To convert the repeating decimal \(0.6\overline{0}\) to a fraction, we proceed as follows:
Let \(x = 0.6\overline{0}\).
-
The repeating part consists of only one digit (the 0), so we multiply \(x\) by 10 to shift the decimal point one place to the right: \[ 10x = 6.\overline{0} \]
-
Now, we write down the equation: \[ 10x = 6.0 + x \] The repeating portion remains \(x\).
-
Subtract \(x\) from both sides: \[ 10x - x = 6 \] Simplifying gives: \[ 9x = 6 \]
-
Now, solve for \(x\): \[ x = \frac{6}{9} \]
-
We can simplify \(\frac{6}{9}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3: \[ x = \frac{6 \div 3}{9 \div 3} = \frac{2}{3} \]
Thus, the repeating decimal \(0.6\overline{0}\) as a fraction in its lowest terms is: \[ \boxed{\frac{2}{3}} \]