Write $0.0 \overline{5}$ As A Fraction.A. $\frac{1}{12}$ B. $\frac{1}{15}$ C. $\frac{1}{18}$ D. $\frac{1}{9}$

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Understanding Repeating Decimals

Repeating decimals are a type of decimal that has a block of digits that repeats indefinitely. For example, the decimal 0.0¯5 is a repeating decimal because the digit 5 repeats indefinitely. In this article, we will explore how to convert repeating decimals to fractions.

The Problem: Converting 0.0¯5 to a Fraction

We are given the repeating decimal 0.0¯5 and asked to convert it to a fraction. To do this, we can use the following steps:

Step 1: Identify the Repeating Block

The repeating block in the decimal 0.0¯5 is the digit 5. This means that the decimal can be written as 0.05555... where the digit 5 repeats indefinitely.

Step 2: Set Up an Equation

Let x = 0.05555... We can multiply both sides of the equation by 10 to get 10x = 0.55555...

Step 3: Subtract the Original Equation from the New Equation

Subtracting the original equation from the new equation, we get:

10x - x = 0.55555... - 0.05555...

This simplifies to:

9x = 0.5

Step 4: Solve for x

Dividing both sides of the equation by 9, we get:

x = 0.5/9

Step 5: Simplify the Fraction

The fraction 0.5/9 can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 1. Therefore, the simplified fraction is:

x = 1/9

Conclusion

In this article, we have shown how to convert the repeating decimal 0.0¯5 to a fraction. By following the steps outlined above, we have arrived at the conclusion that 0.0¯5 can be written as the fraction 1/9.

Why is this Important?

Converting repeating decimals to fractions is an important skill in mathematics because it allows us to perform calculations with repeating decimals more easily. For example, if we need to add or subtract two repeating decimals, we can convert them to fractions and then perform the calculation.

Real-World Applications

Converting repeating decimals to fractions has many real-world applications. For example, in finance, repeating decimals are often used to represent interest rates or investment returns. In science, repeating decimals are used to represent physical constants or measurements.

Common Mistakes to Avoid

When converting repeating decimals to fractions, there are several common mistakes to avoid. These include:

  • Not identifying the repeating block correctly: Make sure to identify the repeating block correctly before setting up the equation.
  • Not setting up the equation correctly: Make sure to multiply both sides of the equation by the correct power of 10.
  • Not subtracting the original equation from the new equation correctly: Make sure to subtract the original equation from the new equation correctly to eliminate the repeating block.
  • Not simplifying the fraction correctly: Make sure to simplify the fraction correctly by dividing both the numerator and the denominator by their greatest common divisor.

Conclusion

In conclusion, converting repeating decimals to fractions is an important skill in mathematics that has many real-world applications. By following the steps outlined above, we can convert repeating decimals to fractions with ease. Remember to avoid common mistakes and to simplify the fraction correctly to arrive at the correct answer.

Final Answer

The final answer is: 19\boxed{\frac{1}{9}}

Understanding Repeating Decimals

Repeating decimals are a type of decimal that has a block of digits that repeats indefinitely. For example, the decimal 0.0¯5 is a repeating decimal because the digit 5 repeats indefinitely. In this article, we will explore how to convert repeating decimals to fractions and answer some common questions about this topic.

Q: What is a repeating decimal?

A: A repeating decimal is a type of decimal that has a block of digits that repeats indefinitely. For example, the decimal 0.0¯5 is a repeating decimal because the digit 5 repeats indefinitely.

Q: Why is it important to convert repeating decimals to fractions?

A: Converting repeating decimals to fractions is important because it allows us to perform calculations with repeating decimals more easily. For example, if we need to add or subtract two repeating decimals, we can convert them to fractions and then perform the calculation.

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

A: To convert a repeating decimal to a fraction, follow these steps:

  1. Identify the repeating block in the decimal.
  2. Set up an equation using the repeating decimal.
  3. Multiply both sides of the equation by the correct power of 10.
  4. Subtract the original equation from the new equation to eliminate the repeating block.
  5. Solve for the variable and simplify the fraction.

Q: What is the greatest common divisor (GCD) and why is it important?

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. The GCD is important because it allows us to simplify fractions by dividing both the numerator and the denominator by their GCD.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, you can use the following methods:

  • List the factors of each number and find the greatest common factor.
  • Use the Euclidean algorithm to find the GCD.
  • Use a calculator or online tool to find the GCD.

Q: What are some common mistakes to avoid when converting repeating decimals to fractions?

A: Some common mistakes to avoid when converting repeating decimals to fractions include:

  • Not identifying the repeating block correctly.
  • Not setting up the equation correctly.
  • Not subtracting the original equation from the new equation correctly.
  • Not simplifying the fraction correctly.

Q: Can I use a calculator to convert repeating decimals to fractions?

A: Yes, you can use a calculator to convert repeating decimals to fractions. Many calculators have a built-in function to convert decimals to fractions.

Q: Are there any real-world applications of converting repeating decimals to fractions?

A: Yes, there are many real-world applications of converting repeating decimals to fractions. For example, in finance, repeating decimals are often used to represent interest rates or investment returns. In science, repeating decimals are used to represent physical constants or measurements.

Q: Can I convert a repeating decimal to a fraction using a different method?

A: Yes, there are several different methods to convert a repeating decimal to a fraction. Some of these methods include:

  • Using a geometric series to convert the repeating decimal to a fraction.
  • Using a continued fraction to convert the repeating decimal to a fraction.
  • Using a calculator or online tool to convert the repeating decimal to a fraction.

Conclusion

In conclusion, converting repeating decimals to fractions is an important skill in mathematics that has many real-world applications. By following the steps outlined above and avoiding common mistakes, you can convert repeating decimals to fractions with ease. Remember to use a calculator or online tool if you need help, and to explore different methods to convert repeating decimals to fractions.

Final Answer

The final answer is: 19\boxed{\frac{1}{9}}