Question

onvert the following repeating decimal to a fraction in simplest form.
., start overline, 8, 2, end overline
.
82

Answer 1

To convert the repeating decimal \(0.\overline{82}\) (where \(82\) repeats) to a fraction, we can follow these steps:

1. Let \( x = 0.\overline{82} \).

2. Multiply both sides of the equation by \(100\) (since there are 2 digits in the repeating part):
\[
100x = 82.\overline{82}
\]

3. Now we have two equations:
\[
x = 0.\overline{82}
\]
\[
100x = 82.\overline{82}
\]

4. Subtract the first equation from the second:
\[
100x - x = 82.\overline{82} - 0.\overline{82}
\]
\[
99x = 82
\]

5. Now solve for \(x\):
\[
x = \frac{82}{99}
\]

6. Check if the fraction \(\frac{82}{99}\) can be simplified. The greatest common divisor (GCD) of \(82\) and \(99\) is \(1\) (since \(82\) is \(2 \times 41\) and \(99\) is \(3^2 \times 11\)), which means the fraction is already in its simplest form.

Thus, the repeating decimal \(0.\overline{82}\) as a fraction in simplest form is:
\[
\frac{82}{99}
\]