Solve The Problem: 5 3/8 - 3 2/5. Reduce The Answer To Its Simplest Form.
H2: Introduction to Mixed Number Subtraction
In the realm of mathematics, subtracting mixed numbers is a fundamental skill that bridges arithmetic and more advanced algebraic concepts. This guide aims to provide a comprehensive understanding of how to effectively subtract mixed numbers, focusing on a step-by-step approach to ensure accuracy and clarity. Specifically, we will address the problem of subtracting 3 2/5 from 5 3/8, reducing the answer to its simplest form. This process involves converting mixed numbers to improper fractions, finding a common denominator, performing the subtraction, and simplifying the result. Understanding these steps is crucial not only for solving mathematical problems but also for developing a strong foundation in numerical literacy. The ability to confidently manipulate fractions and mixed numbers is invaluable in various real-world scenarios, from cooking and measuring to financial calculations and beyond. Therefore, mastering this skill is an investment in your mathematical proficiency and practical problem-solving abilities. This article will walk you through each stage, ensuring you grasp the underlying principles and can apply them to a variety of similar problems. By breaking down the process into manageable steps, we aim to demystify mixed number subtraction and empower you with the tools to tackle these challenges with ease and confidence.
H2: Understanding Mixed Numbers and Improper Fractions
To successfully subtract mixed numbers, it's essential to first grasp the concepts of mixed numbers and improper fractions. A mixed number is a combination of a whole number and a proper fraction, such as 5 3/8. The whole number (5 in this case) represents the integer portion, while the fraction (3/8) represents a part of a whole. An improper fraction, on the other hand, is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). For example, 11/4 is an improper fraction. The key to performing operations with mixed numbers often lies in converting them into improper fractions. This conversion allows us to work with a single fractional value, making addition and subtraction processes more straightforward. To convert a mixed number to an improper fraction, you multiply the whole number by the denominator of the fractional part, add the numerator, and then place the result over the original denominator. For instance, to convert 5 3/8 to an improper fraction, you multiply 5 by 8 (which equals 40), add 3 (resulting in 43), and then place 43 over 8, giving you 43/8. Understanding this conversion process is crucial because it transforms a mixed number, which can be conceptually more complex, into a standard fraction format that is easier to manipulate mathematically. Similarly, you can convert 3 2/5 into an improper fraction by multiplying 3 by 5 (which equals 15), adding 2 (resulting in 17), and placing 17 over 5, giving you 17/5. These conversions are the first step in simplifying the subtraction problem at hand, setting the stage for finding a common denominator and performing the subtraction.
H2: Step 1 Converting Mixed Numbers to Improper Fractions
The initial step in solving our problem, which involves subtracting mixed numbers, is to convert each mixed number into an improper fraction. As discussed earlier, this conversion simplifies the subtraction process by allowing us to work with single fractional values. Our problem is 5 3/8 - 3 2/5. Let's convert 5 3/8 into an improper fraction. Multiply the whole number (5) by the denominator (8), which gives us 40. Then, add the numerator (3) to this result, yielding 43. Place this new numerator (43) over the original denominator (8) to get the improper fraction 43/8. Next, we convert 3 2/5 into an improper fraction using the same method. Multiply the whole number (3) by the denominator (5), which equals 15. Add the numerator (2) to this, giving us 17. Place this result (17) over the original denominator (5) to obtain the improper fraction 17/5. Now that we have converted both mixed numbers into improper fractions, our subtraction problem looks like this: 43/8 - 17/5. This transformation is a critical step because it sets the stage for finding a common denominator, which is necessary to subtract fractions. By expressing both numbers as improper fractions, we eliminate the complexities associated with mixed numbers and prepare the problem for straightforward fraction subtraction. The ability to perform this conversion fluently is fundamental to mastering operations with mixed numbers and fractions, making it a cornerstone of mathematical proficiency.
H2: Step 2 Finding a Common Denominator
After converting the mixed numbers to improper fractions, the next essential step in solving the subtraction problem is to find a common denominator. The reason we need a common denominator is that fractions can only be added or subtracted directly if they have the same denominator. This is because the denominator represents the number of equal parts into which a whole is divided, and we need these parts to be the same size before we can combine them. In our problem, we have 43/8 - 17/5. The denominators are 8 and 5. To find a common denominator, we need to find the least common multiple (LCM) of these two numbers. The least common multiple is the smallest number that is a multiple of both denominators. One way to find the LCM is to list the multiples of each number until we find a common one. The multiples of 8 are 8, 16, 24, 32, 40, 48, and so on. The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, and so on. We see that the smallest number that appears in both lists is 40. Therefore, the least common multiple of 8 and 5 is 40, which will be our common denominator. Now that we have the common denominator, we need to convert both fractions to equivalent fractions with this denominator. To convert 43/8 to a fraction with a denominator of 40, we multiply both the numerator and the denominator by 5 (since 8 * 5 = 40). This gives us (43 * 5) / (8 * 5) = 215/40. Similarly, to convert 17/5 to a fraction with a denominator of 40, we multiply both the numerator and the denominator by 8 (since 5 * 8 = 40). This gives us (17 * 8) / (5 * 8) = 136/40. Now our problem looks like this: 215/40 - 136/40. Finding a common denominator is a crucial step because it allows us to express the fractions in terms of the same-sized parts, making subtraction possible. Without a common denominator, we would be trying to subtract fractions that represent different subdivisions of a whole, which is mathematically unsound. With the fractions now having the same denominator, we are ready to proceed with the subtraction.
H2: Step 3 Subtracting the Fractions
With the fractions now sharing a common denominator, the next logical step is to perform the subtraction. In our problem, we have 215/40 - 136/40. Since the denominators are the same, we can subtract the numerators directly. This means we subtract 136 from 215. Performing this subtraction, 215 - 136 equals 79. Therefore, the result of subtracting the fractions is 79/40. This fraction represents the difference between the two original fractions, expressed as a single fraction with the common denominator. The process of subtracting fractions with a common denominator is straightforward: you simply subtract the numerators while keeping the denominator the same. This is because the denominator indicates the size of the fractional parts, and when the denominators are the same, we are subtracting quantities of the same-sized parts. The result, 79/40, is an improper fraction, as the numerator (79) is greater than the denominator (40). While this is a valid answer, it is often preferable to express the result as a mixed number, which is more intuitive and easier to understand in many contexts. Converting an improper fraction to a mixed number involves dividing the numerator by the denominator and expressing the remainder as a fraction. This will be the next step in simplifying our answer. Subtracting the fractions is a critical step as it provides the numerical difference between the two fractions, but expressing the result in its simplest form, such as a mixed number, enhances its clarity and usability.
H2: Step 4 Simplifying the Answer
After subtracting the fractions, we arrived at the improper fraction 79/40. While this is a correct answer, it is not in its simplest form. To simplify this, we need to convert the improper fraction back into a mixed number. This conversion involves dividing the numerator (79) by the denominator (40). When we divide 79 by 40, we get a quotient of 1 and a remainder of 39. This means that 79/40 is equal to 1 whole and 39 parts out of 40. Therefore, we can express 79/40 as the mixed number 1 39/40. The whole number part of the mixed number is the quotient (1), and the fractional part is the remainder (39) over the original denominator (40). Now, we need to check if the fractional part, 39/40, can be further simplified. To do this, we look for common factors between the numerator (39) and the denominator (40). The factors of 39 are 1, 3, 13, and 39. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. The only common factor between 39 and 40 is 1, which means that the fraction 39/40 is already in its simplest form. Therefore, the simplified answer to our problem, 5 3/8 - 3 2/5, is 1 39/40. Simplifying the answer is a crucial step because it presents the result in a form that is both mathematically correct and easy to interpret. Mixed numbers are often more intuitive than improper fractions, especially in real-world contexts. Additionally, ensuring that the fractional part is in its simplest form (i.e., the numerator and denominator have no common factors other than 1) is a standard practice in mathematics. This final simplification step ensures that our answer is not only accurate but also presented in the most clear and concise manner.
H2: Step 5 Selecting the Correct Option
Having meticulously worked through the steps of converting mixed numbers to improper fractions, finding a common denominator, subtracting the fractions, and simplifying the result, we have arrived at the answer: 1 39/40. Now, the final step is to match our answer with the options provided in the question. The options were:
A. 2 37/40 B. 2 12/25 C. 1 39/40 D. 2 39/40
Comparing our calculated answer, 1 39/40, with the given options, we can clearly see that it matches option C. Therefore, the correct answer is C. 1 39/40. This final step is a critical validation of our work. It confirms that we have not only performed the mathematical operations correctly but also interpreted the question and the answer choices accurately. Selecting the correct option is more than just a formality; it's the culmination of the entire problem-solving process. It demonstrates our ability to synthesize the individual steps into a coherent solution and to confidently identify the correct answer from a set of possibilities. This skill is invaluable in mathematics and beyond, as it reflects a deep understanding of the subject matter and the ability to apply that understanding effectively. In this case, our step-by-step approach has led us to the correct answer, reinforcing the importance of a systematic and thorough approach to problem-solving.
H2: Conclusion
In conclusion, subtracting mixed numbers involves a series of key steps: converting mixed numbers to improper fractions, finding a common denominator, subtracting the fractions, and simplifying the answer. By following these steps methodically, we can accurately solve subtraction problems involving mixed numbers. In the specific problem we addressed, 5 3/8 - 3 2/5, we successfully arrived at the simplified answer of 1 39/40, which corresponds to option C. This process highlights the importance of understanding the underlying concepts of fractions and mixed numbers, as well as the ability to apply mathematical operations with precision. The skill of subtracting mixed numbers is not just a theoretical exercise; it has practical applications in various real-life situations, from cooking and measuring to financial calculations and beyond. Therefore, mastering this skill is an investment in your mathematical proficiency and problem-solving abilities. The step-by-step approach outlined in this guide provides a framework for tackling similar problems with confidence. By breaking down complex problems into manageable steps, we can demystify the process and empower ourselves to solve them effectively. Remember, the key to success in mathematics is not just memorizing formulas, but understanding the principles and applying them consistently. This guide has aimed to provide you with that understanding and the tools to apply it successfully. As you continue your mathematical journey, remember the importance of practice and persistence, and you will find that even the most challenging problems can be solved with the right approach.