To convert the repeating decimal \( 0.6\overline{6} \) (which represents \( 0.6666\ldots \)) to a fraction, we can follow these steps:
-
Let \( x = 0.6666\ldots \).
-
Multiply both sides of the equation by 10 to shift the decimal point: \[ 10x = 6.6666\ldots \]
-
Now, subtract the original equation from this new equation: \[ 10x - x = 6.6666\ldots - 0.6666\ldots \]
This simplifies to: \[ 9x = 6 \]
-
Now, solve for \( x \): \[ x = \frac{6}{9} \]
-
Reduce the fraction to its lowest terms: \[ \frac{6}{9} = \frac{2}{3} \]
Thus, the fraction that is equivalent to \( 0.6\overline{6} \) is \( \frac{2}{3} \).
1 answer