. In our case, the divisor is $-\frac{8}{3}$. The reciprocal of this fraction is $-\frac{3}{8}$. Remember, when finding the reciprocal, we simply swap the numerator and the denominator. So, the division problem $-\frac{16}{21} \div(-\frac{8}{3})$ becomes a multiplication problem: $-\frac{16}{21} \times(-\frac{3}{8})$. This transformation is the cornerstone of fraction division, turning a potentially confusing operation into a familiar one.

Next, we perform the multiplication of the fractions. To multiply fractions, we multiply the numerators together and the denominators together. In our problem, we have: (-\frac{16}{21}) \times(-\frac{3}{8}) = \frac{(-16) \times (-3)}{21 \times 8}. When multiplying negative numbers, a negative times a negative results in a positive. So, (-16) \times (-3) = 48. For the denominators, 21 \times 8 = 168. Thus, our fraction becomes ${\frac{48}{168}}$. This step is a direct application of the rules of fraction multiplication, and it simplifies the problem significantly. However, we are not done yet. The final step is to simplify the resulting fraction to its simplest form. Simplifying fractions involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by it. This process reduces the fraction to its lowest terms, making it easier to understand and use in further calculations. In our case, the fraction ${\frac{48}{168}}$ can be simplified, and we'll see how in the next part.

Simplifying the Result

After multiplying the fractions, we arrived at the fraction ${\frac{48}{168}}$. The final and crucial step is to simplify this fraction to its simplest form. Simplifying a fraction means reducing it to its lowest terms, where the numerator and the denominator have no common factors other than 1. This not only makes the fraction easier to understand but also ensures that it is in its most mathematically elegant form. To simplify ${\frac{48}{168}}$, we need to find the greatest common divisor (GCD) of 48 and 168. The GCD is the largest number that divides both 48 and 168 without leaving a remainder. One way to find the GCD is to list the factors of each number and identify the largest factor they have in common. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The factors of 168 are 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, and 168. By comparing the lists, we can see that the greatest common divisor of 48 and 168 is 24.

Now that we have the GCD, we can simplify the fraction. We divide both the numerator and the denominator by the GCD. So, we divide 48 by 24, which equals 2, and we divide 168 by 24, which equals 7. Therefore, ${\frac{48}{168}}$ simplifies to ${\frac{2}{7}}$. This simplified fraction is in its lowest terms, meaning there are no more common factors between the numerator and the denominator. Simplifying fractions is a fundamental skill in mathematics, and it is essential for various applications. It not only makes fractions easier to work with but also provides a clearer representation of the quantity they represent. In this case, we have successfully simplified the result of our division problem, arriving at the final answer: ${\frac{2}{7}}$. This completes the solution for dividing the fractions $-\frac{16}{21}$ and $-\frac{8}{3}$.

Final Answer in Simplest Form

After performing the division and simplifying the result, we arrive at the final answer. The original problem was to divide $-\frac{16}{21}$ by $-\frac{8}{3}$. We followed the steps of rewriting the division as multiplication by the reciprocal, multiplying the fractions, and simplifying the result. The final answer, in simplest form, is ${\frac{2}{7}}$. This fraction represents the quotient of the two original fractions and is expressed in its most reduced form, where the numerator and denominator have no common factors other than 1. Presenting the final answer in simplest form is crucial in mathematics. It demonstrates a complete understanding of the problem and ensures that the answer is clear, concise, and easy to interpret. In this case, ${\frac{2}{7}}$ is the simplest representation of the division of $-\frac{16}{21}$ by $-\frac{8}{3}$, making it the definitive solution.

In summary, the process of dividing fractions involves several key steps, each of which is essential for arriving at the correct answer. First, we rewrite the division problem as a multiplication problem by taking the reciprocal of the divisor. Second, we multiply the numerators and the denominators of the fractions. Finally, we simplify the resulting fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor. By following these steps diligently, we can confidently solve fraction division problems and express the results in their most accurate and understandable form. The final answer, ${\frac{2}{7}}$, not only solves the specific problem but also reinforces the importance of understanding and applying the fundamental principles of fraction division.

Conclusion

In conclusion, dividing fractions, while it may initially appear complex, is a straightforward process once the underlying principles are understood and applied methodically. In this comprehensive guide, we have walked through the step-by-step solution of dividing $-\frac{16}{21}$ by $-\frac{8}{3}$, demonstrating each stage with clarity and precision. From understanding the basic concept of fractions to simplifying the final result, we have covered all the essential aspects of fraction division. The key to success in dividing fractions lies in rewriting the division as a multiplication by the reciprocal, performing the multiplication correctly, and simplifying the resulting fraction to its lowest terms. This process not only provides the correct answer but also enhances the understanding of the relationships between fractions and the operations performed on them. The final answer, ${\frac{2}{7}}$, represents the simplest form of the division, showcasing the importance of simplification in mathematical problem-solving.

Moreover, the skills acquired in this exercise extend beyond the specific problem. The ability to divide fractions is a fundamental building block for more advanced mathematical concepts, including algebra, calculus, and beyond. Mastering this skill empowers students to tackle a wide range of mathematical challenges with confidence and accuracy. The process of dividing fractions also reinforces critical thinking and problem-solving skills, encouraging learners to break down complex problems into smaller, more manageable steps. This approach is invaluable not only in mathematics but also in various other fields. By understanding the logic behind each step and practicing diligently, anyone can master the art of dividing fractions and apply this knowledge to a multitude of situations. Therefore, the time and effort invested in learning fraction division are well worth it, laying a solid foundation for future mathematical endeavors and beyond. This guide serves as a testament to the power of clear explanations and methodical problem-solving in mathematics, making even the most challenging concepts accessible and understandable.