0.2 over line 90 to fraction al form in its lowest form 

The decimal \( 0.2\overline{90} \) represents \(0.2909090909...\), where "90" is the repeating part of the decimal.

Let's convert it to a fraction:

1. Let \( x = 0.2909090909...\)
2. Rewrite this as \( x = 0.2\overline{90} \)
3. Multiply by 1000 to shift the decimal so that the repeating part aligns:  
   \( 1000x = 290.9090909...\)
4. Subtract the original equation from this new equation:  
   \( 1000x - x = 290.9090909... - 0.2909090909...\)  
   \( 999x = 290.61909\)
   
This setup is incorrect as it doesn't yield a clean fractional result. Let me double-check to identify the right steps here.On second thought, let's carefully rework the process:

Given \( 0.2\overline{90} \), which is a repeating decimal \( 0.2909090909...\), let's convert it to a fraction step-by-step:

1. Let \( x = 0.2909090909...\).
2. Separate the repeating and non-repeating parts by rewriting \( x \) as \( x = 0.2 + 0.0909090909... \).
   - Here, \( 0.2 \) is the non-repeating part, and \( 0.\overline{09} \) is the repeating part.
3. Convert \( 0.2 \) to a fraction:  
   \( 0.2 = \frac{2}{10} = \frac{1}{5} \).
4. Let \( y = 0.\overline{09} \), which can be written as:
   - \( y = \frac{9}{99} = \frac{1}{11} \).
5. Add the two fractions:  
   \[ x = \frac{1}{5} + \frac{1}{11} \].

6. Find a common denominator to add these fractions:  
   \[ x = \frac{11 + 5}{55} = \frac{16}{55} \].

So, the fractional form of \( 0.2\overline{90} \) in its lowest terms is:

\[
\frac{16}{55}
\]

Would you like a further breakdown of any of these steps?

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**Further Questions:**
1. How can other repeating decimals be converted to fractions?
2. What are other ways to verify a fraction is in its simplest form?
3. Why does shifting the decimal place help with conversion of repeating decimals?
4. How does identifying repeating vs. non-repeating parts affect the conversion?
5. Can mixed repeating decimals always be simplified?

**Tip:** Remember that finding the repeating part length can help set up a subtraction equation for conversion.

0.2 over line 90 to fraction al form in its lowest form

Learn how to convert the repeating decimal 0.2 over line 90, which represents 0.2909090909..., into its simplest fraction form. This solution covers repeating decimals, converting mixed repeating decimals, and finding common denominators for fractions. Perfect for students in Grades 8-10.

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