Solving Fractions Evaluate 4/15 - 4/5 + 1/2
In the realm of mathematics, mastering the art of fraction manipulation is a fundamental skill. Fractions, representing parts of a whole, are ubiquitous in various mathematical contexts and real-world applications. This comprehensive guide delves into the intricacies of evaluating the expression 4/15 - 4/5 + 1/2, providing a step-by-step approach to solving this problem and illuminating the underlying principles of fraction arithmetic. Whether you're a student seeking to solidify your understanding or an enthusiast eager to expand your mathematical prowess, this exploration will equip you with the knowledge and confidence to tackle similar fraction-based challenges.
Understanding Fractions: The Building Blocks of Numerical Representation
Before embarking on the solution, it's crucial to establish a solid understanding of fractions themselves. A fraction represents a part of a whole, expressed as a ratio of two numbers: the numerator and the denominator. The numerator indicates the number of parts we're considering, while the denominator signifies the total number of equal parts that constitute the whole. For instance, in the fraction 4/15, the numerator 4 signifies that we're considering 4 parts, and the denominator 15 indicates that the whole is divided into 15 equal parts. Similarly, 4/5 represents 4 parts out of a whole divided into 5 equal parts, and 1/2 represents 1 part out of a whole divided into 2 equal parts.
Fractions can be classified into different categories based on the relationship between the numerator and denominator. A proper fraction has a numerator smaller than the denominator, representing a value less than 1 (e.g., 4/15, 4/5). An improper fraction, on the other hand, has a numerator greater than or equal to the denominator, representing a value greater than or equal to 1 (e.g., 7/4). A mixed number combines a whole number with a proper fraction (e.g., 1 3/4). Understanding these classifications is essential for performing operations with fractions accurately.
To effectively work with fractions, we must grasp the concept of equivalent fractions. Equivalent fractions represent the same value but have different numerators and denominators. For example, 2/4, 3/6, and 4/8 are all equivalent fractions, each representing one-half. We can obtain equivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero number. This principle is crucial for adding and subtracting fractions with different denominators.
The Least Common Multiple (LCM): Finding a Common Ground for Fractions
When adding or subtracting fractions, a fundamental requirement is that they share a common denominator. This common denominator serves as a unifying factor, allowing us to combine the fractions meaningfully. The most efficient common denominator to use is the least common multiple (LCM) of the original denominators. The LCM is the smallest positive integer that is a multiple of all the given denominators. Finding the LCM is a crucial step in simplifying fraction arithmetic.
To determine the LCM of a set of numbers, we can employ various methods, including listing multiples and prime factorization. The listing multiples method involves writing out the multiples of each number until a common multiple is found. For instance, to find the LCM of 15, 5, and 2, we would list the multiples of each: 15, 30, 45...; 5, 10, 15, 20...; and 2, 4, 6, 8, 10.... The smallest number that appears in all three lists is the LCM, which in this case is 30.
The prime factorization method provides a more systematic approach, especially for larger numbers. This method involves expressing each number as a product of its prime factors. Prime factors are prime numbers that divide the given number without leaving a remainder. For example, the prime factorization of 15 is 3 x 5, of 5 is 5, and of 2 is 2. To find the LCM, we take the highest power of each prime factor that appears in any of the factorizations and multiply them together. In this case, the LCM would be 2 x 3 x 5 = 30. Regardless of the method used, finding the LCM is a cornerstone of fraction arithmetic.
Evaluating the Expression: A Step-by-Step Solution
Now, armed with a solid understanding of fractions and the LCM, we can confidently tackle the expression 4/15 - 4/5 + 1/2. Our goal is to simplify this expression to a single fraction, representing the final result. We will follow a step-by-step approach, ensuring accuracy and clarity at each stage.
Step 1: Finding the Least Common Denominator
The first step is to determine the least common denominator (LCD) of the fractions 4/15, 4/5, and 1/2. As we established earlier, the LCM of 15, 5, and 2 is 30. Therefore, the LCD for our fractions is 30. This means we need to convert each fraction into an equivalent fraction with a denominator of 30.
Step 2: Converting to Equivalent Fractions
To convert 4/15 to an equivalent fraction with a denominator of 30, we multiply both the numerator and denominator by 2 (since 15 x 2 = 30). This gives us (4 x 2) / (15 x 2) = 8/30. Similarly, to convert 4/5 to a fraction with a denominator of 30, we multiply both the numerator and denominator by 6 (since 5 x 6 = 30), resulting in (4 x 6) / (5 x 6) = 24/30. Finally, to convert 1/2 to a fraction with a denominator of 30, we multiply both the numerator and denominator by 15 (since 2 x 15 = 30), giving us (1 x 15) / (2 x 15) = 15/30.
Now, our expression becomes 8/30 - 24/30 + 15/30. The fractions now share a common denominator, paving the way for addition and subtraction.
Step 3: Adding and Subtracting Fractions
With a common denominator in place, we can now perform the addition and subtraction operations. To subtract fractions with a common denominator, we subtract the numerators and keep the denominator the same. To add fractions with a common denominator, we add the numerators and keep the denominator the same. Following the order of operations, we perform the subtraction first: 8/30 - 24/30 = (8 - 24) / 30 = -16/30.
Next, we add the remaining fraction: -16/30 + 15/30 = (-16 + 15) / 30 = -1/30.
Step 4: Simplifying the Result
The final step is to simplify the resulting fraction, if possible. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. In this case, -1/30 is already in its simplest form, as 1 and 30 share no common factors other than 1.
Therefore, the final answer to the expression 4/15 - 4/5 + 1/2 is -1/30.
Conclusion: Mastering Fraction Arithmetic
Evaluating expressions involving fractions requires a systematic approach, combining a solid understanding of fraction concepts with proficiency in arithmetic operations. This comprehensive guide has illuminated the process of evaluating the expression 4/15 - 4/5 + 1/2, providing a step-by-step solution that emphasizes clarity and accuracy. By mastering the concepts of equivalent fractions, least common multiples, and fraction arithmetic, you can confidently navigate the world of fractions and tackle more complex mathematical challenges. Remember, practice is key to solidifying your understanding and honing your skills in fraction manipulation. Embrace the challenge, and you'll find that fractions become an integral part of your mathematical toolkit.
Additional practice questions
To further solidify your understanding, consider tackling these additional practice questions:
- Evaluate: 1/3 + 2/5 - 1/6
- Evaluate: 3/8 - 1/4 + 5/12
- Evaluate: 2/7 + 3/14 - 1/2
- Evaluate: 5/9 - 2/3 + 1/6
- Evaluate: 1/2 - 3/4 + 5/8
By working through these examples, you'll reinforce your understanding of fraction arithmetic and gain confidence in your ability to solve similar problems.
