Question

Find the value of $(\mathrm{0.333....}{)}^2$ in decimal form.

• A. 0.999.....
• B. 0.666.....
• C. 0.111.......
• D. 0.333......

Answer 1

The correct option is C 0.111.......

Given operation is $(\mathrm{0.333....}{)}^2$
Now, $\mathrm{0.333....}=3\times \mathrm{0.111...}$............(1)
Let '$x$' be 0.111....
Multilpy $x$ with 10.
$\Rightarrow$ $10x$ = 1.11....
= 1 + 0.111....
= 1 + $x$ $(\because x=\mathrm{0.111....})$
$\Rightarrow$ $9x$ = 1
$\Rightarrow$ $x$ = $\frac{1}{9}$
$\therefore$ 0.111.... = $\frac{1}{9}$................(2)

Substituting the value of 0.111... in (1) we get,
$\mathrm{0.333....}=3\times \frac{1}{9}$
$=\frac{1}{3}$

$\therefore (\mathrm{0.333....}{)}^2=(\frac{1}{3}{)}^2$
$=\frac{1}{9}$
$\because \frac{1}{9}=\mathrm{0.111....}$ [from eq (2)]
$\Rightarrow (\mathrm{0.333....}{)}^2=\mathrm{0.111.....}$