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

0.896896896...

1 answer

Let \( x = 0.896896896\ldots \).

To express this repeating decimal as a fraction, we can identify the repeating part. The repeating block is "896", which has a length of 3. We can manipulate \( x \) as follows:

  1. Multiply \( x \) by \( 1000 \) (since there are 3 digits in the repeating part): \[ 1000x = 896.896896896\ldots \]

  2. Now, we have two equations: \[ x = 0.896896896\ldots \quad (1) \] \[ 1000x = 896.896896896\ldots \quad (2) \]

  3. Subtract equation (1) from equation (2): \[ 1000x - x = 896.896896896\ldots - 0.896896896\ldots \] \[ 999x = 896 \]

  4. Solving for \( x \): \[ x = \frac{896}{999} \]

So, the repeating decimal \( 0.896896896\ldots \) can be expressed as the fraction \( \frac{896}{999} \).