Simplifying Fraction Division A Step-by-Step Guide
In the realm of mathematics, mastering the division of fractions is a fundamental skill. This article delves into the process of simplifying the division of fractions, specifically focusing on the expression $-2 \frac{1}{4} \div -\frac{9}{10}$. We will break down each step, ensuring a clear understanding of how to arrive at the simplest form, expressing the answer as an improper fraction where necessary. This detailed guide will not only provide the solution but also offer a comprehensive understanding of the underlying principles, making it an invaluable resource for students and math enthusiasts alike.
Understanding the Problem
Before diving into the solution, it's crucial to understand the problem at hand. We are tasked with dividing a mixed number, $-2 \frac{1}{4}$, by a negative fraction, $-\frac{9}{10}$. The goal is to simplify this division and express the final answer in its simplest form, preferably as an improper fraction. This involves converting the mixed number into an improper fraction, understanding the rules of dividing fractions, and simplifying the result. Each of these steps is essential in arriving at the correct solution and will be thoroughly explained in the following sections.
Step 1: Converting Mixed Numbers to Improper Fractions
The first crucial step in simplifying $-2 \frac{1}{4} \div -\frac{9}{10}$ is converting the mixed number $-2 \frac{1}{4}$ into an improper fraction. A mixed number consists of a whole number and a proper fraction, while an improper fraction has a numerator that is greater than or equal to its denominator. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator of the fraction and add the numerator. This result becomes the new numerator, and the denominator remains the same.
In our case, we have $-2 \frac{1}{4}$. The whole number is 2, the numerator is 1, and the denominator is 4. Multiplying the whole number (2) by the denominator (4) gives us 8. Adding the numerator (1) to this result gives us 9. Therefore, the improper fraction is $-\frac{9}{4}$. The negative sign is retained because the original mixed number was negative. This conversion is a foundational step, as improper fractions are easier to work with when performing division. Ensuring accuracy in this step is vital for the subsequent calculations and the final result.
Step 2: Dividing Fractions – The Rule of Reciprocals
Now that we've converted the mixed number to an improper fraction, the next step in simplifying $-2 \frac1}{4} \div -\frac{9}{10}$ is understanding how to divide fractions. Dividing fractions might seem daunting at first, but it becomes straightforward once you grasp the key principle{b}$ is $\frac{b}{a}$.
In our problem, we are dividing $-\frac9}{4}$ by $-\frac{9}{10}$. To perform this division, we will multiply $-\frac{9}{4}$ by the reciprocal of $-\frac{9}{10}$. The reciprocal of $-\frac{9}{10}$ is $-\frac{10}{9}$. Thus, our division problem transforms into a multiplication problem{4} \times -\frac{10}{9}$. This transformation is a critical step in simplifying the expression and makes the subsequent calculation much easier. The rule of reciprocals is a cornerstone of fraction division and is essential for solving a wide range of mathematical problems.
Step 3: Multiplying Fractions
With the division transformed into multiplication, we now focus on simplifying $-2 \frac1}{4} \div -\frac{9}{10}$, which has become $-\frac{9}{4} \times -\frac{10}{9}$. Multiplying fractions involves a straightforward process{36}$.
Before moving forward, it's important to note a crucial aspect of multiplying negative numbers. A negative number multiplied by a negative number results in a positive number. This is why the product of -9 and -10 is positive 90. Understanding this rule is essential for accurately multiplying fractions, especially when dealing with negative signs. The result of our multiplication is $\frac{90}{36}$, which now needs to be simplified to its simplest form. Simplification is a key part of working with fractions, ensuring that the final answer is presented in the most concise and understandable manner.
Step 4: Simplifying the Improper Fraction
After multiplying, we have $\frac{90}{36}$, now let's simplify $-2 \frac{1}{4} \div -\frac{9}{10}$. The fraction $\frac{90}{36}$ is an improper fraction because the numerator (90) is greater than the denominator (36). To simplify an improper fraction, we first look for common factors between the numerator and the denominator. In this case, both 90 and 36 are divisible by several numbers, including 2, 3, 6, 9, and 18. The greatest common factor (GCF) of 90 and 36 is 18.
To simplify the fraction, we divide both the numerator and the denominator by their GCF. Dividing 90 by 18 gives us 5, and dividing 36 by 18 gives us 2. Therefore, the simplified fraction is $\frac{5}{2}$. This process of finding the GCF and dividing both the numerator and denominator by it is crucial for expressing fractions in their simplest form. It ensures that the fraction is reduced to its lowest terms, making it easier to understand and work with in future calculations. The simplified improper fraction, $\frac{5}{2}$, is the final answer to our problem.
The Final Answer
After meticulously working through each step, we arrive at the final answer for the problem $-2 \frac{1}{4} \div -\frac{9}{10}$. We converted the mixed number to an improper fraction, applied the rule of reciprocals to transform division into multiplication, performed the multiplication, and simplified the resulting improper fraction. The simplest form of the expression is $\frac{5}{2}$.
Therefore, the correct answer is:
A. $\frac{5}{2}$
This comprehensive guide has not only provided the solution but also explained the underlying mathematical principles, ensuring a thorough understanding of fraction division. Each step, from converting mixed numbers to simplifying improper fractions, is essential for mastering this fundamental mathematical skill.
Conclusion
In conclusion, simplifying the division of fractions, as demonstrated with the expression $-2 \frac{1}{4} \div -\frac{9}{10}$, involves a series of well-defined steps. These steps include converting mixed numbers to improper fractions, applying the rule of reciprocals to change division into multiplication, performing the multiplication, and simplifying the resulting fraction to its lowest terms. Mastering these steps is crucial for success in mathematics, as fraction division is a fundamental concept that appears in various mathematical contexts.
This guide has provided a detailed explanation of each step, ensuring that readers can confidently tackle similar problems in the future. The importance of understanding the underlying principles, such as the rule of reciprocals and the process of finding the greatest common factor, cannot be overstated. By following these steps and practicing regularly, anyone can become proficient in dividing fractions and simplifying complex mathematical expressions. The final answer, $\frac{5}{2}$, serves as a testament to the power of these methods and the clarity they bring to mathematical problem-solving.
