Recurring Decimals Into Fractions Worksheet

Mastering the Art of Converting Recurring Decimals into Fractions: A Comprehensive Worksheet Guide

Recurring decimals, also known as repeating decimals, are decimal numbers where one or more digits repeat infinitely. Understanding how to convert these decimals into fractions is a crucial skill in mathematics, bridging the gap between seemingly endless decimal representations and the elegant simplicity of fractions. This comprehensive guide serves as both an explanation and a virtual worksheet, guiding you through the process step-by-step with various examples and practice problems. By the end, you'll be confidently converting recurring decimals into their fractional equivalents.

Understanding Recurring Decimals

Before we delve into the conversion process, let's solidify our understanding of recurring decimals. A recurring decimal is characterized by a sequence of digits that repeat without end. This repeating sequence is indicated by placing a bar over the repeating digits. For example:

• 0.3333... is written as 0.3̅ (the 3 repeats infinitely)
• 0.142857142857... is written as 0.142857̅ (the sequence 142857 repeats infinitely)
• 0.6666...626666... which is not purely recurring as the pattern does not start from the first decimal place

It's important to note the difference between terminating decimals (like 0.75) which have a finite number of digits and recurring decimals which have an infinite number of repeating digits.

Methods for Converting Recurring Decimals into Fractions

There are several methods for converting recurring decimals into fractions. We'll explore the most common and effective approaches.

Method 1: Using Algebra for Single-Digit Recurring Decimals

This method is particularly useful for decimals with a single repeating digit. Let's illustrate with the example of 0.7̅.

1. Let x equal the recurring decimal: Let x = 0.7̅

2. Multiply by a power of 10 to shift the decimal: Since there's only one repeating digit, we multiply by 10: 10x = 7.7̅

3. Subtract the original equation from the new equation: Subtract the first equation (x = 0.7̅) from the second equation (10x = 7.7̅):

   10x - x = 7.7̅ - 0.7̅ 9x = 7

4. Solve for x: Divide both sides by 9:

   x = 7/9

Therefore, 0.7̅ = 7/9

Method 2: Using Algebra for Multi-Digit Recurring Decimals

This method extends the algebraic approach to decimals with multiple repeating digits. Let's convert 0.142857̅ into a fraction.

1. Let x equal the recurring decimal: Let x = 0.142857̅

2. Determine the number of repeating digits: There are six repeating digits (142857).

3. Multiply by a power of 10 to shift the decimal: We multiply by 10⁶ (1,000,000) to shift the repeating block to the left of the decimal point:

   1,000,000x = 142857.142857̅

4. Subtract the original equation: Subtract the original equation (x = 0.142857̅) from the new equation:

   1,000,000x - x = 142857.142857̅ - 0.142857̅ 999,999x = 142857

5. Solve for x: Divide both sides by 999,999:

   x = 142857/999999

6. Simplify the fraction (if possible): In this case, both the numerator and denominator are divisible by 142857, simplifying to:

   x = 1/7

Therefore, 0.142857̅ = 1/7

Method 3: Using the Place Value Method (for simpler recurring decimals)

This is a more intuitive method, particularly useful for simpler recurring decimals. Let's convert 0.3̅ into a fraction.

1. Identify the repeating digit: The repeating digit is 3.

2. Write the repeating digit as the numerator: The numerator will be 3.

3. The denominator is determined by the number of places the digits repeat: Since only one digit repeats, the denominator is 9 (If it had been two repeating digits, it would be 99, three would be 999 etc.)

Therefore, 0.3̅ = 3/9 = 1/3

Method 4: Dealing with Non-Recurring Parts

Sometimes, you’ll encounter recurring decimals that have a non-recurring part before the recurring section. For example, 0.25̅.

1. Separate the non-recurring and recurring parts: In this case, we separate 0.2 into 0.2 and 0.05̅.
2. Convert each part separately: 0.2 is simply 2/10 or 1/5. To convert 0.05̅, we use the algebraic method or the place value method. Using the algebraic method: Let x = 0.05̅ 100x = 5.05̅ 100x - x = 5 99x = 5 x = 5/99
3. Add the two fractions: Add the fractions from each separate part: 1/5 + 5/99 = 99/495 + 25/495 = 124/495

Therefore, 0.25̅ = 124/495

Practice Worksheet: Converting Recurring Decimals to Fractions

Now it's your turn! Use the methods described above to convert the following recurring decimals into fractions. Remember to simplify your answers where possible.

1. 0.6̅
2. 0.4̅
3. 0.1̅
4. 0.27̅
5. 0.545454... (0.54̅)
6. 0.123̅
7. 0.83333... (0.83̅)
8. 0.16666... (0.16̅)
9. 0.272727... (0.27̅)
10. 0.142857̅
11. 0.58333... (0.583̅)
12. 0.7142857142857... (0.714285̅)
13. 0.125̅
14. 0.09̅
15. 0.9̅

Solutions to the Practice Worksheet

1. 0.6̅ = 2/3
2. 0.4̅ = 4/9
3. 0.1̅ = 1/9
4. 0.27̅ = 3/11
5. 0.54̅ = 6/11
6. 0.123̅ = 41/333
7. 0.83̅ = 5/6
8. 0.16̅ = 1/6
9. 0.27̅ = 3/11
10. 0.142857̅ = 1/7
11. 0.583̅ = 7/12
12. 0.714285̅ = 5/7
13. 0.125̅ = 5/39
14. 0.09̅ = 1/11
15. 0.9̅ = 1

Frequently Asked Questions (FAQ)

Q: What if the recurring decimal has a non-repeating part before the repeating section?

A: You need to handle the non-repeating part separately, converting it to a fraction and then adding it to the fraction representing the repeating part. Refer to Method 4 in the guide for a detailed explanation.

Q: Can all recurring decimals be converted into fractions?

A: Yes, all recurring decimals can be expressed as fractions. This is a fundamental property of rational numbers.

Q: Why are these methods important?

A: Understanding how to convert recurring decimals into fractions is crucial for a deeper grasp of number systems, algebra, and for simplifying calculations in various mathematical contexts. It helps to see the relationship between the two representations.

Q: Are there other methods to convert recurring decimals into fractions?

A: While the methods described are the most common and effective, there might be alternative approaches depending on the complexity of the recurring decimal. However, these methods provide a strong foundation for handling most cases.

Conclusion

Converting recurring decimals into fractions is a valuable skill in mathematics. By mastering the algebraic methods and understanding the place value approach, you can confidently tackle a wide range of recurring decimal conversions. Remember to practice regularly to solidify your understanding and speed up your calculations. With practice, you’ll move from simply understanding the process to effortlessly converting these numbers between their decimal and fractional forms. So, grab your pen and paper, and continue practicing – soon you’ll be a recurring decimal conversion expert!