Question

Find the decimal form of $\frac{1}{3}$.

• A. 0.3
• B. 0.6
• C. 0.333333.......
• D. 0.777

Answer 1

The correct option is C 0.333333.......

Given fraction is $\frac{1}{3}$.
$\frac{1}{3}$ = $\frac{1}{3}$ $\times$ $\frac{10}{10}$
= $\left(\frac{10}{3}\right)$ $\times$ $\frac{1}{10}$
= $\left(3\frac{1}{3}\right)$ $\times$ $\frac{1}{10}$
= $(3+\frac{1}{3})$ $\times$ $\frac{1}{10}$
= $\frac{3}{10}$ + $\frac{1}{30}$.......(1)

Now, $\frac{1}{30}$ = $\frac{1}{3\times 10}$
= $\left(\frac{1}{3}\right)$ $\times$ $\frac{1}{10}$.......(2)
From (1), $\frac{1}{3}$ = $\frac{3}{10}$ + $\frac{1}{30}$; substitute it in eq (2).

$\Rightarrow$ $\frac{1}{30}$ = $(\frac{3}{10}$ + $\frac{1}{30})$ $\times$ $\frac{1}{10}$

= $\frac{3}{100}$ + $\frac{1}{300}$
Now substituting this value in eq(1), we get

$\frac{1}{3}$ = $\frac{3}{10}$ + $(\frac{3}{100}$ + $\frac{1}{300})$

= $(\frac{3}{10}$ + $\frac{3}{100})$ + $\frac{1}{300}$

Similarly finding the value of $\frac{1}{300}=(\frac{1}{3}\times \frac{1}{100})$
and on solving that will give $\frac{3}{1000}$ + $\frac{1}{3000}$

$\Rightarrow$ $\frac{1}{3}$ = $\frac{3}{10}$ + $\frac{3}{100}$ + $(\frac{3}{1000}$ + $\frac{1}{3000})$

This cycle will continue and leads to $\frac{1}{3}$ = $\frac{3}{10}$ + $\frac{3}{100}$ + $\frac{3}{1000}$ + $\frac{3}{10000}$ + ........
= 0.3 + 0.03 + 0.003 + 0.0003 + .....

$\therefore$ $\frac{1}{3}$ = 0.3333.....