Evaluate The Expression 1 8/15 + 3 13/20.

In this article, we will delve into the step-by-step evaluation of the mathematical expression 1 8/15 + 3 13/20. This expression involves the addition of two mixed numbers. To accurately solve this, we'll first convert the mixed numbers into improper fractions, find a common denominator, perform the addition, and then simplify the result back into a mixed number if necessary. This process highlights key concepts in arithmetic, such as fraction manipulation, finding the least common multiple (LCM), and fraction simplification. Mastering these skills is essential for more advanced mathematical problems and real-world applications.

Before we dive into the calculation, it's crucial to understand the difference between mixed numbers and improper fractions. A mixed number is a combination of a whole number and a proper fraction (where the numerator is less than the denominator), such as 1 8/15. An improper fraction, on the other hand, is a fraction where the numerator is greater than or equal to the denominator, such as 23/15. Converting mixed numbers to improper fractions is a necessary step in many arithmetic operations, including addition and subtraction, as it simplifies the process of finding common denominators and performing the calculations. Understanding this conversion is a foundational skill in mathematics, enabling students to tackle more complex problems with confidence.

To evaluate the expression 1 8/15 + 3 13/20, we'll follow a series of steps to ensure accuracy and clarity in our solution. First, we need to convert the mixed numbers into improper fractions. The mixed number 1 8/15 is converted by multiplying the whole number (1) by the denominator (15) and adding the numerator (8), resulting in (1 * 15) + 8 = 23. The improper fraction is thus 23/15. Similarly, for 3 13/20, we calculate (3 * 20) + 13 = 73, giving us the improper fraction 73/20. The expression now becomes 23/15 + 73/20. This conversion is a critical first step because it allows us to work with fractions that have a consistent form, making the subsequent addition process smoother and less prone to errors. Without this conversion, the addition of mixed numbers would involve additional complexities that could easily lead to mistakes. Therefore, mastering this step is crucial for the accurate evaluation of expressions involving mixed numbers.

Once we have the improper fractions 23/15 and 73/20, the next step is to find a common denominator. To add or subtract fractions, they must have the same denominator. The common denominator is a common multiple of the denominators of the fractions being added. The most efficient common denominator to use is the least common multiple (LCM). To find the LCM of 15 and 20, we can list the multiples of each number and identify the smallest multiple they have in common, or we can use the prime factorization method. The prime factorization of 15 is 3 * 5, and the prime factorization of 20 is 2 * 2 * 5. The LCM is found by taking the highest power of each prime factor present in either factorization: 2^2 * 3 * 5 = 60. Therefore, 60 is the least common multiple of 15 and 20. Understanding how to find the LCM is a fundamental skill in fraction arithmetic. It ensures that we are working with the smallest possible equivalent fractions, simplifying the addition or subtraction process and reducing the likelihood of errors in the final result. This step is essential for accurately combining fractions and is a key component of fraction arithmetic.

With the common denominator of 60 determined, we need to convert both fractions to equivalent fractions with this denominator. For the fraction 23/15, we multiply both the numerator and the denominator by 60/15 = 4, resulting in (23 * 4) / (15 * 4) = 92/60. For the fraction 73/20, we multiply both the numerator and the denominator by 60/20 = 3, resulting in (73 * 3) / (20 * 3) = 219/60. Now that both fractions have the same denominator, we can add them by simply adding their numerators while keeping the denominator the same: 92/60 + 219/60 = (92 + 219) / 60 = 311/60. This step is the heart of the addition process, where we combine the equivalent fractions to get a single fraction representing their sum. The ability to convert fractions to equivalent forms with a common denominator and then add them is a crucial skill in arithmetic, providing the foundation for more complex mathematical operations involving fractions.

After adding the fractions, we have the improper fraction 311/60. The final step is to simplify this fraction. First, we convert it back to a mixed number. To do this, we divide the numerator (311) by the denominator (60). The quotient is the whole number part of the mixed number, and the remainder is the numerator of the fractional part. Dividing 311 by 60 gives us a quotient of 5 and a remainder of 11. Thus, the mixed number is 5 11/60. Next, we check if the fractional part 11/60 can be further simplified. The greatest common divisor (GCD) of 11 and 60 is 1, which means the fraction is already in its simplest form. Therefore, the simplified result of the expression 1 8/15 + 3 13/20 is 5 11/60. Simplifying the result is an essential step in mathematics, as it presents the answer in its most concise and understandable form. This final step demonstrates a thorough understanding of fraction manipulation and ensures the solution is both accurate and clearly presented.

In conclusion, we have successfully evaluated the expression 1 8/15 + 3 13/20 by converting the mixed numbers to improper fractions, finding a common denominator, adding the fractions, and simplifying the result back into a mixed number. The final answer is 5 11/60. This process underscores the importance of understanding and applying fundamental arithmetic principles, such as fraction manipulation, finding the least common multiple, and simplifying fractions. Mastering these skills is crucial for success in mathematics and provides a solid foundation for tackling more complex problems in the future. By following these steps carefully and methodically, we can ensure accuracy and clarity in our mathematical solutions, enhancing our overall problem-solving abilities.