Question

Find the decimal form of $\frac{1}{9}$.

• A. 0.1111......
• B. 0.99999.....
• C. 0.191919....
• D. 0.333333

Answer 1

The correct option is A 0.1111......

Given fraction is $\frac{1}{9}$.
$\frac{1}{9}$ = $\frac{1}{9}$ $\times$ $\frac{10}{10}$
= $\left(\frac{10}{9}\right)$ $\times$ $\frac{1}{10}$
= $\left(1\frac{1}{9}\right)$ $\times$ $\frac{1}{10}$
= $(1+\frac{1}{9})$ $\times$ $\frac{1}{10}$
= $\frac{1}{10}$ + $\frac{1}{90}$.......(1)

Now, $\frac{1}{90}$ = $\frac{1}{9\times 10}$
= $\left(\frac{1}{9}\right)$ $\times$ $\frac{1}{10}$.......(2)
From (1), $\frac{1}{9}$ = $\frac{1}{10}$ + $\frac{1}{90}$; substitute it in eq (2).

$\Rightarrow$ $\frac{1}{90}$ = $(\frac{1}{10}$ + $\frac{1}{90})$ $\times$ $\frac{1}{10}$

= $\frac{1}{100}$ + $\frac{1}{900}$
Now substituting this value in eq(1), we get

$\frac{1}{9}$ = $\frac{1}{10}$ + $(\frac{1}{100}$ + $\frac{1}{900})$

= $(\frac{1}{10}$ + $\frac{1}{100})$ + $\frac{1}{900}$

Similarly finding the value of $\frac{1}{900}=(\frac{1}{9}\times \frac{1}{100})$
and on solving that will give $\frac{1}{1000}$ + $\frac{1}{9000}$

$\Rightarrow$ $\frac{1}{9}$ = $\frac{1}{10}$ + $\frac{1}{100}$ + $(\frac{1}{1000}$ + $\frac{1}{9000})$

This cycle will continue and leads to $\frac{1}{1}$ = $\frac{1}{10}$ + $\frac{1}{100}$ + $\frac{1}{1000}$ + $\frac{1}{10000}$ + ........
= 0.1 + 0.01 + 0.001 + 0.0001 + .....

$\therefore$ $\frac{1}{9}$ = 0.1111.....