Write \( 0,2 \dot{6} \) in the form \( \frac{a}{b} \) (where \( a \) and \( b \) are integers). Write your answer as a simplified fraction.

To express \( 0.2\dot{6} \) as a fraction, let's set \( x = 0.2\dot{6} \). This means we can write: \[ x = 0.26666\ldots \] Since the "6" is repeating, we can isolate the repeating part. First, multiply \( x \) by 10 to shift the decimal point one place to the right: \[ 10x = 2.6666\ldots \] Now we can multiply \( x \) by 100 to shift two decimal places, which will allow us to line up the repeating parts when we subtract: \[ 100x = 26.6666\ldots \] Now we have the two equations: 1. \( 10x = 2.6666\ldots \) 2. \( 100x = 26.6666\ldots \) Next, we subtract the first equation from the second: \[ 100x - 10x = 26.6666\ldots - 2.6666\ldots \\ 90x = 24 \] Now, solve for \( x \): \[ x = \frac{24}{90} \] We can simplify this fraction by finding the greatest common divisor (GCD) of 24 and 90. The GCD is 6: \[ \frac{24 \div 6}{90 \div 6} = \frac{4}{15} \] Thus, the simplified form of \( 0.2\dot{6} \) as a fraction is \[ \boxed{\frac{4}{15}} \]