Solving Fraction Addition $\frac{4}{5}+(-\frac{7}{10})$ A Step-by-Step Guide

Introduction

In the realm of mathematics, grasping the fundamentals of fraction arithmetic is pivotal. Fractions are an integral part of our daily lives, appearing in cooking recipes, financial calculations, and various measurement contexts. Mastering the addition and subtraction of fractions is not only essential for academic success but also for practical problem-solving. This article delves into the process of adding fractions, specifically addressing the expression $\frac{4}{5}+(-\frac{7}{10})$. We will break down the steps, explain the underlying concepts, and ensure a clear understanding of how to arrive at the correct solution. Whether you are a student seeking to improve your math skills or someone looking to refresh your knowledge, this guide provides a comprehensive explanation of fraction addition.

Identifying the Problem: Adding Fractions with Unlike Denominators

The given problem requires us to add two fractions: $\frac{4}{5}$ and $-\frac{7}{10}$. A critical aspect to note here is that the fractions have unlike denominators. The denominator is the bottom number in a fraction, representing the total number of parts into which the whole is divided. In our case, the denominators are 5 and 10. To add fractions, they must have the same denominator, which is known as a common denominator. Without a common denominator, we cannot directly add the numerators (the top numbers in a fraction). This is because we are essentially trying to add quantities that are measured in different units. To illustrate, imagine trying to add apples and oranges directly – you first need a common unit, such as "fruit," to combine them meaningfully. Similarly, with fractions, we need a common denominator to ensure we are adding parts of the same whole. The next section will detail the process of finding a common denominator, which is a crucial step in solving the problem.

Finding the Least Common Denominator (LCD)

To add fractions with unlike denominators, the first crucial step is to find the least common denominator (LCD). The LCD is the smallest multiple that the denominators of both fractions share. In our problem, the denominators are 5 and 10. To find the LCD, we can list the multiples of each denominator and identify the smallest multiple they have in common.

Multiples of 5: 5, 10, 15, 20, 25, ...

Multiples of 10: 10, 20, 30, 40, ...

From the lists above, we can see that the smallest multiple that both 5 and 10 share is 10. Therefore, the LCD for our fractions is 10. Alternatively, we can use the prime factorization method to find the LCD. Prime factorization involves breaking down each number into its prime factors. The prime factorization of 5 is simply 5 (since 5 is a prime number), and the prime factorization of 10 is 2 × 5. To find the LCD, we take the highest power of each prime factor that appears in either factorization: 2¹ and 5¹. Multiplying these together (2 × 5) gives us 10, which confirms that the LCD is indeed 10. Understanding how to find the LCD is fundamental to adding and subtracting fractions accurately. In the next step, we will use the LCD to convert our fractions into equivalent fractions with a common denominator.

Converting Fractions to Equivalent Fractions

Once we have identified the least common denominator (LCD), the next step is to convert each fraction into an equivalent fraction with the LCD as its new denominator. An equivalent fraction represents the same value but has a different numerator and denominator. This conversion is essential because we can only add fractions that have the same denominator. In our problem, the LCD is 10. We need to convert both $\frac{4}{5}$ and $-\frac{7}{10}$ to fractions with a denominator of 10.

For $\frac{4}{5}$, we need to determine what number we can multiply the denominator 5 by to get 10. Since 5 × 2 = 10, we multiply both the numerator and the denominator of $\frac{4}{5}$ by 2:

$\frac{4}{5} \times \frac{2}{2} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10}$

So, $\frac{4}{5}$ is equivalent to $\frac{8}{10}$.

For $-\frac{7}{10}$, the denominator is already 10, so no conversion is needed. We can simply rewrite the fraction as $-\frac{7}{10}$. Now that both fractions have the same denominator, we can proceed with the addition. The process of converting fractions to equivalent fractions is a fundamental skill in fraction arithmetic. It allows us to manipulate fractions without changing their value, making addition and subtraction possible. In the following section, we will add the equivalent fractions we have found.

Adding the Fractions with Common Denominators

Now that we have converted the fractions to equivalent forms with a common denominator, we can proceed to add them. Our problem has been transformed from $\frac{4}{5}+(-\frac{7}{10})$ to $\frac{8}{10}+(-\frac{7}{10})$. When adding fractions with the same denominator, we simply add the numerators and keep the denominator the same. In this case, we add 8 and -7 while keeping the denominator 10.

$\frac{8}{10}+(-\frac{7}{10}) = \frac{8 + (-7)}{10}$

Adding 8 and -7 is the same as subtracting 7 from 8, which gives us 1.

$\frac{8 + (-7)}{10} = \frac{1}{10}$

So, the sum of the fractions is $\frac{1}{10}$. This fraction represents the result of our addition. Adding fractions with common denominators is a straightforward process once the fractions have been converted appropriately. In the next section, we will discuss simplifying the result to its lowest terms, which is the final step in solving the problem.

Simplifying the Result

After adding the fractions, the final step is to simplify the result to its lowest terms. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. In other words, there is no number (other than 1) that can divide both the numerator and the denominator evenly. Our result from the previous step is $\frac{1}{10}$. To determine if this fraction can be simplified, we need to find the factors of both the numerator and the denominator.

The factors of 1 are just 1.

The factors of 10 are 1, 2, 5, and 10.

The only common factor between 1 and 10 is 1. Since there are no other common factors, the fraction $\frac{1}{10}$ is already in its simplest form. Therefore, no further simplification is needed. However, if we had obtained a fraction like $\frac{2}{10}$, we would have needed to simplify it by dividing both the numerator and the denominator by their greatest common factor (GCF), which is 2 in this case. This would give us $\frac{1}{5}$. Simplifying fractions is an important skill in mathematics as it ensures that the answer is presented in its most concise and understandable form. In our case, the final simplified result for $\frac{4}{5}+(-\frac{7}{10})$ is $\frac{1}{10}$.

Conclusion

In this article, we have explored the process of adding fractions with unlike denominators, specifically addressing the problem $\frac{4}{5}+(-\frac{7}{10})$. We began by identifying the need for a common denominator and then found the least common denominator (LCD), which was 10. We converted the fractions to equivalent fractions with the LCD as the denominator, resulting in $\frac{8}{10}$ and $-\frac{7}{10}$. Next, we added the numerators while keeping the denominator the same, which gave us $\frac{1}{10}$. Finally, we checked if the result could be simplified, and in this case, $\frac{1}{10}$ was already in its simplest form. Mastering the addition of fractions is a fundamental skill in mathematics, and this step-by-step guide provides a clear understanding of the process. By following these steps, you can confidently solve similar problems and build a strong foundation in fraction arithmetic. The ability to add fractions is not only crucial for academic success but also for various real-life applications, making it an invaluable skill to possess.