To convert the repeating decimal $0.\overline{05}$ to a fraction, we can follow these steps:

Step 1: Let $x = 0.\overline{05}$.

This means that: $x = 0.050505\ldots$

Step 2: Multiply both sides by 100 (since there are two digits in the repeating part).

$100x = 5.050505\ldots$

Step 3: Subtract the original equation $x = 0.050505\ldots$ from this new equation.

$100x - x = 5.050505\ldots - 0.050505\ldots$ This simplifies to: $99x = 5$

Step 4: Solve for $x$.

$x = \frac{5}{99}$

Thus, the fraction form of $0.\overline{05}$ is: $\frac{5}{99}$

Since 5 and 99 have no common factors other than 1, this is the simplest form.

Final Answer:

$0.\overline{05} = \frac{5}{99}$

Would you like more details on this solution?

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Here are some related questions for further exploration:

1. How do you convert a repeating decimal like $0.\overline{12}$ to a fraction?
2. What is the fraction form of $0.\overline{123}$?
3. How can you convert a repeating decimal that starts after a few digits, like $0.1\overline{6}$, to a fraction?
4. How do you find the fraction form of a non-repeating decimal?
5. What is the easiest way to check if two fractions are equivalent?

Tip: When working with repeating decimals, always check the length of the repeating part to decide how much to multiply the decimal by (e.g., 10, 100, etc.).