Converting 5.6 Repeating To A Fraction A Step By Step Guide

Introduction

In the realm of mathematics, the conversion of decimals into fractions is a fundamental skill, especially when dealing with repeating decimals. Repeating decimals, also known as recurring decimals, are decimal numbers that have a digit or a block of digits that repeats indefinitely. Converting these decimals into fractions requires a systematic approach, which involves algebraic manipulation and a clear understanding of place values. In this comprehensive guide, we will delve into the process of converting the repeating decimal $5.\overline{6}$ into its simplest fraction form. This example provides a clear methodology that can be applied to convert any repeating decimal to a fraction. Before diving into the specifics of $5.\overline{6}$, it's essential to grasp the foundational principles that govern the transformation of decimals to fractions. Decimals are essentially fractions with a denominator that is a power of 10. For example, the decimal 0.75 can be easily converted to the fraction ${ \frac{75}{100} }$, which can then be simplified to ${ \frac{3}{4} }$. However, repeating decimals like $5.\overline{6}$ present a unique challenge because the repeating digit extends infinitely, preventing a direct conversion using the same method as terminating decimals. The key to converting repeating decimals lies in setting up an algebraic equation that allows us to eliminate the repeating part. This involves multiplying the decimal by a power of 10 that shifts the repeating block to the left of the decimal point, and then subtracting the original number from this new number. This subtraction effectively cancels out the repeating part, leaving us with a whole number that can be easily expressed as a fraction. By understanding this process, we can systematically convert any repeating decimal into its equivalent fraction, which is a crucial skill in various mathematical contexts, including algebra, calculus, and number theory. This article aims to provide a step-by-step guide that not only solves the specific problem of converting $5.\overline{6}$ but also equips you with the knowledge to tackle similar problems with confidence.

Step-by-Step Conversion of $5.\overline{6}$ to a Fraction

Step 1: Define the Repeating Decimal

The first crucial step in converting a repeating decimal to a fraction is to define the decimal algebraically. Let's denote the given repeating decimal, $5.\overline{6}$, as ${ x }$. This means:
${ x = 5.\overline{6} }$ This notation implies that the digit 6 repeats indefinitely after the decimal point, so we can write it as:
${ x = 5.6666... }$ Defining the repeating decimal in this way allows us to manipulate it algebraically, which is essential for the subsequent steps in the conversion process. This initial step is the foundation upon which the entire conversion process rests. Without a clear algebraic representation, it would be impossible to apply the necessary manipulations to eliminate the repeating part of the decimal. By setting ${ x }$ equal to the decimal, we create a variable that we can work with, making it easier to perform operations such as multiplication and subtraction. This approach is not only effective for converting repeating decimals but also provides a structured way to handle any number with an infinitely repeating pattern. The algebraic representation transforms the abstract concept of an infinitely repeating decimal into a concrete mathematical entity that can be manipulated and solved. Furthermore, this step highlights the importance of precision in mathematical notation. The overline notation ($\overline{6}$) is critical because it clearly indicates that only the digit 6 is repeating, and this understanding is crucial for the next steps where we will use this information to eliminate the repeating part. In summary, defining the repeating decimal algebraically is the necessary first step that sets the stage for a systematic conversion to a fraction, laying the groundwork for the algebraic manipulations that follow.

Step 2: Multiply by a Power of 10

To eliminate the repeating part of the decimal, we need to multiply our defined decimal by a power of 10. The power of 10 we choose depends on the length of the repeating block. In this case, we have only one repeating digit (6), so we will multiply by 10. This gives us:
${ 10x = 56.6666... }$ The purpose of this multiplication is to shift the decimal point one place to the right, which aligns the repeating parts in both the original number and the new number. This alignment is critical because it sets up the next step, where we will subtract the original number from the multiplied number. By shifting the decimal point, we ensure that the repeating digits after the decimal point are identical in both numbers, allowing them to cancel out when we subtract. The choice of 10 as the multiplier is not arbitrary; it is specifically chosen because there is only one repeating digit. If there were two repeating digits, we would multiply by 100, and so on. This approach is universally applicable to any repeating decimal, regardless of the length of the repeating block. Understanding this principle allows us to adapt the method to various scenarios, making it a versatile tool in mathematical problem-solving. The multiplication by a power of 10 is a strategic step that leverages the properties of decimal notation to create a situation where the repeating part can be eliminated. It transforms the problem from one involving an infinite repeating decimal to a manageable algebraic equation. In essence, this step is a clever manipulation that uses the structure of the decimal system to our advantage, paving the way for the subsequent steps that will lead to the final fractional representation. Multiplying by a suitable power of 10 is a fundamental technique in converting repeating decimals to fractions.

Step 3: Subtract the Original Number

The next step involves subtracting the original number from the multiplied number. This crucial operation is designed to eliminate the repeating decimal part. We subtract the equation ${ x = 5.6666... }$ from the equation ${ 10x = 56.6666... }$, which yields:
${ 10x - x = 56.6666... - 5.6666... }$ When we perform the subtraction, the repeating decimal parts cancel each other out, leaving us with whole numbers:
${ 9x = 51 }$ This subtraction step is the heart of the conversion process. It elegantly removes the infinite repeating decimal, transforming the equation into a simple algebraic form that can be easily solved for ${ x }$. The cancellation of the repeating parts is not a coincidence; it is the direct result of multiplying by the appropriate power of 10 in the previous step. By aligning the repeating digits, we set up the subtraction to eliminate them, which is a clever mathematical maneuver. The resulting equation, ${ 9x = 51 }$, is straightforward to solve because it no longer involves any decimal component. This simplification is the key to converting the repeating decimal to a fraction. The subtraction step highlights the power of algebraic manipulation in solving mathematical problems. It demonstrates how a carefully chosen operation can transform a complex problem into a simple one. In this case, the subtraction not only eliminates the repeating decimal but also reveals the underlying fractional relationship. This step underscores the importance of understanding the properties of numbers and how they can be manipulated to achieve a desired outcome. Subtracting the original number from the multiplied number is the pivotal step that transforms the repeating decimal into a manageable algebraic equation.

Step 4: Solve for x

Now that we have the equation ${ 9x = 51 }$, we can easily solve for ${ x }$ by dividing both sides by 9:
${ x = \frac{51}{9} }$ This gives us the fraction representation of the repeating decimal. However, it is not yet in simplest form. Solving for ${ x }$ is a straightforward algebraic step, but it is crucial because it isolates the variable that represents our original repeating decimal. The resulting fraction, ${ \frac{51}{9} }$, is the equivalent fractional representation of $5.\overline{6}$, but it is important to note that fractions should always be expressed in their simplest form. This step is a direct application of the principles of algebra, where we use inverse operations to isolate a variable. Dividing both sides of the equation by 9 is the inverse operation of multiplication, and it allows us to determine the value of ${ x }$. The fraction ${ \frac{51}{9} }$ provides a concrete representation of the repeating decimal, which is a significant step forward in the conversion process. However, the job is not yet complete because the fraction can be simplified further. This underscores the importance of not only finding a solution but also ensuring that the solution is presented in its most simplified form. Solving for ${ x }$ is a critical step in the conversion process, as it transforms the algebraic equation into a fractional representation of the repeating decimal, setting the stage for the final simplification.

Step 5: Simplify the Fraction

To express the fraction in its simplest form, we need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both by it. The GCD of 51 and 9 is 3. Dividing both the numerator and the denominator by 3, we get:
${ x = \frac{51 \div 3}{9 \div 3} = \frac{17}{3} }$ This fraction can also be written as a mixed number:
${ x = 5 \frac{2}{3} }$ Simplifying the fraction is an essential step in the conversion process because it ensures that the answer is presented in its most concise and understandable form. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. Finding the greatest common divisor (GCD) is a key part of this process. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. In this case, the GCD of 51 and 9 is 3, which means that both numbers can be divided by 3 to simplify the fraction. Dividing both the numerator and the denominator by the GCD is a fundamental principle of fraction simplification. This operation maintains the value of the fraction while reducing it to its simplest terms. The simplified fraction, ${ \frac{17}{3} }$, is the most basic fractional representation of the repeating decimal $5.\overline{6}$. Furthermore, converting the improper fraction ${ \frac{17}{3} }$ to a mixed number, ${ 5 \frac{2}{3} }$, provides an alternative representation that can be more intuitive for some people. A mixed number combines a whole number and a proper fraction, making it easy to visualize the quantity. Simplifying the fraction is not just a matter of mathematical convention; it also makes the result more accessible and easier to work with in future calculations. It is a crucial step in ensuring that the answer is presented in its most refined form. Simplifying the fraction is the final step in the conversion process, ensuring that the fractional representation is in its most concise and understandable form.

Conclusion

In conclusion, the repeating decimal $5.\overline{6}$ can be expressed as the fraction ${ \frac{17}{3} }$ or the mixed number ${ 5 \frac{2}{3} }$ in simplest form. This conversion process involves defining the decimal algebraically, multiplying by a power of 10, subtracting the original number, solving for the variable, and simplifying the fraction. This method provides a systematic approach to converting any repeating decimal into its equivalent fraction, a crucial skill in mathematics. Understanding how to convert repeating decimals to fractions is essential for various mathematical applications. It allows us to work with numbers in their most precise form, which is particularly important in fields such as algebra, calculus, and number theory. Fractions provide an exact representation of numbers, while decimals, especially repeating decimals, can sometimes be cumbersome to work with. The conversion process also reinforces key algebraic principles, such as manipulating equations and solving for variables. It demonstrates the power of algebraic techniques in solving problems that might initially seem complex. By following the steps outlined in this guide, anyone can confidently convert repeating decimals to fractions, enhancing their mathematical toolkit. This skill not only helps in academic settings but also has practical applications in everyday life, such as in finance, engineering, and computer science. The ability to convert between decimals and fractions allows for a deeper understanding of numerical relationships and improves problem-solving abilities. The systematic approach presented here ensures that the conversion is accurate and efficient, regardless of the complexity of the repeating decimal. In summary, mastering the conversion of repeating decimals to fractions is a valuable skill that enhances mathematical proficiency and provides a deeper understanding of number systems. The result confirms that the repeating decimal $5.\overline{6}$ is accurately represented by the fraction ${ \frac{17}{3} }$ or the mixed number ${ 5 \frac{2}{3} }$. This exercise underscores the importance of precision and systematic problem-solving in mathematics.

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