Convert the following repeating decimal to a fraction in simplest form.

Question

., start overline, 9, 5, end overline
.
95

Answer 1

To convert the repeating decimal \(0.\overline{95}\) to a fraction, we can follow these steps:

1. Let \(x = 0.\overline{95}\). This means \(x = 0.959595...\).

2. To eliminate the repeating part, we can multiply \(x\) by 100 (since the repeating section has two digits): \[ 100x = 95.959595... \]

3. Now we have two equations: \[ x = 0.959595... \] \[ 100x = 95.959595... \]

4. Next, we subtract the first equation from the second: \[ 100x - x = 95.959595... - 0.959595... \] \[ 99x = 95 \]

5. Now, solve for \(x\): \[ x = \frac{95}{99} \]

6. 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 \(1\).

7. Therefore, \( \frac{95}{99} \) is already in its simplest form.

The repeating decimal \(0.\overline{95}\) as a fraction in simplest form is: \[ \boxed{\frac{95}{99}} \]