Write 0.666 as a fraction.

The correct answer is: If 0.666 (finite) then $\frac{333}{500}$; if 0.666\ldots (repeating) then $\frac{2}{3}$.

Explanation

0.666 written exactly (three decimal places) is a finite decimal; 0.666… with an ellipsis means the digit 6 repeats forever. These two interpretations give different fractions.

Steps (finite 0.666):

1. Write as a fraction: $$0.666=\frac{666}{1000}$$
2. Reduce by the greatest common divisor (2): $$\frac{666}{1000}=\frac{333}{500}$$

Steps (repeating 0.666\ldots):

1. Let $x=0.666\ldots$
2. Multiply by 10: $$10x=6.666\ldots$$
3. Subtract: $$10x-x=6.666\ldots-0.666\ldots\implies9x=6$$
4. Solve: $$x=\frac{6}{9}=\frac{2}{3}$$

Therefore, use $\frac{333}{500}$ for the finite decimal 0.666, or $\frac{2}{3}$ for the repeating decimal 0.666… .