To convert the repeating decimal \( 0.\overline{67} \) to a fraction, we can use the following steps:
Let \( x = 0.\overline{67} \).
This means that:
\[ x = 0.67676767\ldots \]
Next, we multiply both sides of the equation by 100 to move the decimal point two places to the right:
\[ 100x = 67.67676767\ldots \]
Now we have the two equations:
- \( x = 0.67676767\ldots \)
- \( 100x = 67.67676767\ldots \)
We can subtract the first equation from the second:
\[ 100x - x = 67.67676767\ldots - 0.67676767\ldots \]
This simplifies to:
\[ 99x = 67 \]
Now, divide both sides by 99:
\[ x = \frac{67}{99} \]
To ensure that the fraction is in simplest form, we check if 67 and 99 have any common factors.
- The number 67 is a prime number.
- The factors of 99 are \( 1, 3, 9, 11, 33, 99 \).
Since 67 is not a factor of 99, \( \frac{67}{99} \) is already in simplest form.
Thus, the repeating decimal \( 0.\overline{67} \) as a fraction in simplest form is:
\[ \boxed{\frac{67}{99}} \]