Dividing Fractions Step-by-Step Simplify (7y/6) ÷ (z/-2)

In the realm of mathematics, mastering the art of dividing fractions is crucial for building a strong foundation. This article delves into the process of finding the quotient and simplifying the answer for the expression ${\frac{7y}{6} \div \frac{z}{-2}}$. We will break down each step, ensuring a clear and comprehensive understanding of the underlying principles. Whether you're a student grappling with fraction division or simply seeking to refresh your knowledge, this guide will equip you with the necessary skills and insights.

Understanding the Fundamentals of Fraction Division

Before we embark on solving the given expression, it's essential to grasp the fundamental concept of dividing fractions. Dividing fractions might seem daunting at first, but it's essentially a process of multiplying by the reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator. For instance, the reciprocal of ${\frac{a}{b}}$ is ${\frac{b}{a}}$. When dividing fractions, we invert the second fraction (the divisor) and then multiply the two fractions. This seemingly simple rule is the cornerstone of fraction division, and mastering it will pave the way for tackling more complex problems. In our case, we'll be applying this principle to divide ${\frac{7y}{6}}$ by ${\frac{z}{-2}}$, which involves finding the reciprocal of ${\frac{z}{-2}}$ and then multiplying it with ${\frac{7y}{6}}$. Understanding this foundational concept is key to successfully navigating the steps ahead and arriving at the simplified quotient.

Unveiling the Reciprocal Relationship

The concept of a reciprocal is pivotal in the realm of fraction division. A reciprocal, also known as the multiplicative inverse, is the value that, when multiplied by the original number, yields a product of 1. In the context of fractions, this translates to swapping the numerator and the denominator. For example, the reciprocal of ${\frac{2}{3}}$ is ${\frac{3}{2}}$, and their product, ${\frac{2}{3} \times \frac{3}{2}}$, equals 1. Similarly, the reciprocal of 5, which can be written as ${\frac{5}{1}}$, is ${\frac{1}{5}}$. This reciprocal relationship forms the bedrock of fraction division, transforming division problems into multiplication problems. When we divide one fraction by another, we are essentially multiplying the first fraction by the reciprocal of the second. This seemingly simple maneuver simplifies the process and allows us to apply the rules of fraction multiplication, which are generally more straightforward. Recognizing and applying the concept of reciprocals is therefore fundamental to mastering fraction division and unlocking its intricacies.

The Division to Multiplication Transformation

The magic of fraction division lies in its transformation into multiplication. This transformation hinges on the concept of reciprocals, as we've discussed. When faced with a division problem involving fractions, the first step is to identify the divisor, which is the fraction we're dividing by. We then find the reciprocal of this divisor by swapping its numerator and denominator. Once we have the reciprocal, we replace the division operation with multiplication, effectively turning the division problem into a multiplication problem. This transformation is not just a trick; it's a mathematically sound operation based on the properties of reciprocals. By multiplying by the reciprocal, we're essentially undoing the division, achieving the same result through a different operation. This transformation simplifies the process significantly, as multiplication of fractions is generally easier to handle than division. It allows us to apply the straightforward rule of multiplying numerators and denominators, streamlining the solution process. In the context of our problem, ${\frac{7y}{6} \div \frac{z}{-2}}$, we'll transform the division into multiplication by the reciprocal of ${\frac{z}{-2}}$, which sets the stage for the subsequent steps.

Step-by-Step Solution for ${\frac{7y}{6} \div \frac{z}{-2}}$

Now, let's apply our understanding of fraction division to solve the given expression, ${\frac{7y}{6} \div \frac{z}{-2}}$. We will follow a step-by-step approach to ensure clarity and accuracy. Each step will be explained in detail, making the process easy to follow and understand. By the end of this solution, you'll have a firm grasp of how to divide fractions and simplify the resulting expressions.

Step 1 Identifying the Fractions

The first step in tackling any fraction division problem is to clearly identify the fractions involved. In our expression, ${\frac{7y}{6} \div \frac{z}{-2}}$, we have two fractions: the dividend, which is ${\frac{7y}{6}}$, and the divisor, which is ${\frac{z}{-2}}$. The dividend is the fraction being divided, while the divisor is the fraction we are dividing by. It's crucial to distinguish between these two fractions, as the next step involves manipulating the divisor. Misidentifying the fractions can lead to incorrect application of the division rule, resulting in a wrong answer. Therefore, taking a moment to clearly identify the dividend and divisor is a fundamental step towards solving the problem accurately. In this case, ${\frac{7y}{6}}$ is the fraction we're starting with, and ${\frac{z}{-2}}$ is the fraction that dictates how we'll divide it. This distinction sets the stage for the subsequent transformation of the division problem into a multiplication problem.

Step 2 Finding the Reciprocal of the Divisor

Once we've identified the dividend and divisor, the next crucial step is to find the reciprocal of the divisor. As we discussed earlier, the reciprocal of a fraction is obtained by swapping its numerator and denominator. In our case, the divisor is ${\frac{z}{-2}}$. To find its reciprocal, we swap z and -2, resulting in ${\frac{-2}{z}}$. This reciprocal is the key to transforming the division problem into a multiplication problem. It's essential to pay close attention to the signs when finding the reciprocal. The negative sign in ${\frac{z}{-2}}$ remains with the 2 in the reciprocal, resulting in ${\frac{-2}{z}}$. This reciprocal will be multiplied with the dividend in the next step, effectively performing the division operation. Understanding how to find the reciprocal accurately is paramount to successfully dividing fractions. This step is not just about swapping numbers; it's about understanding the mathematical relationship between a number and its inverse, which is a fundamental concept in algebra and beyond.

Step 3 Transforming Division into Multiplication

With the reciprocal of the divisor in hand, we can now perform the pivotal transformation: converting the division operation into multiplication. This transformation is the heart of fraction division and simplifies the process significantly. Instead of dividing ${\frac{7y}{6}}$ by ${\frac{z}{-2}}$, we will now multiply ${\frac{7y}{6}}$ by the reciprocal of ${\frac{z}{-2}}$, which we found to be ${\frac{-2}{z}}$. This change is represented mathematically as: $\frac{7y}{6} \div \frac{z}{-2} = \frac{7y}{6} \times \frac{-2}{z}$ This equation encapsulates the core principle of fraction division: dividing by a fraction is equivalent to multiplying by its reciprocal. By making this transformation, we've effectively converted a potentially complex division problem into a more manageable multiplication problem. This sets the stage for the next step, where we'll apply the rules of fraction multiplication to arrive at the quotient. The ability to make this transformation is a cornerstone of fraction arithmetic and a skill that will prove invaluable in more advanced mathematical contexts.

Step 4 Multiplying the Fractions

Now that we've transformed the division problem into a multiplication problem, we can proceed with multiplying the fractions. The rule for multiplying fractions is straightforward: multiply the numerators together and multiply the denominators together. In our case, we have: $\frac7y}{6} \times \frac{-2}{z}$ Multiplying the numerators gives us $7y \times -2 = -14y$ Multiplying the denominators gives us: $6 \times z = 6z$ Combining these results, we get: $\frac{-14y{6z}$ This fraction represents the quotient of the original division problem, but it may not be in its simplest form. The next step involves simplifying the fraction to its lowest terms, which is crucial for expressing the answer in its most concise and understandable form. The multiplication step itself is a direct application of the rules of fraction arithmetic, but it's important to perform it accurately to ensure the correctness of the final answer. The resulting fraction, ${\frac{-14y}{6z}}$, is a crucial intermediate step that leads us closer to the simplified solution.

Step 5 Simplifying the Result

The final step in solving the problem is simplifying the resulting fraction. Simplification involves reducing the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD). In our case, we have the fraction: $\frac-14y}{6z}$ The GCD of -14 and 6 is 2. Therefore, we divide both the numerator and the denominator by 2 $\frac{-14y \div 2{6z \div 2} = \frac{-7y}{3z}$ This simplified fraction, ${\frac{-7y}{3z}}$, is the final answer to our problem. It represents the quotient of ${\frac{7y}{6}}$ divided by ${\frac{z}{-2}}$ in its simplest form. Simplification is a crucial step in fraction arithmetic as it presents the answer in its most concise and understandable form. It also demonstrates a thorough understanding of fraction properties and the ability to manipulate them effectively. By simplifying the fraction, we ensure that our answer is not only correct but also presented in the most elegant and efficient manner. This final step completes the solution process and provides the definitive answer to the division problem.

Final Answer

In conclusion, by meticulously following the steps outlined above, we've successfully navigated the division of fractions and arrived at the simplified quotient. The final answer to the expression ${\frac{7y}{6} \div \frac{z}{-2}}$ is: $\frac{-7y}{3z}$ This result represents the culmination of our step-by-step solution, which involved understanding the fundamentals of fraction division, identifying reciprocals, transforming division into multiplication, multiplying fractions, and simplifying the result. Each step played a crucial role in arriving at the final answer, highlighting the interconnectedness of mathematical operations. This comprehensive approach not only provides the solution but also reinforces the underlying principles of fraction arithmetic. By mastering these principles, you'll be well-equipped to tackle more complex mathematical problems involving fractions and beyond.