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 the Problem

When dealing with repeating decimals, it's essential to find a way to express them as fractions. This is because fractions are more manageable and easier to work with in mathematical operations. In this problem, we're given the decimal 0.120.\overline{12} and asked to find the fraction that represents it.

The Concept of Repeating Decimals

A repeating decimal is a decimal number that goes on forever in a repeating pattern. In this case, the decimal 0.120.\overline{12} repeats the pattern 1212 indefinitely. To express this as a fraction, we need to find a way to capture the repeating pattern.

Converting Repeating Decimals to Fractions

One way to convert a repeating decimal to a fraction is to use algebraic manipulation. Let's assume that the repeating decimal 0.120.\overline{12} is equal to some fraction ab\frac{a}{b}. We can then set up an equation to solve for ab\frac{a}{b}.

Setting Up the Equation

Let x=0.12x = 0.\overline{12}. Since the decimal repeats every two digits, we can multiply xx by 100100 to shift the decimal two places to the right. This gives us 100x=12.12100x = 12.\overline{12}.

Subtracting the Original Equation

Now, let's subtract the original equation x=0.12x = 0.\overline{12} from the equation 100x=12.12100x = 12.\overline{12}. This will help us eliminate the repeating decimal.

Simplifying the Equation

When we subtract the original equation from the equation 100x=12.12100x = 12.\overline{12}, we get:

100xx=12.120.12100x - x = 12.\overline{12} - 0.\overline{12}

This simplifies to:

99x=1299x = 12

Solving for x

Now, we can solve for xx by dividing both sides of the equation by 9999.

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

Reducing the Fraction

The fraction 1299\frac{12}{99} can be reduced by dividing both the numerator and the denominator by their greatest common divisor, which is 33.

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

Conclusion

Therefore, the fraction that represents the decimal 0.120.\overline{12} is 433\frac{4}{33}.

Answer

The correct answer is:

C. 433\frac{4}{33}

Explanation

The other options are incorrect because they do not accurately represent the decimal 0.120.\overline{12}. Option A, 112\frac{1}{12}, is a different fraction that does not have the same decimal representation. Option B, 325\frac{3}{25}, is also incorrect because it does not match the decimal 0.120.\overline{12}. Option D, 334\frac{33}{4}, is the reciprocal of the correct answer, but it is not the correct representation of the decimal 0.120.\overline{12}.

Real-World Applications

Understanding how to convert repeating decimals to fractions is essential in various real-world applications, such as finance, engineering, and science. For example, in finance, repeating decimals can be used to calculate interest rates and investments. In engineering, repeating decimals can be used to calculate the dimensions of structures and machines. In science, repeating decimals can be used to calculate the values of physical constants and properties.

Tips and Tricks

When dealing with repeating decimals, it's essential to remember the following tips and tricks:

  • Use algebraic manipulation to convert repeating decimals to fractions.
  • Set up an equation to solve for the fraction.
  • Subtract the original equation from the equation to eliminate the repeating decimal.
  • Reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor.
  • Check your answer by converting the fraction back to a decimal.

By following these tips and tricks, you can confidently convert repeating decimals to fractions and solve problems in various real-world applications.

Q: What is a repeating decimal?

A: A repeating decimal is a decimal number that goes on forever in a repeating pattern. For example, the decimal 0.120.\overline{12} repeats the pattern 1212 indefinitely.

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

A: Converting repeating decimals to fractions is essential in various real-world applications, such as finance, engineering, and science. Fractions are more manageable and easier to work with in mathematical operations.

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

A: To convert a repeating decimal to a fraction, you can use algebraic manipulation. Let's assume that the repeating decimal 0.120.\overline{12} is equal to some fraction ab\frac{a}{b}. You can then set up an equation to solve for ab\frac{a}{b}.

Q: What is the step-by-step process for converting a repeating decimal to a fraction?

A: The step-by-step process for converting a repeating decimal to a fraction is as follows:

  1. Let x=0.12x = 0.\overline{12}.
  2. Multiply xx by 100100 to shift the decimal two places to the right. This gives us 100x=12.12100x = 12.\overline{12}.
  3. Subtract the original equation x=0.12x = 0.\overline{12} from the equation 100x=12.12100x = 12.\overline{12}.
  4. Simplify the equation to get 99x=1299x = 12.
  5. Solve for xx by dividing both sides of the equation by 9999.
  6. Reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor.

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 using algebraic manipulation to convert the repeating decimal to a fraction.
  • Not setting up an equation to solve for the fraction.
  • Not subtracting the original equation from the equation to eliminate the repeating decimal.
  • Not reducing the fraction by dividing both the numerator and the denominator by their greatest common divisor.

Q: How do I check my answer when converting a repeating decimal to a fraction?

A: To check your answer when converting a repeating decimal to a fraction, you can convert the fraction back to a decimal. If the decimal representation matches the original repeating decimal, then your answer is correct.

Q: What are some real-world applications of converting repeating decimals to fractions?

A: Some real-world applications of converting repeating decimals to fractions include:

  • Finance: Converting repeating decimals to fractions can be used to calculate interest rates and investments.
  • Engineering: Converting repeating decimals to fractions can be used to calculate the dimensions of structures and machines.
  • Science: Converting repeating decimals to fractions can be used to calculate the values of physical constants and properties.

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

A: Some tips and tricks for converting repeating decimals to fractions include:

  • Use algebraic manipulation to convert repeating decimals to fractions.
  • Set up an equation to solve for the fraction.
  • Subtract the original equation from the equation to eliminate the repeating decimal.
  • Reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor.
  • Check your answer by converting the fraction back to a decimal.

By following these tips and tricks, you can confidently convert repeating decimals to fractions and solve problems in various real-world applications.