Mastering Fraction Arithmetic Step By Step Solutions

When dealing with mixed number subtraction, such as the problem 5 1/3 - 2 3/4, it's essential to approach it systematically to avoid errors. This problem combines both whole numbers and fractions, requiring us to handle each part carefully. Our primary focus here is on providing a step-by-step solution that not only answers the question but also enhances your understanding of fraction arithmetic. This involves converting mixed numbers into improper fractions, finding a common denominator, performing the subtraction, and then simplifying the result back into a mixed number if necessary.

To begin, convert the mixed numbers into improper fractions. A mixed number is a whole number combined with a fraction, like 5 1/3. To convert it, multiply the whole number by the denominator of the fraction and add the numerator. Place this result over the original denominator. For 5 1/3, this becomes (5 * 3) + 1 / 3 = 16/3. Similarly, convert 2 3/4 into an improper fraction: (2 * 4) + 3 / 4 = 11/4. Now, the problem transforms into 16/3 - 11/4, which is easier to handle arithmetically.

Next, to subtract these fractions, a common denominator is needed. The common denominator is the least common multiple (LCM) of the two denominators, 3 and 4. The LCM of 3 and 4 is 12. Convert both fractions to have this denominator. To convert 16/3 to an equivalent fraction with a denominator of 12, multiply both the numerator and the denominator by 4: (16 * 4) / (3 * 4) = 64/12. For 11/4, multiply both the numerator and the denominator by 3: (11 * 3) / (4 * 3) = 33/12. The problem now reads 64/12 - 33/12.

With a common denominator, subtraction can proceed by subtracting the numerators while keeping the denominator the same: 64/12 - 33/12 = (64 - 33) / 12 = 31/12. The result, 31/12, is an improper fraction. To convert it back to a mixed number, divide the numerator by the denominator. 31 divided by 12 is 2 with a remainder of 7. Thus, 31/12 is equivalent to 2 7/12. This is the simplified answer to the original problem, 5 1/3 - 2 3/4.

Moving on to the next problem, 5/6 - 2 3/4, we encounter a scenario where subtracting a larger mixed number from a smaller fraction results in a negative value. This problem emphasizes the importance of understanding how to handle negative fractions and mixed numbers. To accurately solve this, we'll again convert the mixed number into an improper fraction, find a common denominator, and then perform the subtraction, paying close attention to the sign of the result. The key here is to ensure that we correctly interpret the negative sign and what it means in the context of fraction subtraction.

Initially, convert the mixed number 2 3/4 into an improper fraction. As demonstrated earlier, this involves multiplying the whole number by the denominator and adding the numerator: (2 * 4) + 3 / 4 = 11/4. Now the problem is 5/6 - 11/4. This setup clearly shows that we are subtracting a larger value from a smaller one, which will result in a negative answer. Before we can subtract, we need to find a common denominator for 6 and 4. The least common multiple (LCM) of 6 and 4 is 12.

Convert both fractions to have a denominator of 12. For 5/6, multiply both the numerator and denominator by 2: (5 * 2) / (6 * 2) = 10/12. For 11/4, multiply both the numerator and denominator by 3: (11 * 3) / (4 * 3) = 33/12. The problem is now 10/12 - 33/12. Subtracting the numerators gives us (10 - 33) / 12 = -23/12.

The result, -23/12, is an improper fraction and negative. To convert it into a mixed number, divide the absolute value of the numerator by the denominator. 23 divided by 12 is 1 with a remainder of 11. Thus, 23/12 is equivalent to 1 11/12. Since the fraction was negative, the mixed number is -1 11/12. This final answer illustrates the result of subtracting a larger value from a smaller one in the realm of fractions.

Problem number 9, -4 1/3 + 3, presents the addition of a negative mixed number and a positive integer. This type of problem is fundamental in understanding how to operate with both fractions and integers, particularly when dealing with negative numbers. The strategy here involves converting the mixed number into an improper fraction, adding it to the integer, and then simplifying the result. The key is to handle the negative sign correctly and understand how it affects the addition.

First, convert the mixed number -4 1/3 into an improper fraction. Multiply the whole number 4 by the denominator 3 and add the numerator 1, which gives us (4 * 3) + 1 = 13. Since the original mixed number is negative, the improper fraction is -13/3. Now, we are adding this to the integer 3. To add a fraction and an integer, it's helpful to express the integer as a fraction with the same denominator as the other fraction. Thus, 3 can be written as 3/1. To get a common denominator of 3, we multiply both the numerator and the denominator by 3: (3 * 3) / (1 * 3) = 9/3.

Now, the problem is -13/3 + 9/3. Adding the numerators while keeping the denominator the same, we get (-13 + 9) / 3 = -4/3. The result, -4/3, is an improper fraction and negative. To convert it into a mixed number, divide the absolute value of the numerator by the denominator. 4 divided by 3 is 1 with a remainder of 1. Thus, 4/3 is equivalent to 1 1/3. Since the fraction was negative, the mixed number is -1 1/3. This is the simplified answer to the problem, showing the addition of a negative mixed number and a positive integer.

Moving to problem 10, –4/5 + 1 1/5, we tackle the addition of a negative fraction and a positive mixed number. This exercise reinforces the concepts of working with both negative numbers and fractions, requiring careful attention to signs and fractional parts. The solution involves converting the mixed number to an improper fraction, finding a common denominator if necessary (though in this case, it's already present), and then adding the fractions. The focus remains on correctly applying the rules of addition with negative numbers.

First, convert the mixed number 1 1/5 into an improper fraction. Multiply the whole number 1 by the denominator 5 and add the numerator 1: (1 * 5) + 1 = 6. Thus, 1 1/5 is equivalent to 6/5. The problem now becomes -4/5 + 6/5. Since both fractions already have a common denominator of 5, we can proceed directly with the addition.

Add the numerators while keeping the denominator the same: (-4 + 6) / 5 = 2/5. The result is 2/5, which is a positive, proper fraction. This answer clearly demonstrates the process of adding a negative fraction to a positive mixed number, highlighting the importance of understanding how negative numbers interact with fractions in addition.

Problem 11, 2/5 + 4 - 3/4, introduces a combination of addition and subtraction involving fractions and a whole number. This problem is a comprehensive exercise in fraction arithmetic, requiring us to integrate multiple operations. The approach involves dealing with the whole number, finding a common denominator for the fractions, performing the addition and subtraction in the correct order, and simplifying the final result. This question tests not only arithmetic skills but also the ability to manage multiple steps in a single calculation.

To start, consider the whole number 4 as a fraction with a denominator of 1: 4/1. The problem now looks like 2/5 + 4/1 - 3/4. To combine these fractions, we need a common denominator. The least common multiple (LCM) of 5, 1, and 4 is 20. Convert each fraction to have a denominator of 20.

For 2/5, multiply both the numerator and the denominator by 4: (2 * 4) / (5 * 4) = 8/20. For 4/1, multiply both the numerator and the denominator by 20: (4 * 20) / (1 * 20) = 80/20. For 3/4, multiply both the numerator and the denominator by 5: (3 * 5) / (4 * 5) = 15/20. The problem now reads 8/20 + 80/20 - 15/20.

Perform the addition and subtraction from left to right: (8 + 80) / 20 - 15/20 = 88/20 - 15/20. Then, subtract the numerators: (88 - 15) / 20 = 73/20. The result, 73/20, is an improper fraction. Convert it to a mixed number by dividing 73 by 20. The quotient is 3, and the remainder is 13. Thus, 73/20 is equivalent to 3 13/20. This mixed number is the simplified solution to the combined operation problem.

Problem 12, 8/9 + 6 1/3 + 2/9, focuses on the addition of mixed numbers and fractions. This problem reinforces the skill of converting mixed numbers to improper fractions and then adding fractions with both common and uncommon denominators. The methodical approach to this problem highlights the importance of organized steps in solving multi-term fraction additions. The solution involves converting the mixed number, finding a common denominator, summing the fractions, and simplifying the result.

First, convert the mixed number 6 1/3 into an improper fraction. Multiply the whole number 6 by the denominator 3 and add the numerator 1: (6 * 3) + 1 = 19. Therefore, 6 1/3 is equivalent to 19/3. The problem now looks like 8/9 + 19/3 + 2/9. To add these fractions, we need a common denominator. The least common multiple (LCM) of 9 and 3 is 9.

The fractions 8/9 and 2/9 already have the common denominator of 9. Convert 19/3 to a fraction with a denominator of 9 by multiplying both the numerator and the denominator by 3: (19 * 3) / (3 * 3) = 57/9. The problem now is 8/9 + 57/9 + 2/9. Add the numerators while keeping the denominator the same: (8 + 57 + 2) / 9 = 67/9.

The result, 67/9, is an improper fraction. Convert it to a mixed number by dividing 67 by 9. The quotient is 7, and the remainder is 4. Thus, 67/9 is equivalent to 7 4/9. This is the simplified solution, demonstrating how to effectively add mixed numbers and fractions by converting to improper fractions and finding common denominators.

Lastly, problem 13, 5 3/4 - 2 4/5 - 5/6, is a multi-step subtraction problem involving mixed numbers and fractions. This type of problem demands a solid understanding of fraction subtraction and the ability to manage several operations in sequence. The solution requires converting mixed numbers to improper fractions, finding a common denominator, performing the subtractions step by step, and simplifying the result. This exercise serves as a comprehensive review of fraction arithmetic, emphasizing accuracy and attention to detail.

Begin by converting the mixed numbers 5 3/4 and 2 4/5 into improper fractions. For 5 3/4, multiply 5 by 4 and add 3: (5 * 4) + 3 = 23. So, 5 3/4 becomes 23/4. For 2 4/5, multiply 2 by 5 and add 4: (2 * 5) + 4 = 14. Thus, 2 4/5 is 14/5. The problem now is 23/4 - 14/5 - 5/6. To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 4, 5, and 6 is 60.

Convert each fraction to have a denominator of 60. For 23/4, multiply both the numerator and the denominator by 15: (23 * 15) / (4 * 15) = 345/60. For 14/5, multiply both the numerator and the denominator by 12: (14 * 12) / (5 * 12) = 168/60. For 5/6, multiply both the numerator and the denominator by 10: (5 * 10) / (6 * 10) = 50/60. The problem is now 345/60 - 168/60 - 50/60.

Perform the subtractions from left to right: (345 - 168) / 60 - 50/60 = 177/60 - 50/60. Then, subtract the numerators: (177 - 50) / 60 = 127/60. The result, 127/60, is an improper fraction. Convert it to a mixed number by dividing 127 by 60. The quotient is 2, and the remainder is 7. Therefore, 127/60 is equivalent to 2 7/60. This mixed number is the final, simplified answer to the multi-step subtraction problem.