Mastering Mixed Number Arithmetic Step-by-Step Solutions And Explanations

Let's dive into the world of mixed number arithmetic with this first problem: 5\frac{1}{3} - 2\frac{3}{4}. To effectively tackle this subtraction, we need to convert the mixed numbers into improper fractions. This involves multiplying the whole number by the denominator and adding the numerator, then placing the result over the original denominator. So, 5\frac{1}{3} becomes (5 * 3 + 1) / 3 = 16/3, and 2\frac{3}{4} becomes (2 * 4 + 3) / 4 = 11/4. Now, our problem transforms into 16/3 - 11/4. To subtract these fractions, they need a common denominator. The least common multiple (LCM) of 3 and 4 is 12. We convert 16/3 to an equivalent fraction with a denominator of 12 by multiplying both the numerator and denominator by 4, resulting in 64/12. Similarly, we convert 11/4 by multiplying both numerator and denominator by 3, resulting in 33/12. The problem now reads 64/12 - 33/12. Subtracting the numerators gives us 31/12. Finally, we convert this improper fraction back into a mixed number. 31 divided by 12 is 2 with a remainder of 7, so the answer is 2\frac{7}{12}. This meticulous process ensures accuracy and builds a strong foundation for more complex arithmetic problems. Remember, understanding the underlying principles of fraction manipulation is key to mastering these operations. Continue practicing, and you'll find these calculations becoming second nature. Exploring different strategies and methods can also enhance your understanding and efficiency in solving such problems. The key is consistent practice and a clear understanding of the steps involved.

In this problem, we're faced with subtracting a mixed number from a proper fraction: \frac{5}{6} - 2\frac{3}{4}. Our initial step, as with any mixed number operation, is to convert the mixed number into an improper fraction. Following the same procedure as before, we convert 2\frac{3}{4} to (2 * 4 + 3) / 4 = 11/4. Now, our problem is \frac{5}{6} - \frac{11}{4}. To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 6 and 4 is 12. We convert \frac{5}{6} to an equivalent fraction with a denominator of 12 by multiplying both numerator and denominator by 2, yielding 10/12. Similarly, we convert \frac{11}{4} by multiplying both numerator and denominator by 3, resulting in 33/12. The problem now becomes 10/12 - 33/12. Subtracting the numerators gives us -23/12. Since the result is an improper fraction, we convert it back into a mixed number. -23 divided by 12 is -1 with a remainder of -11, so the answer is -1\frac{11}{12}. It's crucial to pay close attention to the signs when dealing with subtraction. The negative sign indicates that the second fraction is larger than the first, resulting in a negative difference. Practicing these types of problems will help solidify your understanding of fraction arithmetic, particularly when negative numbers are involved. Remember, the goal is not just to arrive at the correct answer but also to understand the process thoroughly. This understanding will enable you to tackle more complex problems with confidence. Regular practice and a systematic approach are key to mastering these concepts.

Here, we're adding a whole number to a negative mixed number: -4\frac{1}{3} + 3. To solve this, it's helpful to first convert the mixed number into an improper fraction. So, -4\frac{1}{3} becomes -(4 * 3 + 1) / 3 = -13/3. Now, our problem is -13/3 + 3. To add these, we need to express the whole number 3 as a fraction with the same denominator as -13/3. We can write 3 as 3/1, and to get a denominator of 3, we multiply both the numerator and denominator by 3, resulting in 9/3. Now the problem is -13/3 + 9/3. Adding the numerators, we get -13 + 9 = -4, so the result is -4/3. Finally, we convert this improper fraction back into a mixed number. -4 divided by 3 is -1 with a remainder of -1, so the answer is -1\frac{1}{3}. Understanding how to work with negative numbers and fractions is a critical skill in mathematics. It's essential to pay attention to the signs and ensure that you're applying the correct operations. Visualizing the number line can sometimes be helpful in understanding addition and subtraction with negative numbers. Practice is key to building confidence and accuracy in these types of calculations. The more you work with these concepts, the more intuitive they will become. Remember, patience and persistence are essential when learning new mathematical skills. Keep practicing, and you'll see improvement over time.

In this problem, we have the addition of a negative fraction and a mixed number: -\frac{4}{5} + 1\frac{1}{5}. As with previous problems involving mixed numbers, our first step is to convert 1\frac{1}{5} into an improper fraction. This gives us (1 * 5 + 1) / 5 = 6/5. Now the problem is -\frac{4}{5} + \frac{6}{5}. Since the fractions already have a common denominator, we can simply add the numerators: -4 + 6 = 2. Therefore, the result is 2/5. This problem highlights the importance of understanding how to add fractions with the same denominator and how to work with negative fractions. When the denominators are the same, the operation becomes straightforward, focusing solely on the numerators. Remember to pay close attention to the signs when adding or subtracting. Visualizing the fractions on a number line can be a helpful strategy for understanding the operation. Practice with these types of problems will strengthen your ability to perform fraction arithmetic accurately and efficiently. The key is to break down the problem into manageable steps and apply the rules of fraction addition and subtraction systematically. Regular practice will build your confidence and help you master these essential mathematical skills.

Here, we have a combination of addition and subtraction involving a fraction, a whole number, and another fraction: \frac{2}{5} + 4 - \frac{3}{4}. To solve this, we first need to express the whole number 4 as a fraction. We can write it as 4/1. To combine these fractions, we need a common denominator for 5, 1, and 4. The least common multiple (LCM) of 5, 1, and 4 is 20. Now we convert each term to an equivalent fraction with a denominator of 20. \frac{2}{5} becomes (2 * 4) / (5 * 4) = 8/20. 4/1 becomes (4 * 20) / (1 * 20) = 80/20. And \frac{3}{4} becomes (3 * 5) / (4 * 5) = 15/20. The problem now transforms into 8/20 + 80/20 - 15/20. Adding and subtracting the numerators, we get 8 + 80 - 15 = 73. So the result is 73/20. Finally, we convert this improper fraction back into a mixed number. 73 divided by 20 is 3 with a remainder of 13, so the answer is 3\frac{13}{20}. This problem demonstrates the importance of finding a common denominator when adding or subtracting fractions and highlights the process of converting between improper fractions and mixed numbers. Breaking down the problem into smaller steps makes it easier to manage and reduces the chance of errors. Remember, a systematic approach is crucial for solving complex arithmetic problems. Practice with these types of problems will improve your ability to handle various combinations of operations involving fractions and whole numbers. The key is to understand the underlying principles and apply them consistently.

In this addition problem, we have a combination of fractions and a mixed number: \frac{8}{9} + 6\frac{1}{3} + \frac{2}{9}. The first step is to convert the mixed number, 6\frac{1}{3}, into an improper fraction. This gives us (6 * 3 + 1) / 3 = 19/3. Now our problem is \frac{8}{9} + \frac{19}{3} + \frac{2}{9}. To add these fractions, we need a common denominator. The least common multiple (LCM) of 9 and 3 is 9. We already have 8/9 and 2/9 with a denominator of 9. We need to convert 19/3 to an equivalent fraction with a denominator of 9. We do this by multiplying both the numerator and the denominator by 3, which gives us (19 * 3) / (3 * 3) = 57/9. Now the problem is 8/9 + 57/9 + 2/9. Adding the numerators, we get 8 + 57 + 2 = 67. So the result is 67/9. Finally, we convert this improper fraction back into a mixed number. 67 divided by 9 is 7 with a remainder of 4, so the answer is 7\frac{4}{9}. This problem emphasizes the importance of converting mixed numbers to improper fractions before performing addition or subtraction. Finding the least common multiple and converting fractions to equivalent forms with the common denominator is a fundamental skill in fraction arithmetic. Breaking the problem down into manageable steps helps ensure accuracy. Regular practice with these types of problems will strengthen your understanding and ability to work with fractions and mixed numbers efficiently. Remember, the key is to follow a systematic approach and double-check your calculations to avoid errors.

This final problem involves subtracting mixed numbers and fractions: 5\frac{3}{4} - 2\frac{4}{5} - \frac{5}{6}. Our first step is to convert the mixed numbers into improper fractions. 5\frac{3}{4} becomes (5 * 4 + 3) / 4 = 23/4, and 2\frac{4}{5} becomes (2 * 5 + 4) / 5 = 14/5. Now the problem is 23/4 - 14/5 - \frac{5}{6}. To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 4, 5, and 6 is 60. We convert each fraction to an equivalent fraction with a denominator of 60. 23/4 becomes (23 * 15) / (4 * 15) = 345/60. 14/5 becomes (14 * 12) / (5 * 12) = 168/60. And \frac{5}{6} becomes (5 * 10) / (6 * 10) = 50/60. The problem now transforms into 345/60 - 168/60 - 50/60. Subtracting the numerators, we get 345 - 168 - 50 = 127. So the result is 127/60. Finally, we convert this improper fraction back into a mixed number. 127 divided by 60 is 2 with a remainder of 7, so the answer is 2\frac{7}{60}. This problem exemplifies the comprehensive process of working with mixed numbers and fractions, including conversion to improper fractions, finding the least common multiple, and converting back to mixed numbers. It highlights the importance of meticulous calculations and a step-by-step approach to ensure accuracy. Problems like these are excellent for reinforcing your understanding of fraction arithmetic and building confidence in your mathematical abilities. Remember, practice makes perfect, so continue to work through similar problems to solidify your skills.