To convert the repeating decimal $\overline{0.09}$ (where 0.09 repeats) into a fraction, follow these steps:

Step 1: Let the repeating decimal be $x$.

Let $x = 0.\overline{09}$ (i.e., 0.090909...).

Step 2: Multiply both sides by 100.

Since the repeating part is two digits long, multiply both sides by 100 to shift the decimal point two places to the right.

$100x = 9.090909\overline{09}$

Step 3: Subtract the original equation from the new equation.

Subtract $x = 0.090909\overline{09}$ from $100x = 9.090909\overline{09}$:

$100x - x = 9.090909\overline{09} - 0.090909\overline{09}$

This simplifies to:

$99x = 9$

Step 4: Solve for $x$.

Now, divide both sides by 99:

$x = \frac{9}{99}$

Step 5: Simplify the fraction.

Simplify $\frac{9}{99}$ by dividing both the numerator and the denominator by 9:

$x = \frac{1}{11}$

Thus, the repeating decimal $0.\overline{09}$ is equal to the fraction $\frac{1}{11}$ in simplest form.

Would you like more details or have any questions?

Here are some related questions:

1. How do you convert other repeating decimals to fractions?
2. What if the repeating part starts after some non-repeating digits?
3. Can repeating decimals be converted into mixed numbers?
4. Why do repeating decimals always result in rational numbers?
5. What are examples of repeating decimals that simplify to improper fractions?

Tip: Multiplying by powers of 10 helps eliminate the repeating part in decimal-to-fraction conversions.