Calculating Remaining Paper A Step-by-Step Guide
In this article, we'll dive into a practical mathematical problem involving fractions and subtraction. This scenario, centered around janitors collecting and distributing used paper, provides a relatable context for understanding these concepts. We'll break down the problem step-by-step, ensuring clarity and comprehension. Specifically, we will explore how to calculate the amount of paper remaining after a portion has been given away. This involves working with mixed numbers and performing subtraction, essential skills in everyday calculations. Let's delve into the world of fractions and problem-solving!
Understanding the Problem
In order to solve the problem effectively, it’s important to make sure we have a strong grip on the given information. First and foremost, we need to understand the initial amount of used paper that the janitors collected. According to the problem, the janitors gathered 15 2/7 kilograms of used paper from the Materials Recovery Facility (MRF). This is our starting point, and it's a mixed number, which means it combines a whole number (15) with a fraction (2/7). Mixed numbers can sometimes be tricky to work with directly, so we might need to convert them into improper fractions later on. Next, we need to consider what happened to this initial amount of paper. The janitors didn't keep all of it; instead, they gave a portion of it away. Specifically, they gave 2 3/5 kilograms of used paper to the teachers. This is another mixed number that we'll need to account for in our calculations. The teachers planned to use this paper as covers on the floor to prevent paint spills, which is a practical and environmentally conscious use of recycled materials. Understanding the teachers' use case helps us appreciate the real-world relevance of this mathematical problem. Finally, the core question we need to answer is: how much paper was left after the janitors gave some away to the teachers? This means we're dealing with a subtraction problem. We need to subtract the amount of paper given to the teachers (2 3/5 kilograms) from the initial amount collected (15 2/7 kilograms). To successfully perform this subtraction, we'll need to make sure both amounts are expressed in the same format, likely as improper fractions. By carefully identifying and understanding each piece of information provided in the problem, we set ourselves up for accurate calculations and a clear solution.
Converting Mixed Numbers to Improper Fractions
Before we can subtract the amounts of paper, we need to convert the mixed numbers into improper fractions. This is a crucial step because it allows us to perform arithmetic operations like subtraction more easily. Let’s start with the first mixed number, 15 2/7 kilograms. To convert this to an improper fraction, we follow a simple process. First, we multiply the whole number part (15) by the denominator of the fractional part (7). This gives us 15 * 7 = 105. Next, we add the numerator of the fractional part (2) to the result: 105 + 2 = 107. This new number, 107, becomes the numerator of our improper fraction. The denominator stays the same as the original fraction, which is 7. So, 15 2/7 kilograms is equivalent to 107/7 kilograms as an improper fraction. Now, let’s convert the second mixed number, 2 3/5 kilograms, to an improper fraction using the same method. We multiply the whole number (2) by the denominator (5): 2 * 5 = 10. Then, we add the numerator (3): 10 + 3 = 13. This gives us the numerator of our improper fraction, and the denominator remains 5. Therefore, 2 3/5 kilograms is equivalent to 13/5 kilograms as an improper fraction. By converting both mixed numbers into improper fractions, we have 107/7 kilograms and 13/5 kilograms. These fractions are now in a format that makes subtraction straightforward. We can proceed to the next step, which involves finding a common denominator so that we can subtract the fractions accurately. This conversion is a foundational skill in working with fractions and is essential for solving a wide range of mathematical problems.
Finding a Common Denominator
To subtract two fractions, they must have the same denominator. This common denominator acts as a common unit, allowing us to accurately compare and subtract the numerators. In our problem, we need to subtract 13/5 from 107/7. The denominators are 7 and 5, so we need to find the least common multiple (LCM) of these two numbers. The LCM is the smallest number that both 7 and 5 divide into evenly. One way to find the LCM is to list the multiples of each number until we find a common one. The multiples of 7 are: 7, 14, 21, 28, 35, 42, and so on. The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, and so on. We can see that the smallest number that appears in both lists is 35. Therefore, the LCM of 7 and 5 is 35. This means that 35 will be our common denominator. Now, we need to convert both fractions to have this denominator. To convert 107/7 to an equivalent fraction with a denominator of 35, we need to multiply both the numerator and the denominator by the same number. Since 7 multiplied by 5 equals 35, we multiply both the numerator and the denominator of 107/7 by 5. This gives us (107 * 5) / (7 * 5) = 535/35. So, 107/7 is equivalent to 535/35. Next, we need to convert 13/5 to an equivalent fraction with a denominator of 35. Since 5 multiplied by 7 equals 35, we multiply both the numerator and the denominator of 13/5 by 7. This gives us (13 * 7) / (5 * 7) = 91/35. So, 13/5 is equivalent to 91/35. Now that both fractions have the same denominator, we can proceed with the subtraction. We have 535/35 and 91/35, which are ready to be subtracted.
Subtracting the Fractions
With both fractions now sharing a common denominator of 35, we can proceed with the subtraction. Our goal is to subtract the amount of paper given to the teachers (91/35 kilograms) from the total amount of paper collected (535/35 kilograms). To subtract fractions with a common denominator, we simply subtract the numerators and keep the denominator the same. In this case, we subtract 91 from 535, which gives us 535 - 91 = 444. The denominator remains 35, so our result is 444/35 kilograms. This fraction represents the amount of paper remaining after the janitors gave some to the teachers. However, 444/35 is an improper fraction, which means the numerator is larger than the denominator. While this is a perfectly valid answer, it's often more helpful to express the result as a mixed number so that we can better understand the quantity. To convert 444/35 to a mixed number, we need to divide the numerator (444) by the denominator (35). When we divide 444 by 35, we get 12 as the whole number part and a remainder of 24. This means that 444/35 is equivalent to 12 whole units and 24/35 of another unit. Therefore, the mixed number is 12 24/35. So, the janitors had 12 24/35 kilograms of paper left. This result gives us a clear picture of the amount of paper remaining. We've successfully subtracted the fractions and expressed the answer in a way that's easy to understand.
Expressing the Answer as a Mixed Number
As we determined in the previous section, the remaining amount of paper is represented by the improper fraction 444/35 kilograms. While this is a mathematically correct answer, it’s often more intuitive and practical to express it as a mixed number. A mixed number combines a whole number and a proper fraction, making it easier to visualize and understand the quantity. To convert the improper fraction 444/35 into a mixed number, we perform division. We divide the numerator (444) by the denominator (35) to find out how many whole units we have and what the remainder is. When we divide 444 by 35, we find that 35 goes into 444 twelve times (12 * 35 = 420) with a remainder of 24 (444 - 420 = 24). This means we have 12 whole kilograms of paper. The remainder, 24, becomes the numerator of the fractional part of our mixed number, and the denominator remains the same (35). Therefore, the mixed number is 12 24/35. This tells us that the janitors had 12 whole kilograms of paper and an additional 24/35 of a kilogram remaining. This format is much easier to grasp than the improper fraction 444/35. For example, we can easily see that the janitors had more than 12 kilograms but less than 13 kilograms of paper left. Expressing the answer as a mixed number provides a clearer understanding of the quantity and makes it more relatable to real-world situations. In summary, converting improper fractions to mixed numbers is a valuable skill that helps us interpret and communicate mathematical results more effectively. In this case, it allows us to confidently state that the janitors had 12 24/35 kilograms of paper remaining.
Final Answer and Conclusion
After carefully working through the steps, we've arrived at the final answer to our problem. The janitors had 12 24/35 kilograms of used paper remaining after giving some to the teachers. This answer is expressed as a mixed number, which provides a clear and intuitive understanding of the quantity. We started with the initial amount of paper collected, which was 15 2/7 kilograms. Then, we subtracted the amount given to the teachers, which was 2 3/5 kilograms. To perform this subtraction, we first converted the mixed numbers into improper fractions: 15 2/7 became 107/7, and 2 3/5 became 13/5. Next, we found a common denominator for the fractions, which was 35. We converted both fractions to have this denominator: 107/7 became 535/35, and 13/5 became 91/35. With a common denominator, we subtracted the fractions: 535/35 - 91/35 = 444/35. Finally, we converted the improper fraction 444/35 back into a mixed number, which gave us 12 24/35 kilograms. This entire process highlights the importance of understanding and manipulating fractions in practical situations. From converting mixed numbers to improper fractions to finding common denominators and performing subtraction, each step is crucial for arriving at the correct answer. This problem demonstrates how mathematical skills can be applied to real-world scenarios, such as managing resources in a school or facility. By breaking down the problem into manageable steps, we've shown how even complex calculations can be approached with confidence and accuracy. The final answer, 12 24/35 kilograms, provides a clear and concise solution to the question posed.