Calculating The Cost Of Cloth: A Step-by-Step Guide
In this comprehensive guide, we will delve into the step-by-step process of calculating the cost of cloth. Specifically, we will address the scenario where the cost of a meter of cloth is ₹16 1/2, and we aim to determine the total cost for 8 3/4 meters of the same cloth. This problem involves working with mixed fractions and applying fundamental arithmetic operations. By breaking down the problem into smaller, manageable steps, we will ensure a clear understanding of the solution. Calculating the cost of materials is a common task in various real-life situations, such as tailoring, crafting, and home improvement projects. Therefore, mastering this skill is highly valuable. We will also explore the underlying principles of proportional reasoning, which is a crucial mathematical concept. This concept allows us to understand how quantities change in relation to each other. Through detailed explanations and practical examples, this guide will equip you with the knowledge and skills necessary to solve similar problems confidently. Whether you are a student learning the basics of fractions or someone looking to refine your practical math skills, this guide will provide a solid foundation. Let's embark on this mathematical journey and unravel the intricacies of calculating the cost of cloth.
Understanding Mixed Fractions
Before we proceed with the calculation, it is essential to grasp the concept of mixed fractions. A mixed fraction is a combination of a whole number and a proper fraction. In our case, we have two mixed fractions: ₹16 1/2 (the cost per meter) and 8 3/4 meters (the quantity of cloth). To perform calculations with mixed fractions, we must first convert them into improper fractions. Improper fractions are fractions where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This conversion is crucial because it allows us to apply standard arithmetic operations such as multiplication and division more easily. Converting a mixed fraction to an improper fraction involves a simple two-step process. First, we multiply the whole number part by the denominator of the fractional part. Second, we add the numerator of the fractional part to the result. This sum becomes the new numerator of the improper fraction, while the denominator remains the same. For example, let's convert 16 1/2 to an improper fraction. We multiply 16 (the whole number) by 2 (the denominator), which gives us 32. Then, we add 1 (the numerator), resulting in 33. So, 16 1/2 is equivalent to 33/2 as an improper fraction. Similarly, we will convert 8 3/4 to an improper fraction. This understanding of mixed and improper fractions is the cornerstone of solving our problem accurately. Without this foundational knowledge, the subsequent calculations would be challenging to execute. Therefore, ensure you have a firm grasp of this concept before moving forward. The ability to convert between mixed and improper fractions is a fundamental skill in mathematics, applicable in various contexts beyond just calculating costs.
Converting Mixed Fractions to Improper Fractions
To effectively calculate the cost of 8 3/4 meters of cloth when one meter costs ₹16 1/2, the initial and crucial step is to convert both mixed fractions into improper fractions. This conversion simplifies the multiplication process and ensures accuracy in our final result. Let's begin by converting ₹16 1/2 into an improper fraction. As discussed earlier, we multiply the whole number (16) by the denominator (2), which yields 32. We then add the numerator (1) to this result, giving us 33. Thus, the improper fraction equivalent of 16 1/2 is 33/2. This means that ₹16 1/2 is the same as ₹33/2 per meter. Next, we apply the same process to convert 8 3/4 meters into an improper fraction. We multiply the whole number (8) by the denominator (4), which results in 32. Adding the numerator (3) to this gives us 35. Therefore, 8 3/4 meters is equivalent to 35/4 meters. Now that we have both quantities expressed as improper fractions (33/2 and 35/4), we can proceed with the multiplication to find the total cost. This conversion is not just a mathematical manipulation; it's a way of expressing the quantities in a format that is easier to work with. Improper fractions allow us to perform operations like multiplication and division in a straightforward manner, without the need for additional steps or adjustments. The conversion process also reinforces the understanding of fractions and their relationship to whole numbers. By mastering this skill, you are not only solving this specific problem but also building a stronger foundation for more complex mathematical concepts in the future. The ability to seamlessly convert between mixed and improper fractions is a hallmark of mathematical proficiency.
Calculating the Total Cost
With both the cost per meter and the quantity of cloth expressed as improper fractions, we can now proceed to calculate the total cost. This involves multiplying the cost per meter (₹33/2) by the quantity of cloth (35/4 meters). The fundamental principle of multiplying fractions states that we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, we multiply 33 by 35 to get the new numerator, and we multiply 2 by 4 to get the new denominator. The multiplication of 33 and 35 results in 1155. The multiplication of 2 and 4 results in 8. Therefore, the total cost in fraction form is ₹1155/8. This fraction represents the total cost, but it is an improper fraction, which might not be immediately intuitive. To make the cost more understandable, we need to convert this improper fraction back into a mixed fraction. To do this, we divide the numerator (1155) by the denominator (8). The quotient (the whole number result of the division) becomes the whole number part of the mixed fraction, the remainder becomes the numerator of the fractional part, and the denominator remains the same. When we divide 1155 by 8, we get a quotient of 144 and a remainder of 3. This means that 1155/8 is equivalent to 144 3/8 as a mixed fraction. Therefore, the total cost of 8 3/4 meters of cloth is ₹144 3/8. This result provides a clear understanding of the cost, with ₹144 representing the whole number part and 3/8 representing the fractional part of a rupee. The process of multiplying fractions and converting between improper and mixed fractions is a cornerstone of arithmetic. By mastering these skills, you can confidently tackle a wide range of problems involving fractional quantities.
Converting the Improper Fraction to a Mixed Fraction
After multiplying the fractions and obtaining the total cost as an improper fraction (₹1155/8), the next crucial step is to convert this improper fraction into a mixed fraction. Converting an improper fraction to a mixed fraction makes the result more understandable and relatable in a real-world context. An improper fraction, where the numerator is greater than the denominator, represents a quantity greater than one whole. To convert it to a mixed fraction, we need to determine how many whole units are contained within the fraction and what fractional part remains. The process involves dividing the numerator by the denominator. In our case, we divide 1155 (the numerator) by 8 (the denominator). This division tells us how many whole units (rupees in this context) are present in the total cost. When we perform the division, we find that 1155 divided by 8 gives us a quotient of 144 and a remainder of 3. The quotient, 144, represents the whole number part of the mixed fraction. This means there are 144 whole rupees in the total cost. The remainder, 3, represents the fractional part that is left over after accounting for the whole rupees. This remainder becomes the numerator of the fractional part of the mixed fraction, while the denominator remains the same (8). Therefore, the fractional part is 3/8. Combining the whole number part (144) and the fractional part (3/8), we get the mixed fraction 144 3/8. This means that ₹1155/8 is equivalent to ₹144 3/8. This conversion provides a clear and practical understanding of the total cost: ₹144 and 3/8 of a rupee. Converting improper fractions to mixed fractions is a fundamental skill in mathematics, particularly in situations involving real-world quantities. It allows us to express fractional amounts in a way that is easier to comprehend and use in everyday calculations.
Final Answer: ₹144 3/8
After meticulously performing the calculations, we arrive at the final answer: the cost of 8 3/4 meters of cloth is ₹144 3/8. This answer represents the culmination of our step-by-step approach, which involved converting mixed fractions to improper fractions, multiplying the fractions, and then converting the resulting improper fraction back into a mixed fraction. The final answer, ₹144 3/8, provides a clear and practical understanding of the total cost. It tells us that the cloth costs ₹144 and an additional 3/8 of a rupee. This fractional part can be further understood as 37.5 paise (since 1 rupee = 100 paise, and 3/8 of 100 is 37.5). Therefore, the total cost is ₹144.375. This comprehensive calculation demonstrates the importance of understanding fractions and their operations in real-world scenarios. The ability to work with mixed and improper fractions is a crucial skill in various fields, from everyday budgeting to professional applications in finance and engineering. The process we followed highlights the logical progression of problem-solving in mathematics: breaking down a complex problem into smaller, manageable steps, performing the necessary calculations, and then interpreting the result in a meaningful way. By understanding each step and the underlying principles, you can confidently apply these skills to solve similar problems in the future. The final answer, ₹144 3/8, not only provides the numerical solution but also reinforces the practical application of mathematical concepts in everyday life.
Conclusion
In conclusion, we have successfully calculated the cost of 8 3/4 meters of cloth when one meter costs ₹16 1/2. This calculation involved several key steps, each requiring a solid understanding of fractional arithmetic. We began by recognizing the need to convert mixed fractions into improper fractions to facilitate multiplication. This conversion is a fundamental technique in dealing with mixed numbers, allowing us to perform operations more efficiently. Next, we multiplied the improper fractions representing the cost per meter and the quantity of cloth. This multiplication yielded an improper fraction, which we then converted back into a mixed fraction to provide a more intuitive understanding of the total cost. The final answer, ₹144 3/8, represents the total cost of the cloth. This answer not only provides a numerical value but also demonstrates the practical application of fractions in real-world scenarios. The process we followed highlights the importance of breaking down complex problems into smaller, manageable steps. By systematically addressing each step, we ensured accuracy and clarity in our solution. This approach is applicable to a wide range of mathematical problems and is a valuable skill in problem-solving. Furthermore, this exercise reinforces the significance of understanding the underlying principles of arithmetic. The ability to work with fractions, convert between mixed and improper forms, and apply these concepts to practical situations is crucial for mathematical proficiency. Whether you are a student learning the basics or someone looking to enhance your practical math skills, this detailed guide provides a solid foundation for tackling similar problems in the future. The key takeaway is that with a clear understanding of the concepts and a systematic approach, even seemingly complex problems can be solved with confidence.