The notation \( 3.\overline{2} \) refers to a repeating decimal where the digit "2" repeats infinitely. This can be expressed as:

\[
3.\overline{2} = 3 + 0.\overline{2}
\]

Now let's convert \( 0.\overline{2} \) into a fraction.

1. Let \( x = 0.\overline{2} \).
2. Multiply both sides by 10 to shift the repeating decimal:
   \[
   10x = 2.\overline{2}
   \]
3. Subtract the original \( x = 0.\overline{2} \) from this equation:
   \[
   10x - x = 2.\overline{2} - 0.\overline{2}
   \]
   \[
   9x = 2
   \]
4. Solve for \( x \):
   \[
   x = \frac{2}{9}
   \]

Thus, \( 0.\overline{2} = \frac{2}{9} \).

Now, adding this to 3, we get:
\[
3.\overline{2} = 3 + \frac{2}{9} = \frac{27}{9} + \frac{2}{9} = \frac{29}{9}
\]

Therefore, \( 3.\overline{2} = \frac{29}{9} \).

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How do you convert other repeating decimals into fractions?
2. What is the process for handling non-repeating decimals?
3. How can you convert improper fractions back into mixed numbers?
4. What are some applications of repeating decimals in real life?
5. How do you simplify complex repeating decimals?

**Tip:** For any repeating decimal, setting it equal to a variable and using algebraic manipulation is a key strategy for converting it into a fraction.

Learn how to convert the repeating decimal 3.\overline{2} into a fraction using algebraic manipulation. This problem explores key concepts like repeating decimals, fractions, and the use of algebra to solve. Suitable for students in Grades 7-9.

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