Calculating Land Fraction For Tree Tomatoes In An Orchard

Introduction

In this article, we delve into a fascinating problem involving land allocation within an orchard. A farmer possesses a total of 2 3/4 hectares of land, which is dedicated to cultivating various fruit trees. Our primary objective is to determine the fraction of land that is specifically occupied by tree tomatoes. This seemingly straightforward question requires a systematic approach, involving fractions, subtraction, and careful analysis. Let's embark on this mathematical journey to unravel the solution step by step. Understanding the allocation of land is crucial for farmers to optimize their resources and maximize their yield. Effective land management is a cornerstone of successful agriculture, ensuring that each crop receives the necessary space and resources to thrive. This problem not only provides a practical application of mathematical concepts but also highlights the importance of planning and resource allocation in real-world scenarios. Let's break down the problem, explore the steps involved, and arrive at the final answer.

Problem Statement

The core of our problem lies in determining the precise fraction of land dedicated to tree tomatoes. To achieve this, we must first understand how the land is divided among different types of trees. The farmer's orchard spans 2 3/4 hectares, a significant area that accommodates a variety of fruit trees. The initial piece of information we have is that 1 1/4 hectares are occupied by apple trees. This establishes a baseline, allowing us to calculate the remaining land available for other trees. The problem introduces another layer of complexity by stating that one-fourth of the remaining land is occupied by lemon trees. This means we need to first calculate the land remaining after accounting for apple trees and then determine one-fourth of that area. Finally, the problem states that the rest of the land is occupied by tree tomatoes. This implies that after subtracting the land occupied by apple and lemon trees, the remaining area will represent the fraction of land dedicated to tree tomatoes. This step-by-step breakdown is essential for solving the problem accurately. By carefully considering each piece of information, we can construct a clear pathway to the solution. The challenge lies in accurately performing the arithmetic operations and interpreting the results in the context of the problem. Now, let's proceed with the calculations to determine the fraction of land occupied by tree tomatoes.

Step 1: Calculate the Total Land Area

The first crucial step in solving this problem is to express the total land area as an improper fraction. The farmer has 2 3/4 hectares of land, which is currently represented as a mixed number. To convert a mixed number to an improper fraction, we multiply the whole number (2) by the denominator of the fraction (4) and then add the numerator (3). This result becomes the new numerator, while the denominator remains the same. So, 2 3/4 is converted to (2 * 4 + 3) / 4 = 11/4 hectares. Converting to an improper fraction makes it easier to perform mathematical operations, particularly subtraction and multiplication. This conversion ensures that we are working with consistent units throughout the problem. Understanding the concept of improper fractions is fundamental to solving problems involving mixed numbers. By expressing the total land area as 11/4 hectares, we have a clear starting point for our calculations. This value represents the entirety of the orchard, and we will use it to determine the fractions allocated to different types of trees. Now, let's move on to the next step, which involves calculating the land occupied by apple trees.

Step 2: Calculate the Land Occupied by Apples

Having established the total land area, our next task is to determine the portion occupied by apple trees. The problem states that 1 1/4 hectares of land are dedicated to apple trees. Similar to the previous step, we need to convert this mixed number into an improper fraction. We multiply the whole number (1) by the denominator (4) and add the numerator (1), resulting in (1 * 4 + 1) / 4 = 5/4 hectares. Expressing the land occupied by apples as an improper fraction allows us to easily compare it with the total land area and perform subsequent calculations. This conversion maintains consistency in our units and simplifies the mathematical process. The 5/4 hectares represents a significant portion of the orchard dedicated to apple cultivation. This value will be crucial in determining the remaining land available for other types of trees. By accurately converting the mixed number to an improper fraction, we ensure that our calculations are precise and reliable. Now, let's proceed to the next step, which involves calculating the land remaining after accounting for the apple trees.

Step 3: Calculate the Remaining Land After Apples

Now that we know the total land area (11/4 hectares) and the land occupied by apple trees (5/4 hectares), we can calculate the remaining land. This is achieved by subtracting the land occupied by apples from the total land area. The calculation is as follows: 11/4 - 5/4. Since the fractions have the same denominator, we can simply subtract the numerators: (11 - 5) / 4 = 6/4 hectares. This fraction can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 2. Therefore, 6/4 simplifies to 3/2 hectares. Calculating the remaining land is a crucial step in determining the allocation for other trees. This subtraction provides us with the available area after accounting for the apple orchard. The 3/2 hectares represents the land that can be further divided between lemon trees and tree tomatoes. By accurately performing this subtraction, we ensure that our subsequent calculations are based on a precise value. This step highlights the importance of basic arithmetic operations in solving practical problems. Now, let's move on to the next step, which involves calculating the land occupied by lemon trees.

Step 4: Calculate the Land Occupied by Lemons

With the remaining land calculated as 3/2 hectares, we can now determine the portion occupied by lemon trees. The problem states that one-fourth of the remaining land is occupied by lemon trees. This means we need to calculate 1/4 of 3/2. To find a fraction of a fraction, we multiply the two fractions together: (1/4) * (3/2). Multiplying the numerators gives us 1 * 3 = 3, and multiplying the denominators gives us 4 * 2 = 8. Therefore, the land occupied by lemon trees is 3/8 hectares. Calculating the fraction of the remaining land is a key step in determining the allocation for lemon trees. This multiplication allows us to accurately determine the portion of land dedicated to lemon cultivation. The 3/8 hectares represents a specific fraction of the orchard allocated to lemon trees. By accurately performing this multiplication, we ensure that our subsequent calculations are precise and reliable. This step demonstrates the application of fraction multiplication in a practical context. Now, let's proceed to the final step, which involves calculating the land occupied by tree tomatoes.

Step 5: Calculate the Land Occupied by Tree Tomatoes

We have now determined the total land area (11/4 hectares), the land occupied by apple trees (5/4 hectares), and the land occupied by lemon trees (3/8 hectares). To find the land occupied by tree tomatoes, we need to subtract the land occupied by apple and lemon trees from the total land area. First, let's find a common denominator for the fractions 11/4, 5/4, and 3/8. The least common multiple of 4 and 8 is 8, so we will convert the fractions to have a denominator of 8. 11/4 is equivalent to 22/8, and 5/4 is equivalent to 10/8. Now we can perform the subtraction: 22/8 - 10/8 - 3/8. Subtracting the numerators gives us (22 - 10 - 3) / 8 = 9/8 hectares. Therefore, the land occupied by tree tomatoes is 9/8 hectares. Calculating the land occupied by tree tomatoes is the final step in solving our problem. This subtraction accurately determines the portion of the orchard dedicated to tree tomato cultivation. The 9/8 hectares represents the answer to our question. By accurately performing this subtraction, we have successfully allocated the land among the different types of trees. This step highlights the importance of combining multiple arithmetic operations to solve complex problems. Now, let's summarize our findings and present the final answer.

Final Answer: Fraction of Land Occupied by Tree Tomatoes

After carefully analyzing the problem and performing the necessary calculations, we have arrived at the solution. The land occupied by tree tomatoes is 9/8 hectares. This represents the fraction of the total orchard area that is dedicated to cultivating tree tomatoes. This final answer provides a clear and concise solution to the problem. The 9/8 hectares represents a significant portion of the orchard, highlighting the importance of tree tomatoes in the farmer's cultivation plan. By systematically working through each step, we have successfully determined the land allocation for tree tomatoes. This problem demonstrates the practical application of mathematical concepts in real-world scenarios. The ability to calculate fractions and perform arithmetic operations is crucial for effective land management and resource allocation in agriculture. We hope this detailed explanation has provided a clear understanding of the solution and the steps involved. The farmer has allocated 9/8 hectares of the orchard to tree tomatoes, completing the land allocation plan.

Conclusion

In conclusion, we have successfully navigated the mathematical problem of land allocation within an orchard. By systematically breaking down the problem into smaller steps, we were able to accurately determine the fraction of land occupied by tree tomatoes. This process involved converting mixed numbers to improper fractions, performing subtraction and multiplication of fractions, and simplifying the results. This problem highlights the importance of mathematical skills in practical applications, such as agriculture and land management. The ability to calculate fractions and perform arithmetic operations is essential for making informed decisions about resource allocation and optimization. By understanding the steps involved in solving this problem, we can apply these principles to other similar scenarios. The solution, 9/8 hectares, provides a clear understanding of the land dedicated to tree tomato cultivation. This exercise demonstrates the value of mathematical reasoning and problem-solving in real-world contexts. We hope this exploration has been both informative and insightful, showcasing the power of mathematics in everyday life.