Question

write the following infinite expansion as a fraction. you do not need to simplify.

0.329329329...

Answer 1

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:

1. \( x = 0.329329329... \)
2. \( 1000x = 329.329329329... \)

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 \( \frac{329}{999} \).