Calculating The Value Of The Expression X = 5/6 - 7/8 + 11/12
Understanding the Problem
In this mathematical problem, we aim to calculate the value of the expression: x = 5/6 - 7/8 + 11/12. This involves performing arithmetic operations on fractions, specifically subtraction and addition. To accurately solve this, we need to find a common denominator for the fractions, which will allow us to combine them seamlessly. The fundamental concept here is the addition and subtraction of fractions, which requires a shared base or denominator. Without a common denominator, we cannot directly add or subtract fractions because they represent parts of different wholes. Finding the least common multiple (LCM) of the denominators is the key to simplifying the process and avoiding unnecessarily large numbers. Once we have the common denominator, we can adjust the numerators accordingly and perform the arithmetic operations. This process is crucial in various mathematical contexts, including algebra, calculus, and real-world applications involving proportions and ratios. By mastering fraction arithmetic, we build a strong foundation for more advanced mathematical concepts and problem-solving techniques. Furthermore, understanding how to manipulate fractions efficiently is not only helpful in academic settings but also in practical situations, such as cooking, measuring, and financial calculations. Therefore, a solid grasp of these basic arithmetic operations is essential for mathematical proficiency and everyday applications.
To begin, let's break down the steps involved in solving this expression and delve into the underlying principles of fraction arithmetic. First, we need to identify the denominators of the fractions in the expression. The denominators are 6, 8, and 12. Next, we need to find the least common multiple (LCM) of these denominators. The LCM is the smallest number that is a multiple of each denominator. Finding the LCM is a critical step because it allows us to rewrite the fractions with a common denominator, making it possible to add and subtract them. There are several methods to find the LCM, including listing multiples, prime factorization, and using the greatest common divisor (GCD). For the denominators 6, 8, and 12, we can list the multiples of each number and identify the smallest multiple they have in common. Multiples of 6 include 6, 12, 18, 24, 30, and so on. Multiples of 8 include 8, 16, 24, 32, 40, and so on. Multiples of 12 include 12, 24, 36, 48, and so on. From these lists, we can see that the smallest common multiple is 24. Therefore, the LCM of 6, 8, and 12 is 24. Once we have the LCM, we can rewrite each fraction with the common denominator of 24. This involves multiplying both the numerator and the denominator of each fraction by a factor that will make the denominator equal to 24. This step is essential to ensure that the value of the fraction remains the same while changing its representation. After rewriting the fractions with a common denominator, we can proceed with the addition and subtraction operations. We add or subtract the numerators while keeping the common denominator. This process simplifies the expression into a single fraction. Finally, if necessary, we simplify the resulting fraction by reducing it to its lowest terms. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD). By following these steps, we can accurately calculate the value of the expression and gain a deeper understanding of fraction arithmetic.
Finding the Least Common Multiple (LCM)
The first crucial step in solving this expression is to find the Least Common Multiple (LCM) of the denominators: 6, 8, and 12. The LCM is the smallest positive integer that is divisible by each of these numbers. Finding the LCM allows us to express the fractions with a common denominator, which is necessary for adding and subtracting them. There are several methods to determine the LCM, and one of the most common is the prime factorization method. Prime factorization involves breaking down each number into its prime factors. Prime numbers are numbers that have only two divisors: 1 and themselves. For instance, the prime factors of 6 are 2 and 3, since 6 = 2 × 3. Similarly, the prime factors of 8 are 2 × 2 × 2, which can be written as 2^3. The prime factors of 12 are 2 × 2 × 3, or 2^2 × 3. Once we have the prime factorization of each number, we can find the LCM by taking the highest power of each prime factor that appears in any of the factorizations. In this case, the prime factors are 2 and 3. The highest power of 2 is 2^3 (from the factorization of 8), and the highest power of 3 is 3^1 (from the factorizations of 6 and 12). Therefore, the LCM is the product of these highest powers: LCM(6, 8, 12) = 2^3 × 3^1 = 8 × 3 = 24. This means that 24 is the smallest number that is divisible by 6, 8, and 12. Another method to find the LCM is by listing multiples of each number until a common multiple is found. This method is straightforward but can be time-consuming for larger numbers. We list the multiples of 6: 6, 12, 18, 24, 30, and so on. We list the multiples of 8: 8, 16, 24, 32, 40, and so on. We list the multiples of 12: 12, 24, 36, 48, and so on. By comparing these lists, we can see that the smallest common multiple is 24. This confirms our result from the prime factorization method. Understanding and finding the LCM is a fundamental skill in arithmetic and is essential for working with fractions. It simplifies the process of adding and subtracting fractions, making it easier to solve mathematical expressions. Without a common denominator, it is impossible to directly add or subtract fractions because they represent parts of different wholes. The LCM provides the necessary common ground, allowing us to combine the fractions accurately. In addition to solving fraction problems, the LCM has applications in various other areas of mathematics and real-life situations, such as scheduling events, dividing items into equal groups, and solving algebraic equations. Therefore, mastering the concept of LCM is a valuable skill that extends beyond basic arithmetic.
Rewriting Fractions with a Common Denominator
After determining the LCM to be 24, the next critical step is to rewrite each fraction in the expression with this common denominator. This process involves converting each fraction while maintaining its original value. To do this, we multiply both the numerator and the denominator of each fraction by a factor that will transform the denominator into 24. This technique is based on the fundamental principle that multiplying a fraction by a form of 1 (i.e., a number divided by itself) does not change its value. Let's begin with the first fraction, 5/6. To convert the denominator 6 into 24, we need to multiply it by 4. Therefore, we multiply both the numerator and the denominator of 5/6 by 4: (5 × 4) / (6 × 4) = 20/24. This gives us an equivalent fraction, 20/24, which has the same value as 5/6 but with a denominator of 24. Next, we consider the second fraction, 7/8. To convert the denominator 8 into 24, we need to multiply it by 3. Thus, we multiply both the numerator and the denominator of 7/8 by 3: (7 × 3) / (8 × 3) = 21/24. This gives us the equivalent fraction 21/24, which has the same value as 7/8 but with a denominator of 24. Finally, we look at the third fraction, 11/12. To convert the denominator 12 into 24, we need to multiply it by 2. Therefore, we multiply both the numerator and the denominator of 11/12 by 2: (11 × 2) / (12 × 2) = 22/24. This results in the equivalent fraction 22/24, which has the same value as 11/12 but with a denominator of 24. Now that we have rewritten all the fractions with a common denominator, the expression becomes: x = 20/24 - 21/24 + 22/24. This transformation is crucial because it allows us to perform the addition and subtraction operations on the numerators directly, while keeping the common denominator. Rewriting fractions with a common denominator is a fundamental skill in fraction arithmetic and is essential for solving a wide range of mathematical problems. This technique not only simplifies the process of adding and subtracting fractions but also provides a clear understanding of how fractions can be manipulated without changing their inherent value. In various mathematical contexts, such as algebra, calculus, and real-world applications, the ability to work with fractions and convert them into equivalent forms is invaluable. Furthermore, mastering this skill enhances one's overall mathematical proficiency and problem-solving abilities.
Performing the Arithmetic Operations
With the fractions now rewritten with a common denominator, 24, the next logical step is to perform the arithmetic operations as indicated in the expression: x = 20/24 - 21/24 + 22/24. This involves subtracting 21/24 from 20/24 and then adding 22/24 to the result. When fractions have a common denominator, adding or subtracting them is straightforward. We simply add or subtract the numerators while keeping the denominator the same. Let's begin by subtracting 21/24 from 20/24: 20/24 - 21/24. To perform this subtraction, we subtract the numerators: 20 - 21 = -1. Therefore, the result is -1/24. This indicates that the value of the expression so far is a negative fraction, which is perfectly valid in mathematics. Next, we add 22/24 to the result: -1/24 + 22/24. Again, we add the numerators while keeping the common denominator: -1 + 22 = 21. So, the result is 21/24. This fraction represents the final value of the expression after performing all the indicated operations. It is essential to understand the rules for adding and subtracting integers, especially when dealing with negative numbers. The subtraction of 21 from 20 results in -1 because 21 is greater than 20, and we take the sign of the larger number. Similarly, when adding -1 and 22, we effectively subtract the smaller absolute value from the larger absolute value (22 - 1 = 21) and keep the sign of the larger number, which is positive in this case. Performing arithmetic operations on fractions is a fundamental skill in mathematics, and it is essential for solving various types of problems. Whether it is simple addition and subtraction or more complex operations involving multiplication and division, a solid understanding of fraction arithmetic is crucial. This skill is not only valuable in academic settings but also in practical situations, such as cooking, measuring, and financial calculations. The ability to accurately perform these operations allows for precise calculations and problem-solving in real-world scenarios. Furthermore, mastering fraction arithmetic lays the groundwork for more advanced mathematical concepts, such as algebra, calculus, and statistics. Therefore, it is important to develop a strong foundation in fraction arithmetic to succeed in mathematics and related fields.
Simplifying the Resulting Fraction
After performing the arithmetic operations, we arrived at the fraction 21/24. The final step in solving this problem is to simplify this fraction to its lowest terms. Simplifying a fraction means reducing it to an equivalent fraction where the numerator and the denominator have no common factors other than 1. This is also known as reducing the fraction to its simplest form. To simplify 21/24, we need to find the greatest common divisor (GCD) of the numerator (21) and the denominator (24). The GCD is the largest positive integer that divides both numbers without leaving a remainder. There are several methods to find the GCD, including listing factors, prime factorization, and using the Euclidean algorithm. One common method 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 24 are 1, 2, 3, 4, 6, 8, 12, and 24. By comparing these lists, we can see that the greatest common factor is 3. Therefore, the GCD of 21 and 24 is 3. Alternatively, we can use the prime factorization method to find the GCD. The prime factorization of 21 is 3 × 7. The prime factorization of 24 is 2 × 2 × 2 × 3, or 2^3 × 3. The GCD is the product of the common prime factors raised to the lowest power they appear in either factorization. In this case, the only common prime factor is 3, which appears to the power of 1 in both factorizations. Thus, the GCD is 3. Once we have the GCD, we can simplify the fraction by dividing both the numerator and the denominator by the GCD. So, we divide both 21 and 24 by 3: (21 ÷ 3) / (24 ÷ 3) = 7/8. This gives us the simplified fraction 7/8. The fraction 7/8 is in its simplest form because 7 and 8 have no common factors other than 1. Simplifying fractions is an essential skill in mathematics, as it allows us to express fractions in their most concise and understandable form. Simplified fractions are easier to compare, work with, and interpret. In many contexts, such as problem-solving, calculations, and data representation, it is standard practice to present fractions in their simplest form. Furthermore, simplifying fractions helps to avoid confusion and ensures that the answer is clear and unambiguous. Understanding and applying the concept of GCD is crucial for simplifying fractions efficiently. Whether using the listing factors method, prime factorization, or the Euclidean algorithm, the ability to find the GCD accurately is a valuable skill. In addition to simplifying fractions, the GCD has applications in various other areas of mathematics, such as number theory, algebra, and cryptography. Therefore, mastering the concept of GCD is a valuable skill that extends beyond basic arithmetic.
Final Answer
In conclusion, by following the steps of finding the LCM, rewriting the fractions with a common denominator, performing the arithmetic operations, and simplifying the result, we have successfully calculated the value of the expression x = 5/6 - 7/8 + 11/12. The final simplified answer is x = 7/8. This process demonstrates the importance of understanding and applying the fundamental principles of fraction arithmetic. Each step, from finding the LCM to simplifying the final fraction, plays a crucial role in arriving at the correct solution. The LCM allows us to combine the fractions seamlessly, the common denominator ensures that we are adding and subtracting equivalent parts, the arithmetic operations give us the numerical result, and simplification presents the answer in its most concise and understandable form. Throughout this problem, we have utilized various mathematical techniques and concepts, including prime factorization, greatest common divisor (GCD), and the manipulation of fractions. These skills are not only essential for solving fraction problems but also for tackling more complex mathematical challenges in algebra, calculus, and other fields. The ability to work with fractions effectively is a cornerstone of mathematical proficiency and is valuable in numerous real-world applications. Whether it is calculating proportions in cooking, measuring ingredients for a recipe, or managing finances, fractions are ubiquitous in everyday life. Therefore, mastering fraction arithmetic is a practical skill that has broad applicability beyond the classroom. Furthermore, the problem-solving process demonstrated in this example highlights the importance of a systematic approach to mathematical problems. Breaking down the problem into smaller, manageable steps, such as finding the LCM and rewriting fractions, makes the overall task less daunting and more achievable. This methodical approach can be applied to a wide range of problems, both in mathematics and in other disciplines. By developing a structured problem-solving strategy, individuals can enhance their critical thinking skills and improve their ability to tackle complex challenges. In summary, the calculation of the value of the expression x = 5/6 - 7/8 + 11/12 not only provides a numerical answer but also reinforces essential mathematical concepts and problem-solving skills. The final result, x = 7/8, is a testament to the power of fraction arithmetic and the importance of a systematic approach to mathematical problem-solving. This example serves as a valuable learning experience, demonstrating the interconnectedness of mathematical concepts and the practical applications of these concepts in everyday life.
