Express The Recurring Decimal $0 . \overline{67}$ In Its Simplest Fractional Form.

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Introduction

Recurring decimals are a type of decimal that has a repeating pattern of digits. In this article, we will focus on expressing the recurring decimal $0 . \overline{67}$ in its simplest fractional form. This involves converting the recurring decimal into a fraction, which can be expressed as a ratio of two integers.

Understanding Recurring Decimals

A recurring decimal is a decimal that has a repeating pattern of digits. For example, the decimal $0 . \overline{3}$ has a repeating pattern of the digit 3. Similarly, the decimal $0 . \overline{67}$ has a repeating pattern of the digits 67.

Expressing Recurring Decimals as Fractions

To express a recurring decimal as a fraction, we can use the following steps:

  1. Let $x = 0 . \overline{67}$
  2. Multiply both sides of the equation by 100 to get $100x = 67 . \overline{67}$
  3. Subtract the original equation from the new equation to get $99x = 67$
  4. Divide both sides of the equation by 99 to get $x = \frac{67}{99}$

Simplifying the Fraction

The fraction $\frac{67}{99}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 67 and 99 is 1, so the fraction cannot be simplified further.

Conclusion

In this article, we have expressed the recurring decimal $0 . \overline{67}$ in its simplest fractional form. We have used the steps outlined above to convert the recurring decimal into a fraction, and have simplified the fraction to its simplest form.

Why is it Important to Express Recurring Decimals as Fractions?

Expressing recurring decimals as fractions is important because it allows us to perform mathematical operations on the decimal with ease. For example, if we want to add or subtract two recurring decimals, we can simply add or subtract their corresponding fractions.

Real-World Applications of Recurring Decimals

Recurring decimals have many real-world applications. For example, they are used in finance to calculate interest rates, and in science to calculate physical constants.

Common Types of Recurring Decimals

There are several types of recurring decimals, including:

  • Simple Recurring Decimals: These are decimals that have a single repeating pattern of digits. For example, the decimal $0 . \overline{3}$ is a simple recurring decimal.
  • Complex Recurring Decimals: These are decimals that have multiple repeating patterns of digits. For example, the decimal $0 . \overline{12}$ is a complex recurring decimal.
  • Periodic Recurring Decimals: These are decimals that have a repeating pattern of digits that repeats after a certain number of digits. For example, the decimal $0 . \overline{123456}$ is a periodic recurring decimal.

How to Convert Recurring Decimals to Fractions

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

  1. Let $x = 0 . \overline{ab}$
  2. Multiply both sides of the equation by 10 to get $10x = ab . \overline{ab}$
  3. Subtract the original equation from the new equation to get $9x = ab$
  4. Divide both sides of the equation by 9 to get $x = \frac{ab}{9}$

Conclusion

In this article, we have discussed the concept of recurring decimals and how to express them in their simplest fractional form. We have also discussed the importance of expressing recurring decimals as fractions and have provided examples of real-world applications of recurring decimals. Additionally, we have provided a step-by-step guide on how to convert recurring decimals to fractions.

Frequently Asked Questions

Q: What is a recurring decimal?

A: A recurring decimal is a type of decimal that has a repeating pattern of digits.

Q: How do I express a recurring decimal in its simplest fractional form?

A: To express a recurring decimal in its simplest fractional form, you can use the steps outlined above.

Q: What are the real-world applications of recurring decimals?

A: Recurring decimals have many real-world applications, including finance and science.

Q: What are the different types of recurring decimals?

A: There are several types of recurring decimals, including simple recurring decimals, complex recurring decimals, and periodic recurring decimals.

Q: How do I convert a recurring decimal to a fraction?

A: To convert a recurring decimal to a fraction, you can use the steps outlined above.

References

Q: What is a recurring decimal?

A: A recurring decimal is a type of decimal that has a repeating pattern of digits. For example, the decimal $0 . \overline{3}$ has a repeating pattern of the digit 3.

Q: How do I express a recurring decimal in its simplest fractional form?

A: To express a recurring decimal in its simplest fractional form, you can use the steps outlined above. Let $x = 0 . \overline{ab}$, multiply both sides of the equation by 10 to get $10x = ab . \overline{ab}$, subtract the original equation from the new equation to get $9x = ab$, and divide both sides of the equation by 9 to get $x = \frac{ab}{9}$.

Q: What are the real-world applications of recurring decimals?

A: Recurring decimals have many real-world applications, including finance and science. For example, they are used in finance to calculate interest rates, and in science to calculate physical constants.

Q: What are the different types of recurring decimals?

A: There are several types of recurring decimals, including:

  • Simple Recurring Decimals: These are decimals that have a single repeating pattern of digits. For example, the decimal $0 . \overline{3}$ is a simple recurring decimal.
  • Complex Recurring Decimals: These are decimals that have multiple repeating patterns of digits. For example, the decimal $0 . \overline{12}$ is a complex recurring decimal.
  • Periodic Recurring Decimals: These are decimals that have a repeating pattern of digits that repeats after a certain number of digits. For example, the decimal $0 . \overline{123456}$ is a periodic recurring decimal.

Q: How do I convert a recurring decimal to a fraction?

A: To convert a recurring decimal to a fraction, you can use the steps outlined above. Let $x = 0 . \overline{ab}$, multiply both sides of the equation by 10 to get $10x = ab . \overline{ab}$, subtract the original equation from the new equation to get $9x = ab$, and divide both sides of the equation by 9 to get $x = \frac{ab}{9}$.

Q: What is the difference between a recurring decimal and a non-recurring decimal?

A: A recurring decimal is a type of decimal that has a repeating pattern of digits, while a non-recurring decimal is a type of decimal that does not have a repeating pattern of digits.

Q: Can a recurring decimal be expressed as a finite decimal?

A: No, a recurring decimal cannot be expressed as a finite decimal. By definition, a recurring decimal has a repeating pattern of digits, which means that it cannot be expressed as a finite decimal.

Q: Can a recurring decimal be expressed as a fraction?

A: Yes, a recurring decimal can be expressed as a fraction. To express a recurring decimal as a fraction, you can use the steps outlined above.

Q: What is the significance of recurring decimals in mathematics?

A: Recurring decimals are significant in mathematics because they can be used to represent irrational numbers, which are numbers that cannot be expressed as a finite decimal. Recurring decimals are also used in many mathematical formulas and equations.

Q: Can recurring decimals be used in real-world applications?

A: Yes, recurring decimals can be used in real-world applications. For example, they are used in finance to calculate interest rates, and in science to calculate physical constants.

Q: What are some common examples of recurring decimals?

A: Some common examples of recurring decimals include:

  • 0.3‾0 . \overline{3}

  • 0.12‾0 . \overline{12}

  • 0.123456‾0 . \overline{123456}

Q: How do I determine if a decimal is recurring or non-recurring?

A: To determine if a decimal is recurring or non-recurring, you can look for a repeating pattern of digits. If the decimal has a repeating pattern of digits, it is a recurring decimal. If the decimal does not have a repeating pattern of digits, it is a non-recurring decimal.

Q: Can recurring decimals be used in computer programming?

A: Yes, recurring decimals can be used in computer programming. For example, they can be used to represent irrational numbers, which are numbers that cannot be expressed as a finite decimal.

Q: What are some common uses of recurring decimals in computer programming?

A: Some common uses of recurring decimals in computer programming include:

  • Representing irrational numbers
  • Calculating interest rates
  • Calculating physical constants

Q: Can recurring decimals be used in scientific applications?

A: Yes, recurring decimals can be used in scientific applications. For example, they can be used to calculate physical constants, such as the speed of light.

Q: What are some common examples of recurring decimals in scientific applications?

A: Some common examples of recurring decimals in scientific applications include:

  • 0.3‾0 . \overline{3}

  • 0.12‾0 . \overline{12}

  • 0.123456‾0 . \overline{123456}

Q: How do I convert a recurring decimal to a scientific notation?

A: To convert a recurring decimal to a scientific notation, you can use the steps outlined above. Let $x = 0 . \overline{ab}$, multiply both sides of the equation by 10 to get $10x = ab . \overline{ab}$, subtract the original equation from the new equation to get $9x = ab$, and divide both sides of the equation by 9 to get $x = \frac{ab}{9}$.

Q: What is the difference between a recurring decimal and a non-recurring decimal in scientific notation?

A: A recurring decimal is a type of decimal that has a repeating pattern of digits, while a non-recurring decimal is a type of decimal that does not have a repeating pattern of digits. In scientific notation, a recurring decimal is represented as a fraction, while a non-recurring decimal is represented as a finite decimal.

Q: Can a recurring decimal be expressed as a finite decimal in scientific notation?

A: No, a recurring decimal cannot be expressed as a finite decimal in scientific notation. By definition, a recurring decimal has a repeating pattern of digits, which means that it cannot be expressed as a finite decimal.

Q: Can a recurring decimal be expressed as a fraction in scientific notation?

A: Yes, a recurring decimal can be expressed as a fraction in scientific notation. To express a recurring decimal as a fraction in scientific notation, you can use the steps outlined above.

Q: What is the significance of recurring decimals in scientific notation?

A: Recurring decimals are significant in scientific notation because they can be used to represent irrational numbers, which are numbers that cannot be expressed as a finite decimal. Recurring decimals are also used in many scientific formulas and equations.

Q: Can recurring decimals be used in real-world applications in scientific notation?

A: Yes, recurring decimals can be used in real-world applications in scientific notation. For example, they are used in finance to calculate interest rates, and in science to calculate physical constants.

Q: What are some common examples of recurring decimals in scientific notation?

A: Some common examples of recurring decimals in scientific notation include:

  • 0.3‾0 . \overline{3}

  • 0.12‾0 . \overline{12}

  • 0.123456‾0 . \overline{123456}