Question:
The value of $1.999$... in the form of $\mathrm{p} / \mathrm{q}$, where $p$ and $q$ are integers and
(a) $\frac{19}{10}$
(b) $\frac{1999}{1000}$
(c) 2
(d) $\frac{1}{9}$
Solution:
(c)
Let $x=1.999 . . .$
Now, $\quad 10 x=19.999 \ldots$
On subtracting Eq. (i) from Eq. (ii), we get
$10 x-x=(19.999 \ldots)-(1.9999 \ldots)$
$\Rightarrow \quad 9 x=18$
$\therefore$ $x=\frac{18}{9}=2$