To convert the repeating decimal \(0.6\overline{6}\) (which means \(0.666...\)) into a rational number, we can use the following method:
Let \(x = 0.666...\).
Then, multiplying both sides of the equation by 10 gives us:
\[ 10x = 6.666... \]
Now, we can subtract the first equation from the second:
\[ 10x - x = 6.666... - 0.666... \] \[ 9x = 6 \]
Dividing both sides by 9 gives:
\[ x = \frac{6}{9} = \frac{2}{3} \]
So, the rational number that equals \(0.6\overline{6}\) is \(\frac{2}{3}\).
Looking at the answer choices provided:
- \(6/5\)
- \(6/10\)
- \(4/5\)
- \(2/3\)
The correct answer is \(\frac{2}{3}\).
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