Let \( x = 0.6\overline{6} \).
To eliminate the repeating decimal, we can multiply \( x \) by 10 (since the repeat is one digit long) to shift the decimal point:
\[ 10x = 6.6\overline{6} \]
Now, we can set up the following equation by subtracting the original \( x \) from this equation:
\[ 10x - x = 6.6\overline{6} - 0.6\overline{6} \]
This simplifies to:
\[ 9x = 6 \]
Now, we can solve for \( x \):
\[ x = \frac{6}{9} \]
Next, we reduce the fraction to its lowest terms. The greatest common divisor of 6 and 9 is 3:
\[ x = \frac{6 \div 3}{9 \div 3} = \frac{2}{3} \]
Thus, the fraction that is equivalent to \( 0.6\overline{6} \) is \(\frac{2}{3}\).