To convert the repeating decimal \( 0.454545454545\ldots \) to a rational number, we can use the following method:
Let \( x = 0.454545454545\ldots \).
Now, since the decimal repeats every two digits, we will multiply \( x \) by 100:
\[ 100x = 45.454545454545\ldots \]
Next, we can set up an equation by subtracting the first equation from this new equation:
\[ 100x - x = 45.454545454545\ldots - 0.454545454545\ldots \]
This simplifies to:
\[ 99x = 45 \]
Now, we solve for \( x \):
\[ x = \frac{45}{99} \]
Next, we need to simplify \( \frac{45}{99} \). To do this, we find the greatest common divisor (GCD) of 45 and 99. The factors of 45 are \( 1, 3, 5, 9, 15, 45 \) and the factors of 99 are \( 1, 3, 9, 11, 33, 99 \). The GCD 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 \( 0.454545454545\ldots \) as a rational number in simplest form is
\[ \boxed{\frac{5}{11}}. \]
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