Part A: Converting the Repeating Decimal $8.\overline{3}$ to a Mixed Number

The repeating decimal $8.\overline{3}$ can be rewritten as $8 + 0.\overline{3}$.

To convert $0.\overline{3}$ into a fraction:

1. Let $x = 0.\overline{3}$.
2. Multiply both sides by 10 to eliminate the repeating part:
   $10x = 3.\overline{3}$.
3. Subtract the original $x$ from this new equation:
   $10x - x = 3.\overline{3} - 0.\overline{3}$.
   This simplifies to:
   $9x = 3$.
4. Solving for $x$, we get:
   $x = \frac{3}{9} = \frac{1}{3}$.

So, $8.\overline{3} = 8 + \frac{1}{3}$. Therefore, the mixed number representation is:

Answer for Part A: $8 \frac{1}{3}$.

Among the given options, $8 \frac{1}{3}$ matches this answer.

Part B: Converting $8 \frac{1}{3}$ to an Improper Fraction

To convert the mixed number $8 \frac{1}{3}$ to an improper fraction:

1. Multiply the whole number by the denominator:
   $8 \times 3 = 24$.
2. Add the numerator:
   $24 + 1 = 25$.
3. Place the result over the original denominator:
   $\frac{25}{3}$.

Answer for Part B: $\frac{25}{3}$.

Among the given options, $\frac{25}{3}$ matches this answer.

Summary:

• Part A Answer: $8 \frac{1}{3}$
• Part B Answer: $\frac{25}{3}$

Would you like a detailed explanation of any of these steps, or do you have any further questions?

Related Questions:

1. How do you convert repeating decimals with two repeating digits to fractions?
2. How can mixed numbers be converted back into decimals?
3. What are the steps to simplify fractions?
4. How can you determine if a fraction is in its simplest form?
5. What are the properties of repeating decimals?

Tip: Always double-check your calculations, especially when converting repeating decimals to fractions, as small errors can lead to incorrect results.