Convert 0.333...0.333...0.333... to a fraction.

The repeating decimal \( 0.333\ldots \) can be expressed as a fraction by recognizing it as an infinite series. Here's how to convert it:

1. Let \( x = 0.333\ldots \), meaning \( x = 0.\overline{3} \).

2. Multiply both sides of the equation by 10 to shift the decimal point:
   \[
   10x = 3.333\ldots
   \]

3. Now subtract the original equation from this new equation:
   \[
   10x - x = 3.333\ldots - 0.333\ldots
   \]
   This simplifies to:
   \[
   9x = 3
   \]

4. Solve for \( x \):
   \[
   x = \frac{3}{9} = \frac{1}{3}
   \]

So, \( 0.333\ldots = \frac{1}{3} \).

Would you like more details on this or have any further questions?

Here are some related questions:
1. How do you convert other repeating decimals like 0.666... to fractions?
2. What happens when a repeating decimal has more digits (e.g., 0.123123...)?
3. How would you represent a repeating decimal in terms of a geometric series?
4. Can all repeating decimals be expressed as fractions?
5. What is the difference between repeating and non-repeating decimals in terms of fractions?

**Tip:** Any repeating decimal can be converted into a fraction by algebraic manipulation.

Learn how to convert the repeating decimal 0.333... into a fraction using algebraic methods. This step-by-step guide covers key concepts in algebra, including manipulation of repeating decimals and fractions. Suitable for students in Grades 6-8.

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