If A String Is 6m 8dm 6cm Long And 3m 6dm 5cm Is Cut Off To Make A Rope, What Is The Remaining Length Of The String?

In this article, we delve into a practical problem involving string length and subtraction, relevant to mathematics and everyday life. Our primary goal is to calculate the remaining length of a string after a portion of it has been cut off. The initial length of the string is given as 6 meters 8 decimeters 6 centimeters, and we need to determine the length that remains after cutting off 3 meters 6 decimeters 5 centimeters to make a rope. This seemingly simple problem incorporates multiple units of measurement, requiring a methodical approach to ensure accurate calculations. We'll explore the necessary steps to convert these measurements into a consistent unit, perform the subtraction, and then express the final answer in a meaningful way. By understanding the underlying principles of unit conversion and subtraction, you can apply these skills to a wide range of real-world scenarios. This article aims to provide a clear, step-by-step explanation, making it accessible to anyone regardless of their mathematical background.

Understanding the problem is the first critical step in finding the solution. We are presented with a string of a specific length, and a portion of this string is removed. The task is to determine the length of the string that remains. This involves mathematical concepts such as subtraction and unit conversion. It is essential to recognize that the initial length and the cut length are given in mixed units (meters, decimeters, and centimeters). To perform the subtraction accurately, we need to convert all measurements into a single unit. This could be meters, centimeters, or any other suitable unit. The choice of unit will depend on personal preference and the level of precision required in the final answer. By breaking down the problem into smaller, manageable steps, we can simplify the process and avoid errors. Furthermore, a clear understanding of the problem allows us to better visualize the situation and apply appropriate strategies for solving it.

Converting to a Common Unit is crucial for accurate calculations when dealing with different units of measurement. In this case, we have meters (m), decimeters (dm), and centimeters (cm). To perform the subtraction, we need to convert all measurements into a single unit. The most common and convenient unit for this type of problem is centimeters (cm). Let's begin by converting the initial length of the string, which is 6 meters 8 decimeters 6 centimeters, into centimeters. We know that 1 meter is equal to 100 centimeters, and 1 decimeter is equal to 10 centimeters. Therefore, 6 meters is equal to 6 * 100 = 600 centimeters, and 8 decimeters is equal to 8 * 10 = 80 centimeters. Adding these values together, along with the existing 6 centimeters, gives us a total of 600 + 80 + 6 = 686 centimeters. Next, we need to convert the length of the string that was cut off, which is 3 meters 6 decimeters 5 centimeters, into centimeters. Using the same conversion factors, 3 meters is equal to 3 * 100 = 300 centimeters, and 6 decimeters is equal to 6 * 10 = 60 centimeters. Adding these values together, along with the 5 centimeters, gives us a total of 300 + 60 + 5 = 365 centimeters. Now that both lengths are expressed in the same unit (centimeters), we can proceed with the subtraction. This conversion step is essential to ensure that we are comparing and subtracting like quantities, leading to an accurate result.

With both lengths converted into centimeters, we can now proceed with the subtraction. We have the initial length of the string as 686 centimeters and the length of the string that was cut off as 365 centimeters. To find the remaining length, we simply subtract the cut length from the initial length. This can be written as: Remaining length = Initial length - Cut length. Substituting the values we have: Remaining length = 686 cm - 365 cm. Now, we perform the subtraction: 686 - 365 = 321 centimeters. Therefore, the remaining length of the string is 321 centimeters. This straightforward subtraction is made possible by the previous step of converting all measurements into a common unit. Without this conversion, the subtraction would be more complex and prone to errors. The result, 321 centimeters, represents the length of the string that remains after cutting off 365 centimeters to make a rope. This numerical answer is accurate, but to fully understand its significance, it is helpful to convert it back into a more meaningful unit, such as meters, decimeters, and centimeters. This will allow us to express the remaining length in a way that is easier to visualize and comprehend.

Converting Back to Mixed Units for clarity and practical application is the final step in solving our problem. We have calculated the remaining length of the string to be 321 centimeters. While this answer is correct, it may not be the most intuitive way to express the length. To make it more understandable, we can convert it back into meters, decimeters, and centimeters. We know that 1 meter is equal to 100 centimeters, and 1 decimeter is equal to 10 centimeters. To convert 321 centimeters into meters, we divide by 100. The whole number part of the result will be the number of meters, and the remainder will be the number of centimeters left over. So, 321 cm ÷ 100 = 3 meters with a remainder of 21 centimeters. This means we have 3 meters and 21 centimeters remaining. Now, we need to convert the remaining 21 centimeters into decimeters and centimeters. To find the number of decimeters, we divide 21 by 10. The whole number part of the result will be the number of decimeters, and the remainder will be the number of centimeters left over. So, 21 cm ÷ 10 = 2 decimeters with a remainder of 1 centimeter. Therefore, we have 2 decimeters and 1 centimeter. Combining these results, we find that the remaining length of the string is 3 meters 2 decimeters 1 centimeter. This mixed-unit representation provides a more practical and easily understandable answer. It allows us to visualize the length of the remaining string in terms of familiar units, making it easier to apply this information in real-world situations. This conversion back to mixed units demonstrates the importance of expressing answers in a way that is both accurate and meaningful.

To summarize, let's break down the problem into a step-by-step solution. This will help reinforce the concepts and provide a clear roadmap for solving similar problems in the future.

Step 1: Understand the Problem. We are given the initial length of a string as 6 meters 8 decimeters 6 centimeters, and a portion of the string, 3 meters 6 decimeters 5 centimeters, is cut off. The goal is to find the remaining length of the string. This involves subtraction and unit conversion.

Step 2: Convert to a Common Unit. To perform the subtraction, we need to convert all measurements into a single unit. Centimeters is a convenient choice. We know that 1 meter = 100 centimeters and 1 decimeter = 10 centimeters. Converting the initial length: 6 meters = 6 * 100 = 600 centimeters, 8 decimeters = 8 * 10 = 80 centimeters. Adding these to the existing 6 centimeters gives us 600 + 80 + 6 = 686 centimeters. Converting the cut length: 3 meters = 3 * 100 = 300 centimeters, 6 decimeters = 6 * 10 = 60 centimeters. Adding these to the existing 5 centimeters gives us 300 + 60 + 5 = 365 centimeters.

Step 3: Perform the Subtraction. Now that both lengths are in centimeters, we can subtract the cut length from the initial length: Remaining length = Initial length - Cut length = 686 cm - 365 cm = 321 centimeters.

Step 4: Convert Back to Mixed Units. To express the answer in a more understandable way, we convert 321 centimeters back into meters, decimeters, and centimeters. Converting to meters: 321 cm ÷ 100 = 3 meters with a remainder of 21 centimeters. Converting the remainder to decimeters and centimeters: 21 cm ÷ 10 = 2 decimeters with a remainder of 1 centimeter. Therefore, the remaining length is 3 meters 2 decimeters 1 centimeter.

Step 5: State the Answer. The remaining length of the string is 3 meters 2 decimeters 1 centimeter. This step-by-step approach ensures accuracy and clarity in solving the problem. By breaking down the problem into smaller, manageable steps, we can avoid errors and gain a deeper understanding of the underlying concepts.

This type of problem, involving length calculations and unit conversions, has numerous real-world applications. Understanding how to perform these calculations is essential in various fields and everyday situations. In construction, for example, accurate measurements are crucial for cutting materials to the correct size. Whether it's measuring wood for framing, fabric for upholstery, or pipes for plumbing, the ability to convert between units and perform subtraction is vital. Imagine a carpenter needing to cut a piece of wood that is 5 meters 7 decimeters long from a plank that is 8 meters 2 decimeters 5 centimeters long. They would need to perform a similar calculation to the one we've discussed to determine the remaining length of the plank. In tailoring and sewing, similar calculations are necessary for determining fabric lengths. A seamstress might need to cut a certain length of fabric from a larger piece to create a garment. They would need to accurately measure and subtract lengths to ensure they have enough fabric for their project. In landscaping, measuring distances and calculating lengths is essential for planning gardens, laying paths, and installing fences. A landscaper might need to determine how much fencing material is needed to enclose a yard, which would involve measuring the perimeter and subtracting any gaps for gates or other openings. Even in everyday situations, these skills are useful. For example, when wrapping a gift, you might need to cut a specific length of ribbon from a roll. Or, when arranging furniture in a room, you might need to measure the dimensions of the room and the furniture to ensure everything fits properly. These examples demonstrate the practical importance of understanding length calculations and unit conversions. By mastering these skills, you can confidently tackle a wide range of real-world problems.

When working with length calculations and unit conversions, several common mistakes can occur. Being aware of these potential pitfalls can help you avoid errors and ensure accurate results. One of the most common mistakes is failing to convert all measurements into a single unit before performing calculations. As we've seen in our example, the initial length and the cut length were given in mixed units (meters, decimeters, and centimeters). If we were to subtract the numbers directly without converting them to a common unit, we would get an incorrect result. To avoid this mistake, always make sure to convert all measurements to the same unit before adding or subtracting them. Another common mistake is incorrect unit conversions. It's essential to remember the relationships between different units of length. For example, 1 meter is equal to 100 centimeters, and 1 decimeter is equal to 10 centimeters. Mixing up these conversion factors can lead to significant errors. To avoid this, it's helpful to write down the conversion factors and double-check them before using them. Additionally, pay attention to the direction of the conversion. When converting from a larger unit to a smaller unit (e.g., meters to centimeters), you need to multiply. When converting from a smaller unit to a larger unit (e.g., centimeters to meters), you need to divide. Another potential mistake is errors in subtraction. While subtraction seems straightforward, it's easy to make mistakes, especially when dealing with larger numbers or when borrowing is required. To minimize errors, it's helpful to write out the subtraction problem clearly and double-check your work. You can also use a calculator to verify your answer. Finally, not expressing the answer in the appropriate units can also be a mistake. While a numerical answer in centimeters might be correct, it may not be the most useful or understandable way to express the length in a real-world context. Converting the answer back to mixed units (meters, decimeters, and centimeters) can make it more meaningful and easier to interpret. By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in solving length calculation problems.

In conclusion, calculating the remaining length of a string after a portion has been cut off involves several important steps. We started with a problem presented in mixed units (meters, decimeters, and centimeters), and we systematically worked through the process of finding the solution. The key steps included converting all measurements into a common unit (centimeters), performing the subtraction, and then converting the answer back into mixed units for clarity. This methodical approach is crucial for ensuring accuracy and avoiding errors. By understanding the relationships between different units of length and applying the principles of subtraction, we were able to determine that the remaining length of the string is 3 meters 2 decimeters 1 centimeter. This process not only provides the correct answer but also reinforces the importance of unit conversion in mathematical problem-solving. The skills learned in this exercise are applicable to a wide range of real-world situations, from construction and tailoring to everyday tasks like measuring and cutting materials. By mastering these concepts, you can confidently tackle similar problems involving length calculations and unit conversions. The step-by-step approach we've outlined, including understanding the problem, converting units, performing calculations, and expressing the answer in a meaningful way, can be applied to various mathematical problems beyond just length calculations. Ultimately, the ability to solve these types of problems demonstrates a strong understanding of mathematical principles and their practical applications.