Convert the following repeating decimal to a fraction in simplest form.
., start overline, 0, 5, end overline
. 
05


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.).

Convert the following repeating decimal to a fraction in simplest form: 0.\overline{05}

Learn how to convert the repeating decimal 0.\overline{05} into a fraction with a step-by-step explanation. This problem involves algebraic concepts like manipulating equations and simplifying fractions, making it perfect for students in Grades 7-10.

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