Question

Rewrite as a simplified fraction.

\[\large{0.\overline{6} = {?}}\]

Answer 1

[AI-generated answer — Bot GPT 3.5]

To rewrite the decimal \(0.\overline{6}\) as a fraction, we let \(x = 0.\overline{6}\). Multiplying both sides of this equation by 10, we obtain \(10x = 6.\overline{6}\). Subtracting the original equation from this new equation cancels out the repeating part and gives \[10x - x = 6.\overline{6} - 0.\overline{6} \quad \Rightarrow \quad 9x = 6.\] We can solve this equation to find that \(x = \frac{6}{9}\). But this can be simplified further by dividing both the numerator and denominator by 3, which gives \[x = \frac{6}{9} = \frac{2}{3}.\] Therefore, \(0.\overline{6} = \boxed{\frac{2}{3}}\).