To convert the repeating decimal \( 0.45\overline{45} \) to a fraction, we can use the following steps:
Let \( x = 0.454545\dots \).
Since the repeating part has two digits (45), we can multiply \( x \) by 100 to shift the decimal point two places to the right:
\[ 100x = 45.454545\dots \]
Next, we can set up an equation using the original \( x \):
\[ x = 0.454545\dots \]
Subtract the first equation from the second equation:
\[ 100x - x = 45.454545\dots - 0.454545\dots \] \[ 99x = 45 \]
Now, divide both sides by 99:
\[ x = \frac{45}{99} \]
Next, simplify the fraction. Both 45 and 99 can be divided by their greatest common divisor, which is 9:
\[ \frac{45 \div 9}{99 \div 9} = \frac{5}{11} \]
Thus, the fraction that is equivalent to \( 0.45\overline{45} \) is
\[ \boxed{\frac{5}{11}}. \]
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