The decimal \( 2.009009009 \) is a repeating decimal, with the repeating part being \( 009 \). To express this as a fraction, we can use algebraic techniques.

Let \( x = 2.009009009 \ldots \), where the repeating part is 009.

### Step-by-step solution:
1. Write \( x \) as a sum of a whole number and a repeating decimal:
   \[
   x = 2 + 0.009009009 \ldots
   \]
   Focus on expressing the repeating decimal as a fraction.

2. Let \( y = 0.009009009 \ldots \) (the repeating decimal part). Now, multiply \( y \) by 1000 to shift the decimal point by 3 places:
   \[
   1000y = 9.009009009 \ldots
   \]

3. Subtract the original \( y \) from this equation:
   \[
   1000y - y = 9.009009009 \ldots - 0.009009009 \ldots
   \]
   \[
   999y = 9
   \]

4. Solving for \( y \):
   \[
   y = \frac{9}{999} = \frac{1}{111}
   \]

5. Now, substitute this back into the expression for \( x \):
   \[
   x = 2 + \frac{1}{111}
   \]

6. Convert this into a single fraction:
   \[
   x = \frac{2 \times 111 + 1}{111} = \frac{222 + 1}{111} = \frac{223}{111}
   \]

Thus, \( 2.009009009 \) as a fraction is \( \frac{223}{111} \).

Would you like further details, or have any questions?

Here are 5 related questions:
1. How do you convert a non-repeating decimal to a fraction?
2. What is the general formula for converting repeating decimals into fractions?
3. How can you simplify the fraction \( \frac{223}{111} \)?
4. What are the properties of repeating decimals when expressed as fractions?
5. How do you find the decimal expansion of a fraction?

**Tip:** To simplify fractions, always check for the greatest common divisor (GCD) between the numerator and denominator.

Learn how to convert the repeating decimal 2.009009009 into a fraction with a step-by-step solution. This process involves key algebraic techniques and simplifies the decimal to the fraction 223/111. Suitable for students in Grades 8-10.

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