If Three Meters Of Fabric Cost $6,000, How Much Will Eleven Meters Of The Same Fabric Cost?
In our daily lives, we often encounter situations where we need to calculate the cost of items based on their quantity or length. A common example is buying fabric, where the price is usually given per meter or yard. This article will delve into a practical problem involving calculating the cost of fabric and provide a step-by-step solution. The original question asks: "If three meters of fabric cost $6,000, how much will eleven meters of the same fabric cost?" This is a classic problem involving direct proportion, where the cost increases linearly with the length of the fabric. Understanding direct proportion is crucial for solving various real-world problems, from calculating grocery bills to determining material costs for construction projects. Before diving into the solution, it's essential to grasp the underlying concepts and principles. We will explore the concept of direct proportion, which forms the foundation for solving this type of problem. Direct proportion essentially means that if one quantity increases, the other quantity increases at the same rate. In this case, as the length of the fabric increases, the total cost also increases proportionally. This means that the ratio between the length of the fabric and its cost remains constant. By understanding this fundamental principle, we can set up a simple proportion to solve the problem efficiently. This article will not only provide the solution but also explain the reasoning behind each step, ensuring a clear understanding of the underlying mathematical principles. We will break down the problem into smaller, manageable steps, making it easier to follow and apply to similar situations. Whether you're a student learning about proportions or someone who wants to brush up on their math skills, this article will provide a comprehensive guide to solving fabric cost calculation problems.
To effectively tackle the problem of calculating the cost of eleven meters of fabric, it's crucial to first grasp the concept of direct proportion. In direct proportion, two quantities increase or decrease together at a constant rate. This means that if one quantity doubles, the other quantity also doubles; if one quantity triples, the other quantity triples as well. In our fabric cost scenario, the length of the fabric and the total cost are directly proportional. This is because as you buy more fabric, the cost increases accordingly, and the ratio between the length and the cost remains constant. Mathematically, we can represent this relationship as:
Cost / Length = Constant
This constant represents the price per meter of the fabric. Understanding this relationship is key to solving the problem efficiently. To further illustrate this concept, consider a simple example. If one meter of fabric costs $2, then two meters will cost $4, three meters will cost $6, and so on. You can see that the cost increases proportionally with the length of the fabric. The ratio of cost to length remains constant at $2 per meter. This principle applies to our problem as well. We know that three meters of fabric cost $6,000, and we need to find the cost of eleven meters of the same fabric. The cost per meter will remain the same, allowing us to set up a proportion and solve for the unknown cost. The ability to identify and apply the concept of direct proportion is a valuable skill in mathematics and real-life scenarios. It allows us to make predictions and calculations based on known relationships between quantities. In the context of our fabric cost problem, understanding direct proportion enables us to determine the cost of any length of fabric, given the cost of a specific length. In the next section, we will apply this understanding to set up a proportion and solve for the cost of eleven meters of fabric. We will break down the steps clearly and concisely, ensuring that you can follow along and apply the same method to similar problems in the future. Remember, the key is to recognize the proportional relationship and use it to establish a connection between the known and unknown quantities. By mastering this concept, you will be well-equipped to handle a wide range of problems involving proportional reasoning.
Now that we have a solid understanding of direct proportion, let's apply this concept to our specific problem: If three meters of fabric cost $6,000, how much will eleven meters of the same fabric cost? The first step in solving this problem is to set up a proportion. A proportion is an equation that states that two ratios are equal. In our case, the ratio of the length of the fabric to its cost will be the same for both the given information (three meters costing $6,000) and the unknown cost for eleven meters. We can express this relationship as follows:
(Length 1) / (Cost 1) = (Length 2) / (Cost 2)
Let's define our variables:
- Length 1 = 3 meters
- Cost 1 = $6,000
- Length 2 = 11 meters
- Cost 2 = X (the unknown cost we want to find)
Now, we can substitute these values into our proportion:
3 / 6000 = 11 / X
This equation represents the proportional relationship between the length of the fabric and its cost. The left side of the equation represents the ratio of length to cost for the given information, while the right side represents the same ratio for the unknown cost. By setting these two ratios equal to each other, we can solve for the unknown value, X. Setting up the proportion correctly is crucial for obtaining the correct answer. It ensures that we are comparing the quantities in the right way and that the relationship between them is accurately represented. In this case, we have set up the proportion with the length in the numerator and the cost in the denominator, maintaining consistency on both sides of the equation. Once the proportion is set up, the next step is to solve for the unknown variable. This involves using algebraic techniques to isolate the variable on one side of the equation. In the following section, we will walk through the steps of solving the proportion we have set up to find the cost of eleven meters of fabric. We will use cross-multiplication, a common method for solving proportions, and show how to arrive at the final answer. By mastering the skill of setting up proportions, you will be able to tackle a wide range of problems involving proportional relationships, from scaling recipes to calculating distances on maps. It's a fundamental concept in mathematics and has numerous practical applications in everyday life.
With the proportion set up as 3 / 6000 = 11 / X, the next step is to solve for X, which represents the unknown cost of eleven meters of fabric. A common method for solving proportions is cross-multiplication. This technique involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the two products equal to each other.
Applying cross-multiplication to our proportion, we get:
3 * X = 11 * 6000
Now, we have a simple algebraic equation that we can solve for X. First, let's calculate the product on the right side of the equation:
3 * X = 66000
To isolate X, we need to divide both sides of the equation by 3:
X = 66000 / 3
Performing the division, we find:
X = 22000
Therefore, the cost of eleven meters of the same fabric is $22,000. Solving the proportion using cross-multiplication is a straightforward and efficient method. It allows us to eliminate the fractions and work with a simple linear equation. The key is to remember to multiply the numerator of one fraction by the denominator of the other and then set the products equal. Once we have the equation, we can use basic algebraic operations, such as division, to isolate the unknown variable and find its value. It is always a good practice to check the solution to ensure it makes sense in the context of the problem. In our case, we found that eleven meters of fabric cost $22,000. Since three meters cost $6,000, it is reasonable that eleven meters would cost more. We can also check the proportionality by verifying that the ratio of length to cost is consistent. For three meters, the ratio is 3 / 6000 = 0.0005. For eleven meters, the ratio is 11 / 22000 = 0.0005. The ratios are the same, confirming that our solution is correct. In the next section, we will summarize the solution and highlight the key steps involved in solving this type of problem. We will also discuss the importance of understanding the underlying concepts and applying them to real-world scenarios. By mastering the skill of solving proportions, you will be able to tackle a wide range of problems involving proportional relationships, from scaling recipes to calculating distances on maps.
In this article, we addressed the problem of calculating the cost of eleven meters of fabric, given that three meters of the same fabric cost $6,000. To solve this problem, we used the concept of direct proportion, which states that two quantities increase or decrease together at a constant rate. We began by understanding the problem and identifying the proportional relationship between the length of the fabric and its cost. We then set up a proportion, which is an equation that states that two ratios are equal. In our case, the proportion was:
3 / 6000 = 11 / X
where X represented the unknown cost of eleven meters of fabric. To solve for X, we used the technique of cross-multiplication, which involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the two products equal. This gave us the equation:
3 * X = 11 * 6000
Simplifying the equation, we found:
3 * X = 66000
Dividing both sides by 3, we obtained the solution:
X = 22000
Therefore, eleven meters of the same fabric will cost $22,000. This problem illustrates the importance of understanding proportional relationships and how they can be used to solve real-world problems. By setting up a proportion and using cross-multiplication, we were able to efficiently calculate the unknown cost. The ability to solve problems involving proportions is a valuable skill that can be applied in various contexts, such as calculating unit prices, scaling recipes, and determining distances on maps. In conclusion, mastering the concept of direct proportion and the techniques for solving proportions is essential for developing strong mathematical skills and problem-solving abilities. This article has provided a step-by-step guide to solving a fabric cost calculation problem, highlighting the key concepts and methods involved. By practicing these skills, you will be well-equipped to tackle similar problems in the future and apply them to a wide range of real-life situations. Remember, the key is to understand the underlying principles, set up the proportion correctly, and use the appropriate techniques to solve for the unknown variable. With practice and a solid understanding of the concepts, you can confidently solve problems involving proportional reasoning and make informed decisions in various contexts.