To convert the repeating decimal \(0.45\overline{45}\) into a fraction, we can use the following method:
Let \(x = 0.454545...\).
To eliminate the repeating part, we can multiply both sides of the equation by 100 (because the repeating part has two digits):
\[ 100x = 45.454545... \]
Now we have two equations:
- \(x = 0.454545...\)
- \(100x = 45.454545...\)
Next, we can subtract the first equation from the second:
\[ 100x - x = 45.454545... - 0.454545... \]
This simplifies to:
\[ 99x = 45 \]
Now, we solve for \(x\):
\[ x = \frac{45}{99} \]
To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 9:
\[ x = \frac{45 \div 9}{99 \div 9} = \frac{5}{11} \]
Therefore, the fraction that is equivalent to \(0.45\overline{45}\) is:
\[ \frac{5}{11} \]
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