Subtracting Fractions A Step By Step Guide To 7/20 - 7/8

Subtracting fractions might seem daunting at first, but with a clear understanding of the fundamental principles, it becomes a manageable task. In this article, we will delve into the process of subtracting fractions, specifically addressing the problem of ${\frac{7}{20}-\frac{7}{8}}$. We will break down each step, ensuring clarity and comprehension for anyone looking to master this essential mathematical skill. Fractions, as you know, represent parts of a whole, and subtracting fractions involves finding the difference between these parts. Before we dive into the specifics of our problem, let’s revisit some key concepts that form the foundation of fraction subtraction.

One of the most critical aspects of subtracting fractions is the concept of a common denominator. A common denominator is a shared multiple of the denominators of the fractions involved. Why is this important? Because we can only directly add or subtract fractions when they are expressed with the same denominator. Think of it like trying to compare apples and oranges; you need to convert them into a common unit, like “fruits,” to make a meaningful comparison. Similarly, when subtracting fractions, we need a common denominator to ensure we are dealing with comparable parts of a whole.

The process of finding a common denominator typically involves identifying the least common multiple (LCM) of the denominators. The LCM is the smallest number that is a multiple of both denominators. Once we have the LCM, we can convert each fraction into an equivalent fraction with the LCM as the new denominator. This conversion involves multiplying both the numerator and the denominator of each fraction by a factor that will result in the LCM as the denominator. It’s crucial to multiply both the numerator and the denominator to maintain the fraction's value; we are simply expressing the same fraction in a different form.

Another key concept to grasp is the idea of equivalent fractions. Equivalent fractions are fractions that represent the same value, even though they have different numerators and denominators. For example, ${\frac{1}{2}}$ and ${\frac{2}{4}}$ are equivalent fractions. When we convert fractions to have a common denominator, we are essentially creating equivalent fractions that allow us to perform the subtraction. Understanding equivalent fractions is fundamental to performing accurate calculations and ensuring that the result represents the true difference between the original fractions.

Finally, it’s important to remember the basic structure of a fraction: the numerator (the top number) represents the number of parts we have, and the denominator (the bottom number) represents the total number of parts that make up the whole. When subtracting fractions, we are essentially finding the difference in the number of parts, while the denominator remains the same (once we have a common denominator). This visual understanding can help solidify the concept and make the process more intuitive.

In order to subtract ${\frac{7}{20}}$ and ${\frac{7}{8}}$, the crucial first step involves identifying the common denominator. As we discussed earlier, a common denominator is a shared multiple of the denominators of the fractions, which in this case are 20 and 8. The most efficient way to find the common denominator is to determine the least common multiple (LCM) of these two numbers. The least common multiple is the smallest number that both 20 and 8 can divide into evenly. There are a couple of methods we can use to find the LCM: listing multiples and prime factorization.

Let’s start by listing the multiples of 20 and 8. Multiples of 20 are: 20, 40, 60, 80, 100, and so on. Multiples of 8 are: 8, 16, 24, 32, 40, 48, and so on. By comparing these lists, we can see that the smallest multiple they have in common is 40. Therefore, the least common multiple (LCM) of 20 and 8 is 40. This means that 40 will be our common denominator when we subtract fractions.

Another method to find the LCM is through prime factorization. Prime factorization involves breaking down each number into its prime factors. Prime factors are numbers that are only divisible by 1 and themselves. The prime factorization of 20 is 2 x 2 x 5 (or ${2^2 imes 5}$), and the prime factorization of 8 is 2 x 2 x 2 (or ${2^3}$). To find the LCM using prime factors, we take the highest power of each prime factor that appears in either factorization and multiply them together. In this case, the prime factors are 2 and 5. The highest power of 2 is ${2^3}$ (from the factorization of 8), and the highest power of 5 is ${5^1}$ (from the factorization of 20). Multiplying these together, we get ${2^3 imes 5 = 8 imes 5 = 40}$. Again, we find that the LCM is 40.

Now that we have established that 40 is the common denominator, we need to convert both fractions, ${\frac{7}{20}}$ and ${\frac{7}{8}}$, into equivalent fractions with a denominator of 40. To do this, we need to determine what factor we must multiply the denominator of each fraction by to get 40. For ${\frac{7}{20}}$, we need to multiply the denominator 20 by 2 to get 40. To maintain the value of the fraction, we must also multiply the numerator by the same factor. So, we multiply both the numerator and the denominator of ${\frac{7}{20}}$ by 2, which gives us ${\frac{7 imes 2}{20 imes 2} = \frac{14}{40}}$. For ${\frac{7}{8}}$, we need to multiply the denominator 8 by 5 to get 40. Similarly, we multiply the numerator by 5 as well, giving us ${\frac{7 imes 5}{8 imes 5} = \frac{35}{40}}$.

Now that we have converted both fractions to have a common denominator of 40, we have equivalent fractions ${\frac{14}{40}}$ and ${\frac{35}{40}}$. We are now ready to perform the subtraction. The process of finding the common denominator is a crucial step in subtracting fractions, as it allows us to work with comparable parts of a whole. Understanding how to find the LCM and convert fractions to equivalent fractions is essential for mastering fraction subtraction.

Having identified 40 as the common denominator for ${\frac{7}{20}}$ and ${\frac{7}{8}}$, the next critical step is to convert each fraction into its equivalent fraction with the denominator of 40. This conversion is vital because, as discussed earlier, fractions can only be directly added or subtracted when they share a common denominator. The process of converting fractions to equivalent fractions involves multiplying both the numerator and the denominator by the same factor, thereby maintaining the fraction's value while changing its form.

Let's begin with the fraction ${\frac{7}{20}}$. To convert this fraction to an equivalent form with a denominator of 40, we need to determine what factor to multiply the current denominator, 20, by to obtain 40. By simple division, we find that 40 ÷ 20 = 2. This means we need to multiply the denominator 20 by 2 to get 40. To keep the fraction equivalent, we must also multiply the numerator, 7, by the same factor, 2. Thus, we perform the multiplication: 7 × 2 = 14. So, the equivalent fraction of ${\frac{7}{20}}$ with a denominator of 40 is ${\frac{14}{40}}$.

Now, let's turn our attention to the fraction ${\frac{7}{8}}$. We follow a similar process to convert this fraction to an equivalent form with a denominator of 40. We need to determine what factor to multiply the current denominator, 8, by to obtain 40. Again, we can use division: 40 ÷ 8 = 5. This indicates that we need to multiply the denominator 8 by 5 to get 40. To maintain the fraction’s value, we must also multiply the numerator, 7, by the same factor, 5. Performing this multiplication, we get 7 × 5 = 35. Consequently, the equivalent fraction of ${\frac{7}{8}}$ with a denominator of 40 is ${\frac{35}{40}}$.

At this stage, we have successfully converted both original fractions into their equivalent fractions with a common denominator of 40. Specifically, ${\frac{7}{20}}$ has been transformed into ${\frac{14}{40}}$, and ${\frac{7}{8}}$ has been transformed into ${\frac{35}{40}}$. These equivalent fractions are crucial because they now allow us to perform the subtraction operation directly. By having the same denominator, we can focus solely on the numerators to find the difference between the two fractions.

Understanding how to convert fractions into equivalent fractions is a fundamental skill in fraction arithmetic. This process is not only essential for addition and subtraction but also plays a key role in comparing fractions and simplifying them. The ability to identify the appropriate factor to multiply both the numerator and denominator by is a critical component of this skill. It's important to remember that multiplying both the numerator and denominator by the same factor is equivalent to multiplying the fraction by 1, which does not change its value but only its representation.

With the fractions now expressed in their equivalent forms with a common denominator, ${\frac{14}{40}}$ and ${\frac{35}{40}}$, we are ready to perform the subtraction. Subtracting fractions with a common denominator is a straightforward process: we simply subtract the numerators while keeping the denominator the same. This is because when fractions have the same denominator, they are divided into the same number of parts, making it easy to find the difference in the number of parts.

In our case, we need to subtract ${\frac{35}{40}}$ from ${\frac{14}{40}}$. This means we subtract the numerators: 14 - 35. It's important to note that we are subtracting a larger number (35) from a smaller number (14), which will result in a negative value. The subtraction 14 - 35 equals -21. So, the numerator of our result is -21, and the denominator remains 40, giving us the fraction ${\frac{-21}{40}}$.

The fraction ${\frac{-21}{40}}$ represents the difference between the two original fractions. The negative sign indicates that the second fraction, ${\frac{7}{8}}$, is larger than the first fraction, ${\frac{7}{20}}$. This is a crucial understanding when subtracting fractions: the order matters, and the sign of the result reflects the relative sizes of the fractions being subtracted. If we had subtracted ${\frac{14}{40}}$ from ${\frac{35}{40}}$, we would have obtained the positive result ${\frac{21}{40}}$.

It's worth emphasizing the importance of paying attention to the signs when subtracting fractions, especially when dealing with negative results. A negative fraction simply means that the value is less than zero, and it accurately represents the difference when the larger fraction is subtracted from the smaller one. In practical terms, this could represent a debt, a decrease, or a position below a reference point, depending on the context of the problem.

At this stage, we have successfully performed the subtraction and obtained the fraction ${\frac{-21}{40}}$. However, our task is not yet complete. The final step in solving the problem is to reduce the fraction to its simplest, or reduced, form. Reducing a fraction involves dividing both the numerator and the denominator by their greatest common factor (GCF) to obtain an equivalent fraction with the smallest possible numerator and denominator. This ensures that our answer is expressed in the most concise and easily understandable way.

Having performed the subtraction and obtained the fraction ${\frac{-21}{40}}$, the final step is to reduce this fraction to its simplest terms. Reducing a fraction means expressing it in its lowest terms, where the numerator and the denominator have no common factors other than 1. This process involves finding the greatest common factor (GCF) of the numerator and the denominator and dividing both by it. The greatest common factor is the largest number that can divide both the numerator and the denominator evenly.

In our case, we need to find the GCF of 21 and 40. One way to find the GCF is to list the factors of each number and identify the largest factor they have in common. The factors of 21 are 1, 3, 7, and 21. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. By comparing these lists, we can see that the only common factor is 1. This means that 21 and 40 are relatively prime, and the fraction ${\frac{-21}{40}}$ is already in its simplest form.

Another method to find the GCF is through prime factorization. We already discussed prime factorization earlier when finding the least common multiple (LCM). The prime factorization of 21 is 3 x 7, and the prime factorization of 40 is 2 x 2 x 2 x 5 (or ${2^3 imes 5}$). By comparing the prime factors, we can see that they have no prime factors in common. This confirms that their GCF is 1, and the fraction cannot be reduced further.

Since the greatest common factor of 21 and 40 is 1, the fraction ${\frac{-21}{40}}$ is already in its simplest form. This means that there are no common factors that we can divide both the numerator and the denominator by to obtain a smaller equivalent fraction. Therefore, the reduced form of the fraction is ${\frac{-21}{40}}$.

In conclusion, the result of subtracting fractions ${\frac{7}{20}-\frac{7}{8}}$ is ${\frac{-21}{40}}$. We arrived at this answer by first finding the common denominator of 20 and 8, which is 40. Then, we converted both fractions to equivalent fractions with the common denominator, resulting in ${\frac{14}{40}}$ and ${\frac{35}{40}}$. We then performed the subtraction, obtaining ${\frac{-21}{40}}$. Finally, we checked if the fraction could be reduced to simpler terms, and in this case, it was already in its simplest form. Understanding the process of reducing fractions is essential for expressing answers in the most concise and accurate way.

In summary, to subtract the fractions ${\frac{7}{20}}$ and ${\frac{7}{8}}$, we followed a series of essential steps. First, we identified the common denominator, which is the least common multiple (LCM) of 20 and 8, and found it to be 40. This step is crucial because fractions must have a common denominator before they can be subtracted. Next, we converted each fraction into an equivalent fraction with the common denominator of 40. This involved multiplying both the numerator and the denominator of each fraction by the appropriate factor to achieve the desired denominator. We converted ${\frac{7}{20}}$ to ${\frac{14}{40}}$ and ${\frac{7}{8}}$ to ${\frac{35}{40}}$.

Once we had the equivalent fractions with a common denominator, we performed the subtraction. Subtracting fractions with a common denominator is a straightforward process of subtracting the numerators while keeping the denominator the same. We subtracted 35 from 14, resulting in -21, and kept the denominator as 40, giving us the fraction ${\frac{-21}{40}}$. This negative result indicates that the second fraction, ${\frac{7}{8}}$, is larger than the first fraction, ${\frac{7}{20}}$.

Finally, we examined the resulting fraction, ${\frac{-21}{40}}$, to determine if it could be reduced to its simplest terms. Reducing a fraction involves finding the greatest common factor (GCF) of the numerator and the denominator and dividing both by it. In this case, the GCF of 21 and 40 is 1, meaning the fraction is already in its simplest form. Therefore, no further reduction is possible.

Thus, the final answer to the subtraction problem ${\frac{7}{20}-\frac{7}{8}}$ is ${\frac{-21}{40}}$. This result represents the difference between the two fractions in its most reduced form, providing a clear and concise solution to the problem. The process of subtracting fractions involves several key steps, including finding the common denominator, converting to equivalent fractions, performing the subtraction, and reducing the result to simplest terms. Mastering these steps is essential for proficiency in fraction arithmetic.

Final Answer: ${\boxed{-\frac{21}{40}}}$