Dividing Fractions How To Solve 5/8 ÷ 5/(10w²-21w+2)

Introduction

In the realm of mathematics, dividing fractions is a fundamental operation, crucial for solving a myriad of problems across various fields. This article delves into the intricacies of dividing fractions, focusing on the specific example of ${\frac{5}{8} \div \frac{5}{10w^2-21w+2}}$. We will break down the process step-by-step, ensuring a comprehensive understanding of the underlying principles and techniques involved. Whether you're a student grappling with fraction division or a seasoned mathematician seeking a refresher, this guide will illuminate the path to mastering this essential skill.

Dividing fractions might seem daunting at first, but with a clear understanding of the rules and procedures, it becomes a straightforward task. This article aims to not only provide the solution to the given problem but also to equip you with the knowledge to tackle similar challenges confidently. We will explore the concept of reciprocals, the process of factoring quadratic expressions, and the simplification of fractions, all while keeping the explanation accessible and engaging. So, let's embark on this mathematical journey together and conquer the world of fraction division!

Understanding the Basics of Fraction Division

To effectively divide fractions, it’s crucial to grasp the core concepts and rules that govern this operation. Fraction division, at its heart, is the process of determining how many times one fraction fits into another. Unlike multiplication, where we simply multiply the numerators and denominators, division involves an additional step: finding 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 concept is fundamental because dividing by a fraction is mathematically equivalent to multiplying by its reciprocal. This simple yet powerful rule transforms division problems into multiplication problems, making them much easier to solve.

Consider the general case of dividing two fractions, ${\frac{a}{b}}$ and ${\frac{c}{d}}$. According to the rule, ${\frac{a}{b} \div \frac{c}{d}}$ is the same as ${\frac{a}{b} \times \frac{d}{c}}$. This transformation is the cornerstone of fraction division. Once the division is converted into multiplication, we proceed by multiplying the numerators (a and d) and the denominators (b and c), resulting in ${\frac{ad}{bc}}$. The final step often involves simplifying the resulting fraction to its simplest form, which means reducing it to the lowest possible terms by dividing both the numerator and the denominator by their greatest common divisor (GCD). Understanding these foundational principles is essential before tackling more complex problems involving algebraic expressions in the fractions.

Step-by-Step Solution: ${\frac{5}{8} \div \frac{5}{10w^2-21w+2}}$

Now, let’s apply these principles to the problem at hand: ${\frac{5}{8} \div \frac{5}{10w^2-21w+2}}$. The first step, as we've established, is to convert the division into multiplication by taking the reciprocal of the second fraction. This means we will multiply ${\frac{5}{8}}$ by the reciprocal of ${\frac{5}{10w^2-21w+2}}$, which is ${\frac{10w^2-21w+2}{5}}$. Our problem now looks like this: ${\frac{5}{8} \times \frac{10w^2-21w+2}{5}}$.

Next, we can simplify the expression by canceling out common factors. Notice that there is a factor of 5 in both the numerator and the denominator. We can divide both by 5, which simplifies the expression to ${\frac{1}{8} \times (10w^2-21w+2)}$. Now, we need to address the quadratic expression in the numerator. To simplify further, we will factor the quadratic expression ${10w^2-21w+2}$. Factoring quadratics involves finding two binomials that, when multiplied, give us the original quadratic. In this case, we are looking for two binomials of the form (Ax + B) and (Cx + D) such that (Ax + B)(Cx + D) = ${10w^2-21w+2}$.

After factoring the quadratic expression, we can further simplify the fraction by canceling out any common factors between the numerator and the denominator. This step is crucial for expressing the answer in its simplest form. Once we have simplified the fraction, we will have our final answer. Let's proceed with factoring the quadratic expression in the next section.

Factoring the Quadratic Expression: ${10w^2-21w+2}$

Factoring the quadratic expression ${10w^2-21w+2}$ is a crucial step in simplifying our original problem. Factoring a quadratic expression involves rewriting it as a product of two binomials. Several techniques can be used for this purpose, including trial and error, the quadratic formula, and factoring by grouping. For this particular quadratic, we'll employ the method of factoring by grouping, which is often effective for expressions where the leading coefficient is not 1.

To factor ${10w^2-21w+2}$ by grouping, we first need to find two numbers that multiply to the product of the leading coefficient (10) and the constant term (2), which is 20, and add up to the middle coefficient (-21). These two numbers are -20 and -1. We then rewrite the middle term, -21w, as the sum of -20w and -1w. This gives us the expression ${10w^2 - 20w - 1w + 2}$. Now, we group the first two terms and the last two terms:

${(10w^2 - 20w) + (-1w + 2)}$. Next, we factor out the greatest common factor (GCF) from each group. From the first group, the GCF is 10w, and from the second group, the GCF is -1. Factoring these out, we get ${10w(w - 2) - 1(w - 2)}$. Notice that both terms now have a common factor of (w - 2). We factor this out to obtain the final factored form: ${(10w - 1)(w - 2)}$. Therefore, ${10w^2-21w+2}$ factors into ${(10w - 1)(w - 2)}$. This factored form will allow us to further simplify our fraction in the next step.

Simplifying the Expression and Obtaining the Final Answer

Having factored the quadratic expression ${10w^2-21w+2}$ into ${(10w - 1)(w - 2)}$, we can now substitute this back into our original problem. Recall that we had simplified the expression to ${\frac{1}{8} \times (10w^2-21w+2)}$. Replacing the quadratic with its factored form, we get ${\frac{1}{8} \times (10w - 1)(w - 2)}$. This simplifies to ${\frac{(10w - 1)(w - 2)}{8}}$.

Now, we check to see if there are any further simplifications we can make. We look for common factors between the numerator and the denominator. In this case, there are no common factors between the binomials (10w - 1) and (w - 2) and the number 8. Therefore, the fraction is already in its simplest form. Thus, the final simplified answer to the problem ${\frac{5}{8} \div \frac{5}{10w^2-21w+2}}$ is ${\frac{(10w - 1)(w - 2)}{8}}$.

It's essential to present the answer clearly and concisely. By factoring the quadratic expression and simplifying the fraction, we have arrived at the simplest form of the solution. This demonstrates the importance of understanding both the fundamental rules of fraction division and the techniques for factoring algebraic expressions. In the concluding section, we will summarize the steps taken and emphasize the key concepts learned throughout this process.

Conclusion

In this comprehensive guide, we have successfully navigated the process of dividing fractions, specifically addressing the problem ${\frac{5}{8} \div \frac{5}{10w^2-21w+2}}$. We began by establishing the fundamental principle of fraction division: dividing by a fraction is equivalent to multiplying by its reciprocal. This key concept allowed us to transform the division problem into a multiplication problem, setting the stage for further simplification. We then tackled the algebraic component of the problem, focusing on the quadratic expression in the denominator.

Factoring the quadratic expression ${10w^2-21w+2}$ was a critical step in simplifying the problem. We employed the method of factoring by grouping, which enabled us to rewrite the quadratic as a product of two binomials, ${(10w - 1)(w - 2)}$. This factorization was then substituted back into the original expression, allowing us to simplify the fraction. After substituting, we checked for any common factors between the numerator and the denominator. Finding none, we concluded that the fraction was in its simplest form. The final answer, ${\frac{(10w - 1)(w - 2)}{8}}$, represents the solution to the original division problem.

This exercise highlights the interconnectedness of various mathematical concepts. Fraction division requires a solid understanding of reciprocals and multiplication, while simplifying algebraic fractions often involves factoring techniques. By mastering these skills, you can confidently approach a wide range of mathematical problems. Remember, practice is key to proficiency. Work through similar problems to reinforce your understanding and build your problem-solving abilities. The journey through mathematics is one of continuous learning and discovery, and each problem solved brings you closer to mastery.