Subtracting Fractions A Step-by-Step Guide To Solving 5/7 - 9/11

Introduction to Subtracting Fractions

Fraction subtraction is a fundamental concept in mathematics, and mastering it is crucial for various mathematical operations and real-world applications. In this article, we will delve into the process of subtracting fractions, specifically addressing the problem ${ \frac{5}{7} - \frac{9}{11} }$. We will break down each step, providing a clear and comprehensive explanation to ensure a solid understanding of the underlying principles. Understanding fraction subtraction is not just about following steps; it’s about grasping the concept of dealing with parts of a whole. The ability to subtract fractions efficiently and accurately is essential not only for academic success but also for practical situations involving measurements, proportions, and resource allocation. This article aims to make the process accessible and straightforward, enabling readers to confidently tackle fraction subtraction problems.

Before diving into the specific problem, it’s important to understand the basic principles of fractions. A fraction represents a part of a whole and is written in the form ${ \frac{a}{b} }$, where ${ a }$ is the numerator (the top number) and ${ b }$ is the denominator (the bottom number). The denominator indicates how many equal parts the whole is divided into, and the numerator indicates how many of those parts are being considered. For instance, in the fraction ${ \frac{5}{7} }$, the whole is divided into 7 equal parts, and we are considering 5 of those parts. Similarly, in ${ \frac{9}{11} }$, the whole is divided into 11 equal parts, and we are considering 9 of them. To subtract fractions, it’s imperative that they have the same denominator. This is because we can only directly add or subtract quantities that are expressed in the same units. Think of it like trying to subtract apples from oranges; you first need a common unit, such as fruit. In fractions, the common unit is the denominator. If the fractions have different denominators, we need to find a common denominator before we can perform the subtraction. This involves finding the least common multiple (LCM) of the denominators, which is the smallest number that both denominators can divide into evenly. Finding a common denominator is a critical step because it allows us to express both fractions in terms of the same-sized parts, making the subtraction straightforward. Without a common denominator, the subtraction would be like comparing different-sized slices of a pie, making it impossible to determine the exact difference.

Step-by-Step Solution: ${ \frac{5}{7} - \frac{9}{11} }$

1. Finding the Least Common Denominator (LCD)

The first step in subtracting the fractions ${ \frac{5}{7} }$ and ${ \frac{9}{11} }$ is to find the least common denominator (LCD). The least common denominator is the smallest multiple that both denominators, 7 and 11, share. To find the LCD, we can list the multiples of each denominator and identify the smallest multiple that appears in both lists. Multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, ... Multiples of 11 are: 11, 22, 33, 44, 55, 66, 77, 88, 99, ... We can see that the smallest multiple that both 7 and 11 share is 77. Therefore, the least common denominator (LCD) for these two fractions is 77. Alternatively, since 7 and 11 are both prime numbers (numbers that have only two distinct positive divisors: 1 and themselves), their least common multiple is simply their product. Thus, ${ 7 \times 11 = 77 }$, which confirms that the LCD is 77. Understanding the concept of the least common denominator is essential because it allows us to express fractions in terms of the same-sized parts. This is crucial for performing addition and subtraction operations accurately. Without a common denominator, we would be trying to combine quantities that are measured in different units, which would not yield a meaningful result. The LCD serves as the common unit that allows us to compare and combine fractions effectively. By finding the LCD, we ensure that we are working with fractions that represent parts of the same whole, making the subtraction process both logical and straightforward.

2. Converting Fractions to Equivalent Fractions with the LCD

Once we have determined the least common denominator (LCD) to be 77, the next step is to convert both fractions, ${ \frac{5}{7} }$ and ${ \frac{9}{11} }$, into equivalent fractions with the LCD as their new denominator. An equivalent fraction is a fraction that represents the same value but has a different numerator and denominator. To convert ${ \frac{5}{7} }$ to an equivalent fraction with a denominator of 77, we need to determine what number we must multiply the original denominator (7) by to get the new denominator (77). In this case, ${ 7 \times 11 = 77 }$. Therefore, we multiply both the numerator and the denominator of ${ \frac{5}{7} }$ by 11: ${ \frac{5 \times 11}{7 \times 11} = \frac{55}{77} }$. Similarly, to convert ${ \frac{9}{11} }$ to an equivalent fraction with a denominator of 77, we need to determine what number we must multiply the original denominator (11) by to get the new denominator (77). In this case, ${ 11 \times 7 = 77 }$. Therefore, we multiply both the numerator and the denominator of ${ \frac{9}{11} }$ by 7: ${ \frac{9 \times 7}{11 \times 7} = \frac{63}{77} }$. Now, we have successfully converted both fractions into equivalent fractions with the same denominator: ${ \frac{5}{7} }$ is equivalent to ${ \frac{55}{77} }$, and ${ \frac{9}{11} }$ is equivalent to ${ \frac{63}{77} }$. This conversion is crucial because it allows us to perform the subtraction operation directly, as we are now dealing with fractions that represent parts of the same whole. By expressing the fractions with a common denominator, we ensure that we are comparing and combining equal-sized parts, which is the fundamental principle of fraction arithmetic. The ability to convert fractions into equivalent forms is a key skill in working with fractions and is essential for both addition and subtraction operations.

3. Subtracting the Fractions

After converting the fractions to equivalent forms with the least common denominator, we can now proceed with the subtraction. We have the fractions ${ \frac{55}{77} }$ and ${ \frac{63}{77} }$, which are equivalent to ${ \frac{5}{7} }$ and ${ \frac{9}{11} }$, respectively. The subtraction problem is now: ${ \frac{55}{77} - \frac{63}{77} }$. To subtract fractions with a common denominator, we subtract the numerators and keep the denominator the same. In this case, we subtract 63 from 55: ${ 55 - 63 = -8 }$. So, the result of the subtraction is ${ \frac{-8}{77} }$. This means that the difference between ${ \frac{55}{77} }$ and ${ \frac{63}{77} }$ is ${ \frac{-8}{77} }$. The negative sign indicates that the second fraction, ${ \frac{63}{77} }$, is larger than the first fraction, ${ \frac{55}{77} }$. Therefore, the result of subtracting ${ \frac{9}{11} }$ from ${ \frac{5}{7} }$ is a negative fraction. The subtraction of fractions with a common denominator is a straightforward process, but it’s essential to pay attention to the signs of the numerators. In this case, subtracting a larger number (63) from a smaller number (55) results in a negative number (-8). The negative sign is crucial because it indicates the direction of the difference. Understanding how to subtract fractions with common denominators is a fundamental skill in arithmetic and is essential for solving more complex mathematical problems involving fractions. The ability to accurately perform this operation ensures that we can correctly determine the difference between fractional quantities, which is a key concept in various applications, including measurements, proportions, and problem-solving.

4. Simplifying the Result (if necessary)

After performing the subtraction, we obtained the result ${ \frac{-8}{77} }$. The next step is to determine if this fraction can be simplified further. To simplify a fraction, we look for common factors between the numerator and the denominator and divide both by their greatest common factor (GCF). The numerator is -8, and its factors are: -1, 1, -2, 2, -4, 4, -8, and 8. The denominator is 77, and its factors are: 1, 7, 11, and 77. We need to identify the greatest common factor (GCF) of -8 and 77. The common factors between -8 and 77 are 1 and -1. Since the greatest common factor is 1 (or -1), the fraction ${ \frac{-8}{77} }$ is already in its simplest form. This means that there are no other common factors between the numerator and the denominator that we can divide by to reduce the fraction further. In some cases, simplifying a fraction is necessary to express the result in its most reduced form, which makes it easier to understand and compare with other fractions. For example, if we had obtained a fraction like ${ \frac{4}{8} }$, we would simplify it by dividing both the numerator and the denominator by their greatest common factor, which is 4, resulting in ${ \frac{1}{2} }$. However, in our case, since the fraction ${ \frac{-8}{77} }$ cannot be simplified further, it remains as the final answer. Understanding how to simplify fractions is an important skill in mathematics, as it allows us to express fractions in their most concise and manageable form. This is particularly useful in various applications, such as when working with ratios, proportions, and algebraic expressions. The ability to determine if a fraction can be simplified and to perform the simplification accurately is a key aspect of fraction arithmetic.

Final Answer

Therefore, the final answer to the subtraction problem ${ \frac{5}{7} - \frac{9}{11} }$ is ${ \frac{-8}{77} }$. This result represents the difference between the two fractions and indicates that ${ \frac{9}{11} }$ is ${ \frac{8}{77} }$ greater than ${ \frac{5}{7} }$. In summary, we found the least common denominator (LCD) of 7 and 11, which is 77. We then converted both fractions to equivalent fractions with the LCD as the denominator, resulting in ${ \frac{55}{77} }$ and ${ \frac{63}{77} }$. Next, we subtracted the numerators, giving us ${ \frac{55 - 63}{77} = \frac{-8}{77} }$. Finally, we checked if the fraction could be simplified, but since -8 and 77 have no common factors other than 1, the fraction is already in its simplest form. The negative sign in the result indicates that the second fraction ${ \frac{9}{11} }$ is larger than the first fraction ${ \frac{5}{7} }$. The ability to accurately subtract fractions is a crucial skill in mathematics and has numerous applications in real-world scenarios. Whether it's calculating measurements, dealing with proportions, or solving algebraic equations, a solid understanding of fraction subtraction is essential. This comprehensive solution provides a step-by-step guide to subtracting fractions, emphasizing the importance of finding the LCD, converting fractions to equivalent forms, and simplifying the result. By mastering these steps, individuals can confidently tackle fraction subtraction problems and apply this knowledge to various mathematical contexts.