Repeating Decimals To Fractions Worksheet

Converting Repeating Decimals to Fractions: A Comprehensive Guide with Worksheets

Converting repeating decimals to fractions can seem daunting at first, but with a systematic approach and a little practice, it becomes a manageable and even enjoyable mathematical skill. This comprehensive guide will take you through the process step-by-step, providing explanations, examples, and worksheets to solidify your understanding. We'll explore the underlying principles and offer strategies to tackle various types of repeating decimals, ensuring you can confidently convert them into their fractional equivalents. This is a skill crucial for a variety of mathematical applications, from algebra to calculus.

Understanding Repeating Decimals

A repeating decimal is a decimal number where one or more digits repeat infinitely. These repeating digits are indicated by a bar placed above the repeating sequence. For example:

• 0.333... is written as 0.$\overline{3}$
• 0.142857142857... is written as 0.$\overline{142857}$
• 2.7181818... is written as 2.7$\overline{18}$

The repeating block is called the repetend. Understanding this notation is the first step in converting these decimals to fractions. The key to converting these numbers lies in manipulating algebraic equations to isolate the repeating portion.

Method 1: Using Algebra for Simple Repeating Decimals

This method is best suited for repeating decimals where the repetition starts immediately after the decimal point.

Steps:

1. Assign a variable: Let 'x' represent the repeating decimal.
2. Multiply to shift the decimal: Multiply 'x' by a power of 10 such that the repeating part aligns. The power of 10 will be equal to the number of digits in the repetend. For example, if the repetend has one digit, multiply by 10; if it has two digits, multiply by 100, and so on.
3. Subtract the original equation: Subtract the original equation (x) from the equation obtained in step 2. This will eliminate the repeating part.
4. Solve for x: Solve the resulting equation for 'x', which will now be expressed as a fraction.
5. Simplify the fraction: Reduce the fraction to its simplest form by finding the greatest common divisor (GCD) of the numerator and the denominator.

Example 1: Converting 0.$\overline{3}$ to a fraction:

1. x = 0.$\overline{3}$
2. 10x = 3.$\overline{3}$
3. 10x - x = 3.$\overline{3}$ - 0.$\overline{3}$ => 9x = 3
4. x = 3/9 = 1/3

Example 2: Converting 0.$\overline{12}$ to a fraction:

1. x = 0.$\overline{12}$
2. 100x = 12.$\overline{12}$
3. 100x - x = 12.$\overline{12}$ - 0.$\overline{12}$ => 99x = 12
4. x = 12/99 = 4/33

Method 2: Handling Repeating Decimals with Non-Repeating Parts

This method addresses decimals with a non-repeating part before the repeating section.

Steps:

1. Separate the non-repeating and repeating parts: Identify the non-repeating part (integer and decimal portion before the repetition) and the repeating part.
2. Convert the repeating part: Use Method 1 to convert the repeating part into a fraction.
3. Convert the non-repeating part: Convert the non-repeating part into a fraction (e.g., 0.2 = 2/10 = 1/5).
4. Add the fractions: Add the fraction representing the non-repeating part and the fraction representing the repeating part. Remember to find a common denominator before adding.

Example 3: Converting 0.2$\overline{3}$ to a fraction:

1. Non-repeating part: 0.2 = 2/10 = 1/5
2. Repeating part: 0.$\overline{3}$ = 1/3 (from Example 1)
3. Adding the fractions: 1/5 + 1/3 = (3 + 5)/15 = 8/15

Method 3: Dealing with More Complex Repeating Patterns

Some repeating decimals have longer repeating blocks or more complex patterns. The algebraic approach remains the same, but the multiplications and subtractions become slightly more involved.

Example 4: Converting 0.$\overline{123}$ to a fraction:

1. x = 0.$\overline{123}$
2. 1000x = 123.$\overline{123}$
3. 1000x - x = 123.$\overline{123}$ - 0.$\overline{123}$ => 999x = 123
4. x = 123/999 = 41/333

Worksheet 1: Simple Repeating Decimals

Convert the following repeating decimals to fractions:

1. 0.$\overline{7}$
2. 0.$\overline{4}$
3. 0.$\overline{1}$
4. 0.$\overline{6}$
5. 0.$\overline{9}$
6. 0.$\overline{21}$
7. 0.$\overline{37}$
8. 0.$\overline{81}$
9. 0.$\overline{123}$
10. 0.$\overline{54321}$

Worksheet 2: Repeating Decimals with Non-Repeating Parts

Convert the following repeating decimals to fractions:

1. 0.1$\overline{2}$
2. 0.3$\overline{7}$
3. 0.2$\overline{14}$
4. 0.12$\overline{3}$
5. 0.5$\overline{678}$
6. 0.1$\overline{9}$
7. 0.9$\overline{9}$ (Hint: This one has a surprising result!)
8. 0.27$\overline{27}$
9. 0.4$\overline{123}$
10. 1.2$\overline{345}$

Frequently Asked Questions (FAQ)

Q: What if the repeating part doesn't start immediately after the decimal point?

A: Use Method 2, separating the non-repeating part and the repeating part, then treating them individually before combining the resulting fractions.

Q: Can I use a calculator to help me?

A: While a calculator can help with arithmetic, understanding the underlying algebraic principles is crucial. A calculator can verify your answer but shouldn't replace the process of solving the problem.

Q: What is the significance of converting repeating decimals to fractions?

A: It provides a more precise representation of the number, useful in various mathematical operations and contexts where decimal approximations might lead to inaccuracies. Fractions also offer a clearer understanding of the underlying mathematical relationships.

Q: Are there other methods for converting repeating decimals to fractions?

A: While algebraic manipulation is the most common and widely applicable method, other techniques exist, particularly for specific types of repeating decimals. However, the algebraic approach is generally the most robust and versatile.

Conclusion

Converting repeating decimals to fractions is a valuable skill that enhances your understanding of number systems and algebraic manipulation. Mastering this skill involves understanding the concept of repeating decimals, employing a systematic algebraic approach, and practicing with various examples. The worksheets provided offer ample opportunity for practice, enabling you to develop confidence and proficiency in converting repeating decimals to their equivalent fraction forms. Remember to always simplify your final fraction to its lowest terms. With consistent effort and the techniques outlined above, you can confidently navigate the world of repeating decimals and their fractional representations.