Don José Needs To Buy Nails For 2 Projects; He Needs 3/4 For One And 3/9 For The Other. How Many Nails Does He Need To Buy In Total?

Introduction

In this practical mathematical scenario, we delve into the arithmetic problem faced by Don José, who needs to purchase nails for two distinct projects. This problem serves as a great example of how fractions are used in everyday situations. Understanding fractions and their addition is a fundamental concept in mathematics, and this situation provides a relatable context for grasping these skills. For his first project, Don José requires 3/4 of a certain unit of nails, while for the second project, he needs 3/9 of the same unit. The core question we aim to solve is: how many nails, in total, does Don José need to buy to complete both projects? This involves adding two fractions, which requires finding a common denominator and then summing the numerators. Let's break down the problem step by step to arrive at the correct solution.

Understanding the Problem: Don José's Dilemma

To fully grasp the mathematical challenge, let's dissect the problem statement. Don José is embarking on two separate projects that necessitate the use of nails. For the first project, the nail requirement is quantified as 3/4. This means that if we were to divide the total nails needed for a project into four equal parts, Don José needs three of those parts. Now, for the second project, the nail requirement is specified as 3/9. Here, if we divide the total nails into nine equal parts, Don José needs three of those parts. The critical question that arises is: what is the cumulative amount of nails Don José needs to purchase to successfully execute both projects? To answer this, we must perform an addition of fractions. This seemingly simple problem underscores the practical applications of mathematical concepts in real-life scenarios. Understanding how to add fractions is not just an academic exercise; it's a skill that empowers us to solve tangible problems, like the one Don José is facing. Whether it's carpentry, cooking, or any other field that involves proportions, the ability to work with fractions is invaluable. Therefore, let's proceed to solve this problem with clarity and precision, ensuring Don José has the exact number of nails he needs for his projects.

Step 1: Identifying the Fractions

The initial step in solving Don José's nail conundrum is to clearly identify the fractions involved. The problem explicitly states that Don José needs 3/4 of a unit of nails for the first project and 3/9 of the same unit for the second project. These fractions, 3/4 and 3/9, represent the proportion of nails required for each project relative to a whole unit. In mathematical terms, a fraction consists of two primary components: the numerator and the denominator. The numerator, which is the number above the fraction bar, indicates the number of parts we have. In the fraction 3/4, the numerator is 3, signifying that we have three parts. The denominator, which is the number below the fraction bar, represents the total number of equal parts the whole is divided into. In the fraction 3/4, the denominator is 4, indicating that the whole is divided into four equal parts. Similarly, in the fraction 3/9, the numerator is 3, and the denominator is 9, meaning we have three parts out of a total of nine equal parts. To successfully add these fractions and determine the total number of nails Don José needs, we must first find a common denominator. This crucial step allows us to combine the fractions accurately and arrive at the correct answer. Recognizing and understanding the components of fractions is the foundation for performing various mathematical operations with them, including addition, subtraction, multiplication, and division.

Step 2: Finding the Least Common Denominator (LCD)

Before we can add the fractions 3/4 and 3/9, a crucial step is to find the least common denominator (LCD). The LCD is the smallest common multiple of the denominators of the fractions we are trying to add. In this case, the denominators are 4 and 9. To find the LCD, we can list the multiples of each denominator and identify the smallest multiple they share. Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, ... Multiples of 9 are: 9, 18, 27, 36, 45, ... By examining these lists, we can see that the smallest multiple that both 4 and 9 share is 36. Therefore, the LCD for the fractions 3/4 and 3/9 is 36. Alternatively, we can use the prime factorization method to find the LCD. Prime factorization involves breaking down each number into its prime factors. The prime factorization of 4 is 2 x 2, or 2². The prime factorization of 9 is 3 x 3, or 3². To find the LCD, we take the highest power of each prime factor present in the factorizations and multiply them together. In this case, we have 2² and 3², so the LCD is 2² x 3² = 4 x 9 = 36. Understanding and finding the LCD is a fundamental step in adding fractions with different denominators. It allows us to express the fractions with a common base, making the addition process straightforward and accurate. Once we have the LCD, we can proceed to convert each fraction to an equivalent fraction with the LCD as the new denominator.

Step 3: Converting Fractions to Equivalent Fractions

With the least common denominator (LCD) of 36 identified, the next critical step is to convert the fractions 3/4 and 3/9 into equivalent fractions with 36 as the denominator. This conversion is essential because fractions can only be added directly if they share a common denominator. To convert 3/4 to an equivalent fraction with a denominator of 36, we need to determine what number to multiply both the numerator and the denominator of 3/4 by so that the new denominator is 36. We can find this number by dividing the LCD, 36, by the original denominator, 4. 36 ÷ 4 = 9. So, we multiply both the numerator and the denominator of 3/4 by 9: (3 x 9) / (4 x 9) = 27/36. Therefore, 3/4 is equivalent to 27/36. Now, let's convert 3/9 to an equivalent fraction with a denominator of 36. We divide the LCD, 36, by the original denominator, 9. 36 ÷ 9 = 4. So, we multiply both the numerator and the denominator of 3/9 by 4: (3 x 4) / (9 x 4) = 12/36. Therefore, 3/9 is equivalent to 12/36. By converting the fractions to equivalent forms with a common denominator, we have prepared them for addition. The fractions 27/36 and 12/36 now represent the same proportions as 3/4 and 3/9, respectively, but with a common base that allows for straightforward addition. This process of finding equivalent fractions is a fundamental skill in working with fractions and is crucial for solving various mathematical problems involving fractional quantities.

Step 4: Adding the Equivalent Fractions

Now that we have successfully converted the fractions 3/4 and 3/9 into equivalent fractions with a common denominator of 36, we can proceed to the addition step. We have the equivalent fractions 27/36 and 12/36, which represent the nail requirements for Don José's two projects. To add fractions with a common denominator, the process is quite simple: we add the numerators while keeping the denominator the same. In this case, we add the numerators 27 and 12. 27 + 12 = 39. So, the sum of the numerators is 39. The denominator remains 36. Therefore, the sum of the fractions 27/36 and 12/36 is 39/36. This result, 39/36, represents the total amount of nails Don José needs for both projects, expressed as a fraction of the whole unit. However, 39/36 is an improper fraction, meaning the numerator is greater than the denominator. While 39/36 is a correct answer, it's often preferable to express it as a mixed number or simplify it if possible. To convert an improper fraction to a mixed number, we divide the numerator by the denominator and express the remainder as a fraction of the denominator. Alternatively, we can simplify the improper fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). The next step will explore how to simplify this result to make it more understandable and practical.

Step 5: Simplifying the Result

After adding the equivalent fractions, we arrived at the result 39/36, which represents the total amount of nails Don José needs. However, this fraction is an improper fraction, meaning the numerator (39) is greater than the denominator (36). While mathematically correct, it's often more practical to express the result as a mixed number or a simplified fraction. Let's first convert the improper fraction 39/36 into a mixed number. To do this, we divide the numerator (39) by the denominator (36). 39 ÷ 36 = 1 with a remainder of 3. This means that 39/36 is equal to 1 whole unit and 3/36 of another unit. So, the mixed number is 1 3/36. Now, let's simplify the fractional part of the mixed number, 3/36. To simplify a fraction, we divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 3 and 36 is 3. Dividing both the numerator and the denominator by 3, we get: (3 ÷ 3) / (36 ÷ 3) = 1/12. Therefore, the simplified fractional part is 1/12. Combining the whole number and the simplified fraction, we get the final answer as a mixed number: 1 1/12. This means Don José needs 1 full unit of nails and 1/12 of another unit to complete both projects. Expressing the answer in its simplest form not only makes it easier to understand but also provides a more practical quantity for Don José to purchase. In this form, it is clear that he needs slightly more than one full unit of nails.

Conclusion: Don José's Total Nail Requirement

In conclusion, by meticulously working through the problem, we've determined the total number of nails Don José needs for his two projects. Starting with the initial fractions representing the nail requirements for each project, 3/4 and 3/9, we navigated the process of finding a common denominator, converting the fractions, adding them, and finally, simplifying the result. Our journey through the mathematical steps has led us to the final answer: Don José needs 1 1/12 units of nails. This means he requires one full unit of nails and an additional 1/12 of a unit. This solution not only answers the initial question but also demonstrates the practical application of fraction addition in everyday scenarios. Understanding fractions and how to manipulate them is a fundamental skill in mathematics, with relevance extending far beyond the classroom. From carpentry to cooking, working with proportions is a common task, and the ability to add and simplify fractions is invaluable. Don José's nail problem serves as a tangible example of how these skills are essential for solving real-world challenges. By breaking down the problem into manageable steps and applying the principles of fraction addition, we've successfully determined the precise amount of nails Don José needs, ensuring he is well-equipped for his projects. This exercise highlights the importance of mathematical literacy and its role in practical problem-solving.

Fractions, Addition of Fractions, Least Common Denominator (LCD), Equivalent Fractions, Mixed Number, Simplifying Fractions