Let \( x = 0.4545454545\ldots \).
To convert this repeating decimal to a fraction, we first observe the repeated part, which is "45". This consists of two digits.
To eliminate the repeating decimal, we can multiply \( x \) by 100 (since there are two digits that repeat):
\[ 100x = 45.4545454545\ldots \]
Now, we can set up an equation using the original \( x \):
\[ x = 0.4545454545\ldots \] \[ 100x = 45.4545454545\ldots \]
Next, we subtract the first equation from the second:
\[ 100x - x = 45.4545454545\ldots - 0.4545454545\ldots \]
This simplifies to:
\[ 99x = 45 \]
Now, solve for \( x \):
\[ x = \frac{45}{99} \]
Next, we simplify the fraction \( \frac{45}{99} \). The greatest common divisor (GCD) of 45 and 99 can be found. The prime factorization of both numbers is:
- \( 45 = 3^2 \times 5 \)
- \( 99 = 3^2 \times 11 \)
The GCD is \( 3^2 = 9 \).
Now, we divide both the numerator and the denominator by their GCD (9):
\[ \frac{45 \div 9}{99 \div 9} = \frac{5}{11} \]
Thus, the decimal \( 0.4545454545\ldots \) can be expressed as the fraction:
\[ \boxed{\frac{5}{11}} \]
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