Subtract Sums Of Mixed Fractions A Step-by-Step Guide
Introduction
In this article, we will tackle a mathematical problem involving the subtraction of sums of mixed fractions. Specifically, we aim to subtract the sum of 3 1/3 and 2 2/3 from the sum of 2 3/4 and 5 1/3. This problem requires a solid understanding of mixed fraction arithmetic, including addition and subtraction. Mixed fractions, which combine whole numbers and proper fractions, are commonly encountered in various real-life situations, making proficiency in their manipulation essential. Mastering such problems enhances not only mathematical skills but also problem-solving abilities applicable across different domains. This comprehensive guide will walk you through the step-by-step process, ensuring clarity and accuracy in your calculations.
Understanding Mixed Fractions
Before diving into the problem, let's clarify what mixed fractions are and how to work with them. A mixed fraction is a combination of a whole number and a proper fraction (a fraction where the numerator is less than the denominator). For example, 3 1/3 is a mixed fraction, where 3 is the whole number part and 1/3 is the fractional part. To perform arithmetic operations like addition and subtraction, mixed fractions are often converted into improper fractions. An improper fraction is one where the numerator is greater than or equal to the denominator. Converting mixed fractions to improper fractions simplifies calculations and ensures accurate results. This conversion process involves multiplying the whole number by the denominator of the fractional part, adding the numerator, and placing the result over the original denominator. This fundamental concept is crucial for solving the given problem efficiently and accurately, laying the groundwork for more complex mathematical operations involving fractions.
Step 1: Convert Mixed Fractions to Improper Fractions
To begin solving the problem, the first crucial step is to convert all mixed fractions into improper fractions. This conversion is necessary because arithmetic operations such as addition and subtraction are more straightforward with improper fractions. Let's start by converting 3 1/3 to an improper fraction. To do this, multiply the whole number part (3) by the denominator (3) and add the numerator (1), resulting in (3 * 3) + 1 = 10. Place this result over the original denominator (3), giving us the improper fraction 10/3. Next, we convert 2 2/3. Multiply the whole number part (2) by the denominator (3) and add the numerator (2), resulting in (2 * 3) + 2 = 8. Place this over the original denominator (3), giving us 8/3. Similarly, for 2 3/4, we multiply 2 by 4 and add 3, resulting in (2 * 4) + 3 = 11. Place this over the original denominator (4), giving us 11/4. Finally, for 5 1/3, we multiply 5 by 3 and add 1, resulting in (5 * 3) + 1 = 16. Place this over the original denominator (3), giving us 16/3. Now that all mixed fractions are converted to improper fractions, we can proceed with the addition and subtraction operations more easily.
Step 2: Calculate the Sum of 3 1/3 and 2 2/3
Now that we have converted the mixed fractions into improper fractions, we can proceed to calculate the sum of 3 1/3 and 2 2/3. As we determined in the previous step, 3 1/3 is equivalent to 10/3, and 2 2/3 is equivalent to 8/3. To add these two fractions, we simply add their numerators since they share a common denominator. Thus, we have 10/3 + 8/3. Adding the numerators, we get 10 + 8 = 18. Therefore, the sum is 18/3. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3. Dividing 18 by 3 gives us 6, and dividing 3 by 3 gives us 1. So, the simplified fraction is 6/1, which is equal to the whole number 6. This result represents the sum of the first two mixed fractions in their simplest form, making it easier to proceed with the next steps of the problem. The ability to simplify fractions is a crucial skill in mathematics, as it allows for more manageable calculations and clearer understanding of the quantities involved.
Step 3: Calculate the Sum of 2 3/4 and 5 1/3
Next, we need to calculate the sum of 2 3/4 and 5 1/3. From Step 1, we know that 2 3/4 is equivalent to 11/4 and 5 1/3 is equivalent to 16/3. To add these fractions, we first need to find a common denominator. The least common multiple (LCM) of 4 and 3 is 12, so we will convert both fractions to have a denominator of 12. To convert 11/4 to a fraction with a denominator of 12, we multiply both the numerator and the denominator by 3, resulting in (11 * 3) / (4 * 3) = 33/12. Similarly, to convert 16/3 to a fraction with a denominator of 12, we multiply both the numerator and the denominator by 4, resulting in (16 * 4) / (3 * 4) = 64/12. Now we can add the two fractions: 33/12 + 64/12. Adding the numerators, we get 33 + 64 = 97. Thus, the sum is 97/12. This improper fraction can be left as is for the subtraction step, or it can be converted back into a mixed fraction. Dividing 97 by 12 gives us 8 with a remainder of 1, so the mixed fraction form is 8 1/12. This result represents the sum of the second pair of mixed fractions and is a crucial component in solving the overall problem.
Step 4: Subtract the First Sum from the Second Sum
Now that we have calculated both sums, we can proceed to subtract the first sum from the second sum. In Step 2, we found that the sum of 3 1/3 and 2 2/3 is 6, which can be written as 6/1. In Step 3, we found that the sum of 2 3/4 and 5 1/3 is 97/12. Therefore, we need to subtract 6/1 from 97/12. To perform this subtraction, we need a common denominator. The least common multiple (LCM) of 1 and 12 is 12, so we will convert 6/1 to a fraction with a denominator of 12. To do this, we multiply both the numerator and the denominator of 6/1 by 12, resulting in (6 * 12) / (1 * 12) = 72/12. Now we can subtract the fractions: 97/12 - 72/12. Subtracting the numerators, we get 97 - 72 = 25. Therefore, the result of the subtraction is 25/12. This improper fraction can be converted back into a mixed fraction for clarity. Dividing 25 by 12 gives us 2 with a remainder of 1, so the mixed fraction form is 2 1/12. This final result represents the solution to the original problem, which is the difference between the two sums of mixed fractions.
Step 5: Simplify the Result (If Necessary)
In the previous step, we found that the result of subtracting the sum of 3 1/3 and 2 2/3 from the sum of 2 3/4 and 5 1/3 is 25/12, which is equivalent to the mixed fraction 2 1/12. To ensure our answer is in its simplest form, we need to check if the fractional part, 1/12, can be further simplified. In this case, 1 and 12 have no common factors other than 1, which means the fraction 1/12 is already in its simplest form. Therefore, the mixed fraction 2 1/12 is also in its simplest form. The final simplified result is 2 1/12. This step underscores the importance of always checking for simplification, as it ensures the answer is presented in its most concise and understandable form. Simplification not only makes the answer cleaner but also reflects a thorough understanding of fractional arithmetic.
Conclusion
In this detailed walkthrough, we successfully subtracted the sum of 3 1/3 and 2 2/3 from the sum of 2 3/4 and 5 1/3. The problem required us to convert mixed fractions to improper fractions, find common denominators, perform addition and subtraction, and simplify the final result. By following each step methodically, we arrived at the solution 2 1/12. This exercise highlights the importance of understanding and applying fundamental arithmetic operations with fractions. Proficiency in these skills is crucial for success in more advanced mathematical concepts. Furthermore, the ability to break down complex problems into manageable steps is a valuable skill that extends beyond mathematics, aiding in problem-solving in various aspects of life. Mastering these techniques not only enhances mathematical competence but also fosters critical thinking and analytical abilities.
