Question

Convert the following repeating decimal to a fraction in simplest form.
., 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:
\[
\boxed{\frac{95}{99}}
\]