Question

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



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:

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