Which Fraction Represents The Decimal $0.\overline{12}$?A. $\frac{1}{12}$B. $ 3 25 \frac{3}{25} 25 3 ​ [/tex]C. $\frac{4}{33}$D. $\frac{33}{4}$

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Understanding Recurring Decimals

In mathematics, a recurring decimal is a decimal number that has a repeating pattern of digits. This repeating pattern can be represented as an infinite geometric series, which can be summed to find the fraction equivalent of the recurring decimal. In this article, we will explore how to convert a recurring decimal to a fraction and apply this concept to solve the given problem.

The Problem: Converting a Recurring Decimal to a Fraction

The problem asks us to find the fraction that represents the recurring decimal $0.\overline{12}$. To solve this problem, we need to understand the concept of recurring decimals and how to convert them to fractions.

Step 1: Representing the Recurring Decimal as an Infinite Geometric Series

A recurring decimal can be represented as an infinite geometric series. In this case, the recurring decimal $0.\overline{12}$ can be written as:

0.12‾=12100+1210000+121000000+…0.\overline{12} = \frac{12}{100} + \frac{12}{10000} + \frac{12}{1000000} + \ldots

This is an infinite geometric series with a first term of $\frac{12}{100}$ and a common ratio of $\frac{1}{100}$.

Step 2: Summing the Infinite Geometric Series

To find the sum of an infinite geometric series, we can use the formula:

S=a1−rS = \frac{a}{1 - r}

where $a$ is the first term and $r$ is the common ratio.

In this case, the first term $a$ is $\frac{12}{100}$ and the common ratio $r$ is $\frac{1}{100}$. Plugging these values into the formula, we get:

S=121001−1100S = \frac{\frac{12}{100}}{1 - \frac{1}{100}}

Simplifying the expression, we get:

S=1210099100S = \frac{\frac{12}{100}}{\frac{99}{100}}

S=1299S = \frac{12}{99}

Step 3: Simplifying the Fraction

The fraction $\frac{12}{99}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

1299=433\frac{12}{99} = \frac{4}{33}

Conclusion

In conclusion, the fraction that represents the recurring decimal $0.\overline{12}$ is $\frac{4}{33}$. This is the correct answer to the problem.

Comparison with Other Options

Let's compare the correct answer $\frac{4}{33}$ with the other options:

  • Option A: $\frac{1}{12}$ is not equal to $\frac{4}{33}$.
  • Option B: $\frac{3}{25}$ is not equal to $\frac{4}{33}$.
  • Option D: $\frac{33}{4}$ is not equal to $\frac{4}{33}$.

Therefore, the correct answer is option C: $\frac{4}{33}$.

Final Answer

The final answer is:

  • C. $\frac{4}{33}$

Understanding Recurring Decimals and Fraction Representation

In our previous article, we explored how to convert a recurring decimal to a fraction. In this article, we will answer some frequently asked questions related to recurring decimals and fraction representation.

Q: What is a recurring decimal?

A recurring decimal is a decimal number that has a repeating pattern of digits. This repeating pattern can be represented as an infinite geometric series, which can be summed to find the fraction equivalent of the recurring decimal.

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

To convert a recurring decimal to a fraction, you need to represent the recurring decimal as an infinite geometric series. Then, you can use the formula for the sum of an infinite geometric series to find the fraction equivalent.

Q: What is the formula for the sum of an infinite geometric series?

The formula for the sum of an infinite geometric series is:

S=a1−rS = \frac{a}{1 - r}

where $a$ is the first term and $r$ is the common ratio.

Q: How do I find the first term and the common ratio of an infinite geometric series?

To find the first term and the common ratio of an infinite geometric series, you need to look at the recurring decimal. The first term is the first occurrence of the repeating pattern, and the common ratio is the ratio of the repeating pattern to the previous occurrence.

Q: Can I simplify a fraction that represents a recurring decimal?

Yes, you can simplify a fraction that represents a recurring decimal by dividing both the numerator and the denominator by their greatest common divisor.

Q: What are some common recurring decimals and their fraction equivalents?

Here are some common recurring decimals and their fraction equivalents:

  • 0.1‾=190.\overline{1} = \frac{1}{9}

  • 0.2‾=290.\overline{2} = \frac{2}{9}

  • 0.3‾=39=130.\overline{3} = \frac{3}{9} = \frac{1}{3}

  • 0.4‾=490.\overline{4} = \frac{4}{9}

  • 0.5‾=590.\overline{5} = \frac{5}{9}

  • 0.6‾=69=230.\overline{6} = \frac{6}{9} = \frac{2}{3}

  • 0.7‾=790.\overline{7} = \frac{7}{9}

  • 0.8‾=890.\overline{8} = \frac{8}{9}

  • 0.9‾=99=10.\overline{9} = \frac{9}{9} = 1

Q: Can I use a calculator to convert a recurring decimal to a fraction?

Yes, you can use a calculator to convert a recurring decimal to a fraction. However, keep in mind that the calculator may not always give the simplest fraction equivalent.

Q: What are some real-world applications of recurring decimals and fraction representation?

Recurring decimals and fraction representation have many real-world applications, including:

  • Finance: Recurring decimals are used to calculate interest rates and investments.
  • Science: Recurring decimals are used to calculate physical constants and measurements.
  • Engineering: Recurring decimals are used to calculate dimensions and tolerances.

Conclusion

In conclusion, recurring decimals and fraction representation are important concepts in mathematics. By understanding how to convert a recurring decimal to a fraction, you can apply this knowledge to real-world problems and applications.

Final Answer

The final answer is:

  • C. $\frac{4}{33}$

This is the correct answer to the problem.