Question

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



0.666...0.666...0.666...

Answer 1

To express the repeating decimal \( 0.666\ldots0.666\ldots0.666\ldots \) as a fraction, we need to identify the repeating part.

Let's denote the decimal expansion as \( x \):
\[
x = 0.666\ldots
\]

The digit "6" repeats after the decimal indefinitely. To convert this repeating decimal into a fraction, we can set up the following equation:

1. Multiply \( x \) by 10 (to shift the decimal point to the right):
\[
10x = 6.666\ldots
\]

2. Now, set up the equation using the original \( x \):
\[
10x = 6 + x
\]

3. Rearranging this gives:
\[
10x - x = 6
\]
\[
9x = 6
\]

4. Solving for \( x \) yields:
\[
x = \frac{6}{9}
\]

This fraction can be simplified. However, you mentioned that simplification is not needed, so we can simply state the infinite decimal expansion as a fraction without simplification:

\[
\frac{6}{9}
\]

Thus, the repeating decimal \( 0.666\ldots0.666\ldots0.666\ldots \) can be expressed as the fraction \( \frac{6}{9} \).