To express $0.\overline{32}$ (the repeating decimal $0.323232\ldots$) as a fraction, we can follow these steps:

Step 1: Let $x = 0.\overline{32}$

We define $x$ as the repeating decimal: $x = 0.323232\ldots$

Step 2: Eliminate the repeating decimal

Since the repeating part is 2 digits long ("32"), we multiply both sides of the equation by 100 (because the repeat length is 2 digits): $100x = 32.323232\ldots$

Step 3: Subtract the original equation from the new equation

Now subtract $x = 0.323232\ldots$ from $100x = 32.323232\ldots$: $100x - x = 32.323232\ldots - 0.323232\ldots$ $99x = 32$

Step 4: Solve for $x$

Now, solve for $x$ by dividing both sides of the equation by 99: $x = \frac{32}{99}$

Final Answer:

The fraction equivalent of $0.\overline{32}$ is: $\frac{32}{99}$

This fraction is already in its simplest form because 32 and 99 have no common factors other than 1.

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Would you like to see more examples, or need further clarification? Here are some related questions:

1. How can we convert $0.\overline{6}$ into a fraction?
2. What is the fractional representation of $0.\overline{142857}$?
3. How would you convert $0.\overline{123}$ into a fraction?
4. What is the general method for converting repeating decimals to fractions?
5. What do we mean when a decimal is called "repeating"?

Tip: If the decimal repeats for $n$ digits, multiply by $10^n$ to eliminate the repeating part before subtracting.