Adding Fractions Step-by-Step Solution For 5/7 + 4/21

In the realm of mathematics, fraction addition stands as a fundamental concept, serving as a cornerstone for more advanced topics. Mastering the art of adding fractions is crucial for various mathematical applications, from everyday calculations to complex problem-solving. This article delves into a step-by-step guide on how to solve the fraction addition problem $\frac{5}{7} + \frac{4}{21}$, providing clarity and a comprehensive understanding of the underlying principles. Fraction addition is a crucial skill in mathematics, providing the foundation for more advanced concepts. Understanding how to add fractions accurately and efficiently is essential for success in algebra, calculus, and beyond. This article aims to provide a comprehensive guide to mastering fraction addition, focusing on solving the specific problem of $\frac{5}{7} + \frac{4}{21}$. By breaking down the process into manageable steps and providing clear explanations, we will empower you to confidently tackle similar problems. Whether you are a student learning the basics or someone looking to refresh your skills, this guide will offer valuable insights and strategies for mastering fraction addition.

Before diving into the addition process, it's crucial to grasp the essence of fractions. A fraction represents a part of a whole, comprising two key components: the numerator and the denominator. The numerator signifies the number of parts we're considering, while the denominator indicates the total number of equal parts that make up the whole. To effectively add fractions, especially those with different denominators, a clear understanding of these components is essential. Mastering the concept of equivalent fractions is also paramount, as it allows us to manipulate fractions without altering their inherent value. This understanding lays the groundwork for successfully navigating the complexities of fraction addition. Fractions are a fundamental concept in mathematics, representing a part of a whole. A fraction consists of two main components the numerator and the denominator. The numerator indicates the number of parts we are considering, while the denominator represents the total number of equal parts that make up the whole. For example, in the fraction $\frac{5}{7}$, the numerator is 5, and the denominator is 7. This means we are considering 5 parts out of a total of 7 equal parts. A solid understanding of fractions is crucial for performing operations such as addition, subtraction, multiplication, and division. When adding fractions, it is essential to have a clear grasp of what each component represents. This understanding allows us to manipulate fractions correctly and arrive at the correct answer. In the context of our problem, $\frac{5}{7} + \frac{4}{21}$, understanding the individual fractions is the first step towards finding the solution.

Equivalent Fractions

Equivalent fractions are fractions that, despite having different numerators and denominators, represent the same value. For instance, $\frac{1}{2}$ and $\frac{2}{4}$ are equivalent fractions. Recognizing and creating equivalent fractions is a fundamental skill in fraction arithmetic. It allows us to manipulate fractions to have common denominators, which is essential for addition and subtraction. The process of finding equivalent fractions involves multiplying or dividing both the numerator and the denominator by the same non-zero number. This ensures that the value of the fraction remains unchanged while the representation is altered. In the context of adding fractions with different denominators, finding equivalent fractions is a crucial step. It enables us to rewrite the fractions with a common denominator, making the addition process straightforward. Mastering this skill is essential for effectively solving fraction addition problems. Equivalent fractions are fractions that have the same value but different numerators and denominators. For example, $\frac{1}{2}$, $\frac{2}{4}$, and $\frac{4}{8}$ are all equivalent fractions. The ability to identify and create equivalent fractions is crucial for adding and subtracting fractions, especially when they have different denominators. The fundamental principle behind equivalent fractions is that multiplying or dividing both the numerator and the denominator by the same non-zero number does not change the value of the fraction. This is because you are essentially multiplying or dividing the fraction by 1. For example, to find an equivalent fraction of $\frac{1}{2}$, you can multiply both the numerator and the denominator by 2: $\frac{1 \times 2}{2 \times 2} = \frac{2}{4}$. Similarly, you can multiply by any other non-zero number to obtain another equivalent fraction. In the context of our problem, $\frac{5}{7} + \frac{4}{21}$, we will need to find equivalent fractions to add these fractions together. The concept of equivalent fractions is a cornerstone of fraction arithmetic and is essential for mastering fraction addition and subtraction.

To add fractions, they must share a common denominator. This common denominator acts as the unifying foundation, allowing us to combine the numerators meaningfully. If fractions have different denominators, we must first find a common denominator before proceeding with the addition. The most efficient common denominator to use is the least common multiple (LCM) of the original denominators. The LCM is the smallest number that is a multiple of both denominators. Finding a common denominator is a critical step in adding fractions. It allows us to express the fractions with the same unit size, making it possible to combine them. This process involves identifying a common multiple of the denominators, which will serve as the new denominator for both fractions. The least common multiple (LCM) is often the most convenient choice, as it results in the smallest possible numbers, simplifying the calculations. However, any common multiple will work, although using a larger multiple may result in the need for simplification at the end. The goal is to rewrite the fractions with the common denominator without changing their values, which is achieved by multiplying both the numerator and the denominator by the same factor. This step is essential for accurately adding fractions and is a fundamental skill in arithmetic.

Least Common Multiple (LCM)

The least common multiple (LCM) is the smallest multiple shared by two or more numbers. In the context of fraction addition, the LCM of the denominators serves as the ideal common denominator. Finding the LCM can be achieved through various methods, such as listing multiples or prime factorization. Using the LCM as the common denominator simplifies the subsequent addition process and often reduces the need for simplification at the end. The least common multiple (LCM) is a fundamental concept in number theory and plays a crucial role in fraction arithmetic. The LCM of two or more numbers is the smallest positive integer that is divisible by all the given numbers. In the context of adding fractions, the LCM of the denominators is used as the common denominator. This allows us to rewrite the fractions with a common base, making the addition process straightforward. There are several methods to find the LCM, including listing multiples, prime factorization, and using the formula LCM(a, b) = |a \times b| / GCD(a, b), where GCD is the greatest common divisor. Listing multiples involves writing out the multiples of each number until a common multiple is found. Prime factorization involves breaking down each number into its prime factors and then taking the highest power of each prime factor that appears in any of the factorizations. The choice of method depends on the numbers involved and personal preference. Using the LCM as the common denominator simplifies the addition process and often reduces the need for simplification at the end, as it results in the smallest possible common denominator. In the context of our problem, $\frac{5}{7} + \frac{4}{21}$, finding the LCM of 7 and 21 is a crucial step towards solving the problem.

Finding the LCM of 7 and 21

To find the LCM of 7 and 21, we can list the multiples of each number: Multiples of 7: 7, 14, 21, 28,... Multiples of 21: 21, 42, 63,... The smallest multiple that appears in both lists is 21. Therefore, the LCM of 7 and 21 is 21. This means that 21 is the ideal common denominator for our fractions. Another method to find the LCM is to use prime factorization. The prime factorization of 7 is simply 7, as 7 is a prime number. The prime factorization of 21 is $3 \times 7$. To find the LCM, we take the highest power of each prime factor that appears in either factorization: The highest power of 3 is $3^1$, and the highest power of 7 is $7^1$. Thus, the LCM is $3 \times 7 = 21$. This confirms our earlier finding that the LCM of 7 and 21 is 21. Using the LCM as the common denominator will simplify our fraction addition problem. Finding the LCM of 7 and 21 is a critical step in solving the problem $\frac{5}{7} + \frac{4}{21}$. The LCM is the smallest number that both 7 and 21 divide into evenly. There are several methods to find the LCM, including listing multiples and prime factorization. One way to find the LCM is to list the multiples of each number until a common multiple is found. The multiples of 7 are 7, 14, 21, 28, and so on. The multiples of 21 are 21, 42, 63, and so on. The smallest multiple that appears in both lists is 21. Therefore, the LCM of 7 and 21 is 21. Another method to find the LCM is to use prime factorization. The prime factorization of 7 is simply 7, as 7 is a prime number. The prime factorization of 21 is $3 \times 7$. To find the LCM, we take the highest power of each prime factor that appears in either factorization. In this case, the prime factors are 3 and 7. The highest power of 3 is $3^1$, and the highest power of 7 is $7^1$. Thus, the LCM is $3 \times 7 = 21$. This confirms our earlier finding that the LCM of 7 and 21 is 21. Using the LCM as the common denominator will simplify our fraction addition problem, making the subsequent steps easier to manage. In our case, the LCM of 7 and 21 is 21, which means we will use 21 as the common denominator for adding the fractions.

Once we've identified the common denominator, the next step is to convert each fraction to an equivalent fraction with this denominator. This involves multiplying both the numerator and the denominator of each fraction by a factor that will transform the original denominator into the common denominator. This process ensures that the value of the fraction remains unchanged while allowing us to add them together. Converting fractions to a common denominator is a crucial step in adding fractions with different denominators. This process ensures that we are adding fractions that represent parts of the same whole, making the addition meaningful. To convert a fraction, we need to multiply both the numerator and the denominator by the same factor. This factor is determined by dividing the common denominator by the original denominator. The result is the number we need to multiply both the numerator and the denominator by. This step is essential for ensuring that the fractions are equivalent, meaning they represent the same value but have different numerators and denominators. The process of converting fractions to a common denominator is a fundamental skill in fraction arithmetic and is necessary for accurately adding and subtracting fractions.

Converting 5/7 to a Fraction with a Denominator of 21

To convert $\frac{5}{7}$ to an equivalent fraction with a denominator of 21, we need to determine the factor by which we must multiply the denominator 7 to obtain 21. Dividing 21 by 7 gives us 3. Therefore, we multiply both the numerator and the denominator of $\frac{5}{7}$ by 3: $\frac{5 \times 3}{7 \times 3} = \frac{15}{21}$ Now, $\frac{5}{7}$ is equivalent to $\frac{15}{21}$, and we have successfully converted it to a fraction with the desired denominator. This conversion is crucial for adding it to the other fraction, $\frac{4}{21}$, which already has the common denominator. The process of converting $\frac{5}{7}$ to a fraction with a denominator of 21 is a key step in solving our problem. We need to find a number that, when multiplied by 7, gives us 21. This number is 3, since $7 \times 3 = 21$. To maintain the fraction's value, we must multiply both the numerator and the denominator by the same number. Therefore, we multiply both the numerator 5 and the denominator 7 by 3: $\frac{5 \times 3}{7 \times 3}$ This gives us $\frac{15}{21}$, which is equivalent to $\frac{5}{7}$. We have now successfully converted $\frac{5}{7}$ to an equivalent fraction with a denominator of 21. This conversion allows us to add it to the other fraction in our problem, $\frac{4}{21}$, which already has the common denominator. This step is essential for accurately adding the two fractions together.

Now that both fractions have the same denominator, we can proceed with the addition. To add fractions with a common denominator, we simply add the numerators and keep the denominator the same. This process combines the parts represented by the fractions, giving us the sum. Adding fractions with a common denominator is a straightforward process. Once the fractions have the same denominator, we can simply add the numerators together and keep the denominator the same. This is because the denominator represents the size of the parts, and adding the numerators tells us how many of those parts we have in total. The result is a new fraction that represents the sum of the original fractions. This step is a fundamental operation in fraction arithmetic and is essential for solving many mathematical problems. The process is intuitive and easy to apply once the fractions have a common denominator.

Adding 15/21 and 4/21

To add $\frac{15}{21}$ and $\frac{4}{21}$, we add the numerators (15 and 4) and keep the denominator (21) the same: $\frac{15 + 4}{21} = \frac{19}{21}$ Therefore, the sum of $\frac{15}{21}$ and $\frac{4}{21}$ is $\frac{19}{21}$. This result represents the combined value of the two fractions, expressed as a single fraction with the common denominator. This is the final step in adding the fractions, providing us with the solution to our problem. Adding $\frac{15}{21}$ and $\frac{4}{21}$ is the final step in solving our problem. Since the fractions have a common denominator, we simply add the numerators and keep the denominator the same. The numerators are 15 and 4, so we add them together: $15 + 4 = 19$ The denominator remains 21. Therefore, the sum of $\frac{15}{21}$ and $\frac{4}{21}$ is $\frac{19}{21}$. This means that $\frac{5}{7} + \frac{4}{21} = \frac{19}{21}$. The result, $\frac{19}{21}$, is a proper fraction, meaning the numerator is less than the denominator. This fraction cannot be simplified further, as 19 is a prime number and does not share any common factors with 21 other than 1. Thus, $\frac{19}{21}$ is the final answer to our problem.

After adding fractions, it's essential to check if the resulting fraction can be simplified. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. To simplify a fraction, we divide both the numerator and the denominator by their greatest common divisor (GCD). However, in our case, $\frac{19}{21}$ is already in its simplest form because 19 is a prime number and does not share any common factors with 21 other than 1. Simplifying fractions is an important step in fraction arithmetic. It ensures that the answer is expressed in its most concise form. A fraction is considered simplified when the numerator and the denominator have no common factors other than 1. This means that the fraction cannot be reduced further. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator and then divide both by the GCD. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. If the GCD is 1, then the fraction is already in its simplest form. Simplifying fractions makes them easier to understand and compare. It also ensures that the answer is expressed in the most standard form. While not always necessary, simplifying fractions is a good practice to follow.

In conclusion, solving the fraction addition problem $\frac{5}{7} + \frac{4}{21}$ involves several key steps: understanding fractions, finding a common denominator (LCM), converting fractions to equivalent forms, adding the numerators, and simplifying the result if necessary. By following these steps methodically, we arrive at the solution: $\frac{19}{21}$. This process not only provides the answer but also reinforces the fundamental principles of fraction arithmetic, which are essential for mathematical proficiency. Mastering these concepts enables confident navigation of more complex mathematical challenges. Mastering fraction addition is a crucial skill in mathematics. It lays the foundation for more advanced concepts and is essential for solving a wide range of problems. In this article, we have demonstrated a step-by-step approach to solving the fraction addition problem $\frac{5}{7} + \frac{4}{21}$. We began by understanding the concept of fractions and equivalent fractions. Then, we found the least common multiple (LCM) of the denominators, which is 21. Next, we converted $\frac{5}{7}$ to an equivalent fraction with a denominator of 21, resulting in $\frac{15}{21}$. We then added the fractions, $\frac{15}{21} + \frac{4}{21}$, which gave us $\frac{19}{21}$. Finally, we checked if the fraction could be simplified, but since 19 is a prime number and does not share any common factors with 21, the fraction is already in its simplest form. Therefore, the solution to the problem $\frac{5}{7} + \frac{4}{21}$ is $\frac{19}{21}$. This step-by-step approach can be applied to any fraction addition problem, making it a valuable tool for mastering fraction arithmetic. By understanding the underlying principles and practicing these steps, you can confidently tackle fraction addition problems of any complexity. The ability to add fractions accurately and efficiently is a fundamental skill that will serve you well in mathematics and beyond.

The correct answer is b. 19/21.