Write \( 0 . \overline{67} \) as a fraction.

To convert \( 0 . \overline{67} \) into a fraction, let's start by setting it equal to a variable: Let \( x = 0.676767...\). To eliminate the repeating decimal, we can multiply both sides by 100 (since the repeat length is 2): \[ 100x = 67.676767... \] Next, we can subtract the original equation \( x = 0.676767... \) from this new equation: \[ 100x - x = 67.676767... - 0.676767... \] This simplifies to: \[ 99x = 67 \] Now, solving for \( x \) gives us: \[ x = \frac{67}{99} \] Thus, \( 0 . \overline{67} \) as a fraction is \( \frac{67}{99} \). Now let's sprinkle in some fun facts! Did you know that repeating decimals like \( 0.\overline{67} \) are just a way of expressing rational numbers? In fact, it can often be surprising to discover how many fractions can be represented as repeating decimals – it opens up a whole new world of math magic! Also, if you run into a really tricky repeating decimal, a handy approach is to always set up two equations (like we did), which allow you to eliminate the repeating part. Just keep track of the decimal places when you multiply, and you’ll save yourself from confusion!