To express the repeating decimal \(0.\overline{21}\) as a fraction, we can set:
\[ x = 0.\overline{21} \]
This means:
\[ x = 0.21212121\ldots \]
Next, we notice that the digits "21" repeat every two decimal places, so we can multiply both sides of the equation by \(100\) (since there are two digits in the repeating part):
\[ 100x = 21.212121\ldots \]
Now, we can write our system of equations:
- \( x = 0.212121\ldots \)
- \( 100x = 21.212121\ldots \)
Next, we subtract the first equation from the second:
\[ 100x - x = 21.212121\ldots - 0.212121\ldots \]
This simplifies to:
\[ 99x = 21 \]
Now, we can solve for \(x\):
\[ x = \frac{21}{99} \]
So, the infinite decimal \(0.\overline{21}\) can be expressed as the fraction \(\frac{21}{99}\).
While simplification is not required, note that \(21\) and \(99\) have a common factor of \(3\):
\[ \frac{21 \div 3}{99 \div 3} = \frac{7}{33} \]
Thus, the fraction \(\frac{21}{99}\) is equivalent to \(\frac{7}{33}\), but since we are not required to simplify, the answer is:
\[ \frac{21}{99} \]
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