To convert the repeating decimal \(0.66666666\ldots\) (which can be written as \(0.\overline{6}\)) to a fraction, follow these steps:

1. **Let \(x = 0.\overline{6}\):**
   \[
   x = 0.66666666\ldots
   \]

2. **Multiply both sides by 10 to shift the decimal point one place to the right:**
   \[
   10x = 6.66666666\ldots
   \]

3. **Subtract the original equation from this new equation to eliminate the repeating part:**
   \[
   10x - x = 6.66666666\ldots - 0.66666666\ldots
   \]
   \[
   9x = 6
   \]

4. **Solve for \(x\) by dividing both sides by 9:**
   \[
   x = \frac{6}{9}
   \]

5. **Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 3:**
   \[
   x = \frac{6 \div 3}{9 \div 3} = \frac{2}{3}
   \]

So, \(0.\overline{6}\) as a fraction is \(\frac{2}{3}\).

Would you like any further details or have any questions?

Here are five related questions:
1. How do you convert \(0.\overline{3}\) to a fraction?
2. What is \(0.\overline{9}\) as a fraction, and why is it equal to 1?
3. How do you express a decimal with two repeating digits (e.g., \(0.\overline{12}\)) as a fraction?
4. What is the decimal expansion of the fraction \(\frac{1}{7}\)?
5. How can you check if a fraction's decimal form will repeat?

**Tip:** Repeating decimals can always be expressed as fractions with a denominator that has factors only of 9 or multiples of 9 (for example, \(\frac{1}{3} = 0.\overline{3}\)).

Learn how to convert the repeating decimal 0.666... (0.\overline{6}) to a fraction (2/3) with a detailed step-by-step guide. This problem covers converting repeating decimals to fractions using multiplication and subtraction methods. Suitable for Grades 6-8 students.

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