An algebra problem by Pratik Joshi

We have to find $1.999...$ , so let's say:

$1.999...=x$

• Multiplying this expression by $10$ , we get:

$19.999...=10x$

• Subtracting the $1^{st}$ expression from the $2^{nd}$ we get:

$10x-x=(19.999...)-(1.999...) \\ \implies 9x=18 \\ \implies x=\dfrac{18}9 \\ \implies x=\boxed 2$

9/9=1
1/9=0.1111111.......
Therefore, 0.111111111....... x 9 =0.999999999.......
1/9 x 9=9/9 =1
0.9999999..........=1 :)