1. The problem is to convert the repeating decimal $0.666666\ldots$ into a rational number. 2. Let $x = 0.666666\ldots$. 3. Multiply both sides by 10 to shift the decimal point: $$10x = 6.666666\ldots$$ 4. Subtract the original equation from this new equation: $$10x - x = 6.666666\ldots - 0.666666\ldots$$ $$\cancel{10x} - \cancel{x} = 6.666666\ldots - 0.666666\ldots$$ $$9x = 6$$ 5. Solve for $x$: $$x = \frac{6}{9}$$ 6. Simplify the fraction by dividing numerator and denominator by their greatest common divisor 3: $$x = \frac{\cancel{6}^2}{\cancel{9}^3} = \frac{2}{3}$$ 7. Therefore, the repeating decimal $0.666666\ldots$ as a rational number is $\frac{2}{3}$.