To convert the repeating decimal \( 0.\overline{72} \) into a fraction, we can use the following steps.
Let \( x = 0.\overline{72} \).
This means that: \[ x = 0.72727272\ldots \]
Next, we can multiply both sides of the equation by 100 (since the repeating part has 2 digits): \[ 100x = 72.727272\ldots \]
Now, we can set up the equation: \[ 100x = 72 + x \]
Now, subtract \( x \) from both sides: \[ 100x - x = 72 \] \[ 99x = 72 \]
Now, solve for \( x \): \[ x = \frac{72}{99} \]
Next, we simplify the fraction \( \frac{72}{99} \). We can find the greatest common divisor (GCD) of 72 and 99, which is 9.
Now divide both the numerator and the denominator by the GCD: \[ \frac{72 \div 9}{99 \div 9} = \frac{8}{11} \]
Thus, the repeating decimal \( 0.\overline{72} \) as a fraction in simplest form is: \[ \boxed{\frac{8}{11}} \]