To convert the repeating decimal $0.4\overline{82}$ into a simplified fraction, follow these steps:

1. Set the repeating decimal equal to a variable: Let $x = 0.4\overline{82}$.

2. Eliminate the repeating part: Since the repeating part has two digits (82), multiply $x$ by 100: $100x = 48.2\overline{82}$

3. Set up a second equation: Multiply $x$ by 10 to account for the non-repeating part: $10x = 4.82\overline{82}$

4. Subtract the second equation from the first: $100x - 10x = 48.2\overline{82} - 4.82\overline{82}$ This simplifies to: $90x = 43.4$

5. Solve for $x$: $x = \frac{43.4}{90}$

6. Convert the numerator into an integer: Multiply both the numerator and denominator by 10 to eliminate the decimal: $x = \frac{434}{900}$

7. Simplify the fraction: Find the greatest common divisor (GCD) of 434 and 900. The GCD is 2. $x = \frac{434 \div 2}{900 \div 2} = \frac{217}{450}$

Thus, the simplified fraction for $0.4\overline{82}$ is:

$\boxed{\frac{217}{450}}$

Questions to Consider:

1. Do you need further explanation of any step in the conversion process?
2. Would you like to see how to convert other repeating decimals to fractions?
3. Are you interested in how to identify the GCD for simplification?
4. Would you like to know about the properties of repeating decimals?
5. Do you want to explore different methods for converting decimals to fractions?

Tip:

To convert repeating decimals to fractions, identify the length of the repeating block and use appropriate multipliers to set up your equations.