To convert the repeating decimal \(0.\overline{95}\) to a fraction, we can follow these steps:
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Let \(x = 0.\overline{95}\). This means \(x = 0.959595...\).
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To eliminate the repeating part, we can multiply \(x\) by 100 (since the repeating section has two digits): \[ 100x = 95.959595... \]
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Now we have two equations: \[ x = 0.959595... \] \[ 100x = 95.959595... \]
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Next, we subtract the first equation from the second: \[ 100x - x = 95.959595... - 0.959595... \] \[ 99x = 95 \]
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Now, solve for \(x\): \[ x = \frac{95}{99} \]
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We need to simplify the fraction \( \frac{95}{99} \). First, we find the greatest common divisor (GCD) of 95 and 99:
- The prime factors of 95 are \(5\) and \(19\).
- The prime factors of 99 are \(3\) and \(11\).
- Since they have no common factors, the GCD is \(1\).
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Therefore, \( \frac{95}{99} \) is already in its simplest form.
The repeating decimal \(0.\overline{95}\) as a fraction in simplest form is: \[ \boxed{\frac{95}{99}} \]