Silicone Sheet Shrinkage And Repeating Decimals Converting -6/11
In the fascinating world of materials science, the phenomenon of shrinkage during drying is a common observation. This is especially relevant in the context of flexible materials like silicone sheets, which find widespread applications in industries ranging from manufacturing to medicine. This article delves into the specifics of how substances shrink when they dry, focusing on a scenario where a silicone sheet contracts over time. We will analyze a situation where a silicone sheet shrinks by -6/11 inch for every linear foot during drying. Our primary goal is to determine the repeating decimal that is equivalent to this fraction, providing a clear and comprehensive explanation for readers of all backgrounds.
This topic combines practical material behavior with fundamental mathematical concepts, specifically the conversion of fractions to repeating decimals. By understanding this conversion, we not only address the immediate problem but also reinforce the relationship between fractions and decimals, a core concept in mathematics. Furthermore, the context of material shrinkage provides a real-world application, making the mathematical concept more relatable and engaging.
We will break down the problem step by step, starting with a clear explanation of what the fraction -6/11 represents in terms of the silicone sheet's shrinkage. Then, we will walk through the process of converting this fraction into a decimal. This will involve long division, a method that might seem straightforward but requires careful attention to detail. The result will be a repeating decimal, and we will explain what a repeating decimal is and how it is represented mathematically. Finally, we will compare our result with the given options to identify the correct answer, solidifying our understanding through a practical application. This detailed approach ensures that readers not only get the answer but also grasp the underlying principles and processes.
The problem states that a silicone sheet shrinks by -6/11 inch for every linear foot as it dries. Let's dissect this statement to ensure we fully grasp its meaning. The negative sign is crucial; it indicates that the sheet is shrinking, meaning its length is decreasing. The fraction 6/11 represents the amount of this decrease in inches for each foot of the original length. In other words, for every foot of silicone sheet, the material contracts by 6/11 of an inch during the drying process. This shrinkage is a natural consequence of the material losing moisture and the molecules within the silicone rearranging themselves into a more compact structure. Understanding the magnitude of this shrinkage is essential in various applications, such as manufacturing where precise dimensions are critical. Engineers and material scientists need to account for this shrinkage to ensure that the final product meets the required specifications. For example, if a silicone sheet is used in a mold, the mold dimensions must be slightly larger to compensate for the shrinkage that will occur as the silicone dries.
Now, to work with this value more effectively and to compare it with decimal options, we need to convert the fraction -6/11 into its decimal equivalent. This conversion process is a fundamental mathematical operation, and it allows us to express the shrinkage in a different format, one that might be more intuitive for some people. Decimals provide a way to represent fractional parts using the base-10 number system, which is the system we commonly use in everyday calculations. Converting a fraction to a decimal involves dividing the numerator (6) by the denominator (11). The resulting decimal can either be terminating (ending after a finite number of digits) or repeating (having a pattern of digits that repeats indefinitely). In this case, we will find that -6/11 results in a repeating decimal, which introduces an interesting mathematical concept that we will explore further.
To convert the fraction -6/11 to a decimal, we perform long division. This process involves dividing the numerator (6) by the denominator (11). Since we have a negative fraction, we know the resulting decimal will also be negative. We can focus on the division of 6 by 11 and then apply the negative sign at the end. Let's begin the long division: We start by dividing 6 by 11. Since 11 doesn't go into 6, we add a decimal point and a zero to 6, making it 6.0. Now we divide 60 by 11. 11 goes into 60 five times (5 x 11 = 55), so we write 5 after the decimal point in our quotient. We subtract 55 from 60, which leaves us with a remainder of 5. We add another zero to the remainder, making it 50, and bring it down. Now we divide 50 by 11. 11 goes into 50 four times (4 x 11 = 44), so we write 4 after the 5 in our quotient. We subtract 44 from 50, which leaves us with a remainder of 6. We add another zero, making it 60, and bring it down. Notice that we are back to dividing 60 by 11, which is the same as our first step. This indicates that the division process will start repeating. We already know that 11 goes into 60 five times, giving us a 5 in the quotient, and the process will continue to repeat the 5 and 4 pattern indefinitely. Therefore, the decimal representation of 6/11 is 0.545454... or 0.54 with a bar over the 54 to indicate that the digits 54 repeat infinitely. Since our original fraction was -6/11, the equivalent decimal is -0.545454... or -0.54.
The process of long division is a cornerstone of arithmetic, and it's essential for understanding how fractions and decimals are related. By performing long division, we are essentially breaking down the fraction into its decimal components. The repeating pattern that we observe in this case is a characteristic feature of certain fractions. Not all fractions result in repeating decimals; some have terminating decimal representations. However, fractions whose denominators have prime factors other than 2 and 5 often result in repeating decimals. In the case of 11, which is a prime number, the decimal representation repeats. Understanding these patterns and the reasons behind them enhances our understanding of number systems and arithmetic operations. Furthermore, this skill is not just limited to mathematical exercises; it has practical applications in various fields, including engineering, finance, and computer science, where accurate conversions between fractions and decimals are frequently required.
Now that we have converted the fraction -6/11 to the repeating decimal -0.54, we can identify the correct answer from the given options. Let's review the options:
A. -0.54 B. -054 C. -0.54 D. -0.8
Comparing our result (-0.54) with the options, we can see that option A, -0.54, matches our calculated repeating decimal. This notation indicates that the digits 54 repeat indefinitely. Option B, -054, is simply the integer -54 and is not a decimal at all. Option C, -0.54, represents a terminating decimal, not a repeating one, and is therefore incorrect. Option D, -0.8, is a terminating decimal and is significantly different from our calculated value. Thus, the correct answer is option A, -0.54.
This step highlights the importance of accurately performing the conversion and understanding the notation used for repeating decimals. The overline notation is crucial because it distinguishes a repeating decimal from a terminating decimal. For example, -0.54 is a terminating decimal that ends at the hundredths place, while -0.54 means that the digits 54 repeat infinitely. Misinterpreting this notation can lead to incorrect answers. In practical applications, understanding the difference between terminating and repeating decimals can be critical. For instance, in precise calculations, using a truncated repeating decimal (like -0.54 instead of -0.54) can introduce errors. Therefore, recognizing and correctly representing repeating decimals is a fundamental skill in mathematics and various related fields. This exercise reinforces this understanding and provides a clear example of how to apply this knowledge to solve a real-world problem involving material shrinkage.
In this article, we explored the phenomenon of a silicone sheet shrinking during drying and mathematically determined the repeating decimal equivalent to the shrinkage fraction. We started with the given information that the silicone sheet shrank by -6/11 inch for every linear foot and walked through the process of converting this fraction to a decimal. This involved performing long division, which revealed the repeating pattern of the digits 54. We then expressed the result as -0.54, where the overline indicates the repeating nature of the digits.
We also discussed the significance of the negative sign, which signifies the shrinkage or decrease in length of the silicone sheet. Understanding the context of the problem is crucial, as it helps interpret the mathematical results in a meaningful way. The shrinkage of materials during drying is a practical concern in many industries, and being able to quantify this shrinkage using fractions and decimals is an essential skill for engineers and material scientists.
Furthermore, we highlighted the importance of accurately converting fractions to decimals and correctly interpreting the notation for repeating decimals. The difference between a terminating decimal like -0.54 and a repeating decimal like -0.54 is significant, and using the wrong representation can lead to errors in calculations. This exercise demonstrates how fundamental mathematical concepts, such as fraction-to-decimal conversion and the representation of repeating decimals, are applicable in real-world scenarios.
By breaking down the problem step by step, from understanding the context to performing the calculations and identifying the correct answer, we aimed to provide a comprehensive and clear explanation. This approach not only solves the specific problem but also reinforces the underlying mathematical principles and their practical applications. The ability to convert fractions to repeating decimals is a valuable skill that extends beyond the classroom and is relevant in various fields where precise measurements and calculations are necessary.
