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.