Converting Repeating Decimals To Fractions: Easy Worksheet
Table of Contents :
Converting repeating decimals to fractions can seem like a daunting task at first, but with the right techniques, it becomes a straightforward process. Understanding how to transform these decimals into fractions can open up new avenues for mathematical understanding and problem-solving. In this guide, we will break down the steps to convert repeating decimals into fractions, provide a helpful worksheet, and share examples to solidify your understanding.
What is a Repeating Decimal?
A repeating decimal is a decimal fraction that eventually repeats a digit or a group of digits indefinitely. For example, the decimal 0.666... can also be written as 0.6̅, indicating that the digit '6' repeats forever. Similarly, 0.333... is often written as 0.3̅. Recognizing these repeating parts is crucial for conversion into fractions.
Steps to Convert Repeating Decimals to Fractions
Converting a repeating decimal to a fraction involves a few clear steps. Let's explore them:
Step 1: Identify the Decimal
First, write down the repeating decimal. For our example, let’s use 0.6̅.
Step 2: Set Up an Equation
Let’s assign a variable to the repeating decimal:
- Let ( x = 0.666... )
Step 3: Eliminate the Repeating Part
To eliminate the repeating decimal, multiply both sides of the equation by a power of 10 that corresponds to the number of repeating digits. Since '6' repeats, we will multiply by 10:
- ( 10x = 6.666... )
Step 4: Subtract the Original Equation
Now we subtract the original equation from this new equation:
- ( 10x - x = 6.666... - 0.666... )
This simplifies to:
- ( 9x = 6 )
Step 5: Solve for ( x )
Divide both sides by 9 to find ( x ):
- ( x = \frac{6}{9} )
Step 6: Simplify the Fraction
Finally, simplify the fraction:
- ( \frac{6}{9} = \frac{2}{3} )
So, 0.666... = (\frac{2}{3})!
Examples of Converting Repeating Decimals
Let's provide a few more examples for clarity:
Example 1: Converting 0.3̅
- Let ( x = 0.333... )
- Multiply by 10: ( 10x = 3.333... )
- Subtract: ( 10x - x = 3.333... - 0.333... ) leading to ( 9x = 3 )
- Solve: ( x = \frac{3}{9} = \frac{1}{3} )
Example 2: Converting 0.12̅
- Let ( x = 0.121212... )
- Multiply by 100 (2 digits repeat): ( 100x = 12.1212... )
- Subtract: ( 100x - x = 12.1212... - 0.1212... ) leading to ( 99x = 12 )
- Solve: ( x = \frac{12}{99} = \frac{4}{33} )
Table of Common Repeating Decimals and Their Fraction Equivalents
<table> <tr> <th>Repeating Decimal</th> <th>Fraction Equivalent</th> </tr> <tr> <td>0.3̅</td> <td>(\frac{1}{3})</td> </tr> <tr> <td>0.6̅</td> <td>(\frac{2}{3})</td> </tr> <tr> <td>0.1̅</td> <td>(\frac{1}{9})</td> </tr> <tr> <td>0.12̅</td> <td>(\frac{4}{33})</td> </tr> <tr> <td>0.2̅</td> <td>(\frac{2}{9})</td> </tr> <tr> <td>0.142̅</td> <td>(\frac{1}{7})</td> </tr> </table>
Important Note
"Some repeating decimals may involve more complex repeating sections, and it's essential to multiply by the correct power of 10 to capture the entire repeating portion."
Practice Worksheet
Now that you understand how to convert repeating decimals to fractions, here are a few practice problems for you:
- Convert 0.7̅ to a fraction.
- Convert 0.8̅ to a fraction.
- Convert 0.9̅ to a fraction.
- Convert 0.5̅ to a fraction.
- Convert 0.45̅ to a fraction.
Answers:
- (\frac{7}{9})
- (\frac{8}{9})
- (\frac{1}{1}) (or simply 1)
- (\frac{5}{9})
- (\frac{5}{11})
Conclusion
Converting repeating decimals to fractions is an essential skill in mathematics. By following the outlined steps, you can easily transform these decimals into simpler fraction forms. Remember, practice is key! Use the examples and worksheet provided to hone your skills. The more you practice, the more confident you'll become in converting these tricky numbers into manageable fractions. 🌟 Keep practicing and you'll soon master this mathematical technique!