To convert the repeating decimal \(0.6\) (where \(6\) repeats indefinitely) into a fraction, we can use the following algebraic approach:
Let \(x = 0.666...\).
Then, we can multiply both sides of the equation by \(10\) (since there's one digit that repeats):
\[ 10x = 6.666... \]
Now, we can subtract the original equation from this new equation:
\[ 10x - x = 6.666... - 0.666... \]
This simplifies to:
\[ 9x = 6 \]
Now, solve for \(x\):
\[ x = \frac{6}{9} \]
We can simplify this fraction:
\[ x = \frac{2}{3} \]
Thus, the fraction that is equivalent to the repeating decimal \(0.666...\) is \(\frac{2}{3}\).