To express the repeating decimal \( 0.329329329... \) as a fraction, we can follow these steps:
Let \( x = 0.329329329... \).
Notice that the decimal has a repeating block of three digits (329). To eliminate the repeating part, multiply \( x \) by \( 1000 \) (since the block has three digits):
\[ 1000x = 329.329329329... \]
Now, we have two equations:
- \( x = 0.329329329... \)
- \( 1000x = 329.329329329... \)
Next, we subtract the first equation from the second:
\[ 1000x - x = 329.329329329... - 0.329329329... \]
This simplifies to:
\[ 999x = 329 \]
Now, to express \( x \) as a fraction, we divide both sides by 999:
\[ x = \frac{329}{999} \]
Thus, the infinite expansion \( 0.329329329... \) can be written as the fraction \( \frac{329}{999} \).