Solving $-\frac{8}{5} \div(-\frac{12}{25})$ A Step-by-Step Guide

Fraction division can seem daunting at first, but with a clear understanding of the underlying principles, it becomes a straightforward process. At its core, dividing by a fraction is the same as multiplying by its reciprocal. This means that instead of dividing by a fraction, we flip the second fraction (the divisor) and then multiply the two fractions together. This simple rule is the key to unlocking the world of fraction division. Before we dive into the specific problem of -8/5 ÷ (-12/25), let's solidify our understanding with a broader perspective on fraction division. Think of fractions as representing parts of a whole. When we divide one fraction by another, we are essentially asking how many times the second fraction fits into the first. For instance, if we divide 1/2 by 1/4, we are asking how many quarters are there in a half. The answer, of course, is two. This intuitive understanding can help you visualize the process of fraction division. Now, let's delve deeper into the concept of reciprocals. The reciprocal of a fraction is simply the fraction flipped. The numerator becomes the denominator, and the denominator becomes the numerator. For example, the reciprocal of 2/3 is 3/2. The magic of reciprocals lies in the fact that when you multiply a fraction by its reciprocal, the result is always 1. This property is crucial in fraction division because multiplying by the reciprocal effectively undoes the division. Understanding this fundamental concept is crucial before we tackle more complex problems. Fraction division is not just an abstract mathematical concept; it has practical applications in everyday life. From cooking and baking to measuring and construction, fractions are an integral part of many activities. Therefore, mastering fraction division is not only beneficial for academic success but also for real-world problem-solving. This foundational understanding of fractions and their reciprocals will help us confidently tackle the given problem and more complex calculations in the future. Keep in mind that practice is key to mastering any mathematical concept. The more you work with fractions and division, the more comfortable and confident you will become. So, let's move on to the specific problem at hand and see how these principles apply in practice.

Step-by-Step Solution of -8/5 ÷ (-12/25)

Let's break down the problem -8/5 ÷ (-12/25) into manageable steps. This step-by-step approach will not only help us arrive at the correct answer but also deepen our understanding of the process. Our primary goal is to divide two negative fractions. The first crucial step in solving this problem is to remember the golden rule of fraction division: dividing by a fraction is the same as multiplying by its reciprocal. This means we need to find the reciprocal of the second fraction, which is -12/25. To find the reciprocal, we simply flip the fraction, swapping the numerator and the denominator. So, the reciprocal of -12/25 is -25/12. Now, we can rewrite the original division problem as a multiplication problem: -8/5 × (-25/12). Notice that the division symbol has been replaced with a multiplication symbol, and the second fraction has been replaced with its reciprocal. This transformation is the heart of fraction division. Next, we turn our attention to multiplying the two fractions. When multiplying fractions, we multiply the numerators together and the denominators together. In this case, we have (-8) × (-25) in the numerator and (5) × (12) in the denominator. Before we perform the multiplication, let's consider the signs. We are multiplying a negative number by a negative number. A negative times a negative equals a positive, so our final answer will be positive. This is a crucial detail to keep in mind, as it helps us avoid sign errors. Now, let's multiply the numerators: (-8) × (-25) = 200. And let's multiply the denominators: (5) × (12) = 60. So, our fraction becomes 200/60. However, we are not quite done yet. The fraction 200/60 can be simplified. Both the numerator and the denominator have common factors. Simplifying fractions is essential to expressing the answer in its most concise form. We look for the greatest common factor (GCF) of 200 and 60. The GCF is the largest number that divides both numbers evenly. In this case, the GCF of 200 and 60 is 20. We divide both the numerator and the denominator by 20: (200 ÷ 20) / (60 ÷ 20) = 10/3. Therefore, the simplified fraction is 10/3. This is an improper fraction, meaning the numerator is greater than the denominator. We can convert this improper fraction to a mixed number, which is a whole number and a fraction. To do this, we divide 10 by 3. 10 divided by 3 is 3 with a remainder of 1. So, the mixed number is 3 1/3. In summary, -8/5 ÷ (-12/25) = 10/3 or 3 1/3. This step-by-step solution demonstrates the process of fraction division, emphasizing the importance of reciprocals, sign rules, and simplification. By understanding each step, you can confidently tackle similar problems.

The Significance of Reciprocals in Division

In the realm of mathematics, the concept of reciprocals plays a pivotal role, especially in the context of division. Understanding the significance of reciprocals is not just about memorizing a rule; it's about grasping a fundamental mathematical principle. As we've seen in the problem -8/5 ÷ (-12/25), the key to dividing fractions lies in multiplying by the reciprocal of the divisor. But why does this work? To truly appreciate the power of reciprocals, let's delve into their deeper meaning and explore their applications beyond simple fraction division. The reciprocal of a number is, in essence, its multiplicative inverse. This means that when you multiply a number by its reciprocal, the result is always 1. This property is what makes reciprocals so crucial in division. Think of division as the inverse operation of multiplication. When we divide a number by another number, we are essentially asking,