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

Question

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

GPT-4o mini · Answer

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: $$ \frac{82}{99} $$