Let \( x = 0.454545454545 \ldots \).
To eliminate the repeating decimal, we can multiply both sides of the equation by 100, since the repeating part (45) has a length of 2 digits:
\[ 100x = 45.454545454545 \ldots \]
Now we have two equations:
- \( x = 0.454545454545 \ldots \)
- \( 100x = 45.454545454545 \ldots \)
Next, we can subtract the first equation from the second:
\[ 100x - x = 45.454545454545 \ldots - 0.454545454545 \ldots \]
This simplifies to:
\[ 99x = 45 \]
Now, we can solve for \( x \):
\[ x = \frac{45}{99} \]
Next, we simplify the fraction \( \frac{45}{99} \). We can find the greatest common divisor (GCD) of 45 and 99, which is 9.
Now we divide both the numerator and the denominator by their GCD:
\[ \frac{45 \div 9}{99 \div 9} = \frac{5}{11} \]
Thus, the repeating decimal \( 0.454545454545 \ldots \) can be expressed as the rational number in simplest form:
\[ \boxed{\frac{5}{11}} \]
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