Convert The Recurring Decimal To A Fraction (show All Work):$\[ 0.2144444\ldots = 0.21\dot{4} \\]

Introduction

Recurring decimals, also known as repeating decimals, are decimals that have a block of digits that repeats indefinitely. In this article, we will show how to convert a recurring decimal to a fraction using a step-by-step approach. We will use the example of the recurring decimal $0.2144444\ldots$ to illustrate the process.

Understanding Recurring Decimals

A recurring decimal is a decimal that has a block of digits that repeats indefinitely. For example, the decimal $0.2144444\ldots$ has a block of digits $44$ that repeats indefinitely. We can represent a recurring decimal as a fraction using the following notation:

$$0.\overline{a_1a_2a_3\ldots a_n} = \frac{a_1a_2a_3\ldots a_n}{10^n - 1}$$

where $a_1a_2a_3\ldots a_n$ is the block of digits that repeats indefinitely.

Converting a Recurring Decimal to a Fraction

To convert a recurring decimal to a fraction, we can use the following steps:

Step 1: Identify the Recurring Block of Digits

The first step is to identify the block of digits that repeats indefinitely. In the example of the recurring decimal $0.2144444\ldots$, the block of digits $44$ repeats indefinitely.

Step 2: Represent the Recurring Decimal as a Fraction

We can represent the recurring decimal as a fraction using the following notation:

$$0.\overline{a_1a_2a_3\ldots a_n} = \frac{a_1a_2a_3\ldots a_n}{10^n - 1}$$

In this case, we have:

$$0.\overline{44} = \frac{44}{10^2 - 1}$$

Step 3: Simplify the Fraction

We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of $44$ and $99$ is $11$, so we can simplify the fraction as follows:

$$\frac{44}{99} = \frac{4}{9}$$

Example: Converting $0.2144444\ldots$ to a Fraction

Let's use the example of the recurring decimal $0.2144444\ldots$ to illustrate the process of converting a recurring decimal to a fraction.

Step 1: Identify the Recurring Block of Digits

The block of digits $44$ repeats indefinitely in the decimal $0.2144444\ldots$.

Step 2: Represent the Recurring Decimal as a Fraction

We can represent the recurring decimal as a fraction using the following notation:

$$0.\overline{44} = \frac{44}{10^2 - 1}$$

Step 3: Simplify the Fraction

We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of $44$ and $99$ is $11$, so we can simplify the fraction as follows:

$$\frac{44}{99} = \frac{4}{9}$$

Conclusion

Converting a recurring decimal to a fraction involves identifying the recurring block of digits, representing the recurring decimal as a fraction, and simplifying the fraction. We have shown how to convert the recurring decimal $0.2144444\ldots$ to a fraction using a step-by-step approach.

Common Recurring Decimals and Their Fractions

Here are some common recurring decimals and their fractions:

• $0.\overline{1} = \frac{1}{9}$
• $0.\overline{2} = \frac{2}{9}$
• $0.\overline{3} = \frac{3}{9} = \frac{1}{3}$
• $0.\overline{4} = \frac{4}{9}$
• $0.\overline{5} = \frac{5}{9}$
• $0.\overline{6} = \frac{6}{9} = \frac{2}{3}$
• $0.\overline{7} = \frac{7}{9}$
• $0.\overline{8} = \frac{8}{9}$
• $0.\overline{9} = \frac{9}{9} = 1$

Tips and Tricks

Here are some tips and tricks for converting recurring decimals to fractions:

• Use the notation $0.\overline{a_1a_2a_3\ldots a_n}$ to represent a recurring decimal.
• Identify the recurring block of digits and represent the recurring decimal as a fraction using the notation $\frac{a_1a_2a_3\ldots a_n}{10^n - 1}$.
• Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).
• Use a calculator or computer program to simplify the fraction if necessary.

Conclusion

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Introduction

Converting recurring decimals to fractions can be a challenging task, but with the right approach and tools, it can be done easily. In this article, we will provide a Q&A guide to help you understand the process of converting recurring decimals to fractions.

Q: What is a recurring decimal?

A: A recurring decimal is a decimal that has a block of digits that repeats indefinitely. For example, the decimal $0.2144444\ldots$ has a block of digits $44$ that repeats indefinitely.

Q: How do I identify the recurring block of digits?

A: To identify the recurring block of digits, you need to look for a pattern in the decimal. In the example of the recurring decimal $0.2144444\ldots$, the block of digits $44$ repeats indefinitely.

Q: How do I represent a recurring decimal as a fraction?

A: To represent a recurring decimal as a fraction, you can use the notation $0.\overline{a_1a_2a_3\ldots a_n} = \frac{a_1a_2a_3\ldots a_n}{10^n - 1}$.

Q: How do I simplify the fraction?

A: To simplify the fraction, you need to divide both the numerator and the denominator by their greatest common divisor (GCD). In the example of the recurring decimal $0.2144444\ldots$, the GCD of $44$ and $99$ is $11$, so we can simplify the fraction as follows:

$$\frac{44}{99} = \frac{4}{9}$$

Q: What are some common recurring decimals and their fractions?

A: Here are some common recurring decimals and their fractions:

• $0.\overline{1} = \frac{1}{9}$
• $0.\overline{2} = \frac{2}{9}$
• $0.\overline{3} = \frac{3}{9} = \frac{1}{3}$
• $0.\overline{4} = \frac{4}{9}$
• $0.\overline{5} = \frac{5}{9}$
• $0.\overline{6} = \frac{6}{9} = \frac{2}{3}$
• $0.\overline{7} = \frac{7}{9}$
• $0.\overline{8} = \frac{8}{9}$
• $0.\overline{9} = \frac{9}{9} = 1$

Q: How do I use a calculator or computer program to simplify the fraction?

A: To use a calculator or computer program to simplify the fraction, you can follow these steps:

1. Enter the numerator and denominator of the fraction into the calculator or computer program.
2. Use the calculator or computer program to find the greatest common divisor (GCD) of the numerator and denominator.
3. Divide both the numerator and denominator by the GCD to simplify the fraction.

Q: What are some tips and tricks for converting recurring decimals to fractions?

A: Here are some tips and tricks for converting recurring decimals to fractions:

• Use the notation $0.\overline{a_1a_2a_3\ldots a_n}$ to represent a recurring decimal.
• Identify the recurring block of digits and represent the recurring decimal as a fraction using the notation $\frac{a_1a_2a_3\ldots a_n}{10^n - 1}$.
• Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).
• Use a calculator or computer program to simplify the fraction if necessary.

Conclusion

Converting recurring decimals to fractions can be a challenging task, but with the right approach and tools, it can be done easily. By following the steps outlined in this article and using the notation $0.\overline{a_1a_2a_3\ldots a_n}$, you can convert any recurring decimal to a fraction. Remember to identify the recurring block of digits, represent the recurring decimal as a fraction, and simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).