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

Question

0.683‾

Answer 1

To express the repeating decimal \( 0.683\overline{683} \) as a fraction, we can set it equal to a variable:

Let \[ x = 0.683683683\ldots \]

Now we can multiply both sides of the equation by \( 1000 \) (since the repeating part consists of 3 digits):

\[ 1000x = 683.683683683\ldots \]

Now we have two equations:

1. \( x = 0.683683683\ldots \)
2. \( 1000x = 683.683683683\ldots \)

Next, we can subtract the first equation from the second:

\[ 1000x - x = 683.683683683\ldots - 0.683683683\ldots \] \[ 999x = 683 \]

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

\[ x = \frac{683}{999} \]

Therefore, the repeating decimal \( 0.683\overline{683} \) as a fraction is

\[ \frac{683}{999} \]