Write the following infinite decimal expansion as a fraction. You do not need to simplify.

Question

0.681‾0.\overline{681}0.
681

Answer 1

To convert the repeating decimal \( 0.\overline{681} \) into a fraction, we can use the following method:

Let \( x = 0.\overline{681} \).

To eliminate the repeating part, we can multiply \( x \) by \( 1000 \) (because the repeating part is three digits long):

\[ 1000x = 681.\overline{681} \]

Now, if we subtract the original \( x \) from this equation:

\[ 1000x - x = 681.\overline{681} - 0.\overline{681} \]

This simplifies to:

\[ 999x = 681 \]

Now, we can solve for \( x \):

\[ x = \frac{681}{999} \]

So, the infinite decimal expansion \( 0.\overline{681} \) as a fraction is:

\[ \frac{681}{999} \]

You do not need to simplify it, so the final answer is:

\[ \frac{681}{999} \]