To convert $0.\overline{32}$ (which means the decimal $0.323232...$ repeating) into a fraction, we can follow these steps:

Step 1: Set up an equation

Let $x = 0.\overline{32}$, meaning: $x = 0.32323232\ldots$

Step 2: Eliminate the repeating decimal

To eliminate the repeating part, multiply both sides of the equation by 100 (since the repeating part is two digits long): $100x = 32.32323232\ldots$

Step 3: Subtract the two equations

Now subtract the original equation from the new equation: $100x - x = 32.32323232\ldots - 0.32323232\ldots$ $99x = 32$

Step 4: Solve for $x$

Now, divide both sides by 99: $x = \frac{32}{99}$

Step 5: Simplify the fraction

The fraction $\frac{32}{99}$ is already in its simplest form, since 32 and 99 have no common factors other than 1.

So, the fraction equivalent to $0.\overline{32}$ is: $\boxed{\frac{32}{99}}$

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Do you want more details or have any questions about the steps? Here are some related questions you might find interesting:

1. How do you convert a repeating decimal like $0.\overline{57}$ into a fraction?
2. What is the method to convert a non-repeating decimal into a fraction?
3. How do you simplify fractions involving large numbers?
4. Can all repeating decimals be expressed as fractions?
5. What happens if the repeating decimal is only one digit long?

Tip: For repeating decimals, multiplying by powers of 10 (like 10, 100, 1000) helps shift the decimal point to align with the repeating part.