Solution 1.1:7c

If we look more closely at this number, we see that the combination 001 is repeated from the second decimal place onwards,

and this reveals that the number is rational.

By multiplying a certain number of times by 10 we can move the decimal place step by step to the right

\displaystyle \insteadof[right]{10000x}{x}{}=0\,\textrm{.}\,2\ 001\ 001\ 001\,\ldots

\displaystyle \insteadof[right]{10000x}{10x}{}=2\,\textrm{.}\,\underline{001}\ \underline{001}\ \underline{001}\ 1\ldots

\displaystyle \insteadof[right]{10000x}{100x}{}=20\,\textrm{.}\,01\ 001\ 001\ 1\ldots

\displaystyle \insteadof[right]{10000x}{1000x}{}=200\,\textrm{.}\,1\ 001\ 001\ 1\ldots

\displaystyle \insteadof[right]{10000x}{10000x}{}=2001\,\textrm{.}\,\underline{001}\ \underline{001}\ 1\ldots

In this list, we see that 10x and 10000x have the same decimal expansion, which means that

\displaystyle 10000x-10x = 2001\textrm{.}\underline{001}\ \underline{001}\ \underline{001}\,\ldots - 2\textrm{.}\underline{001}\ \underline{001}\ \underline{001}\,\ldots

\displaystyle \phantom{10000x-10x}{} = 1999\,\mbox{.}\quad(decimal parts cancel)

As \displaystyle 10000x-10x = 9990x then

\displaystyle 9990x = 1999\quad\Leftrightarrow\quad x = \frac{1999}{9990}\,\mbox{.}