Question

The decimal form of $\frac{10}{11}$ is ____

• A. 0.090909.....
• B. 0.909090....
• C. 0.00909...
• D. 0.11111

Answer 1

The correct option is B 0.909090....

Given fraction is $\frac{10}{11}$.
$\frac{10}{11}=10\times \frac{1}{11}$......(1)
Now,
$\frac{1}{11}$ = $\frac{1}{11}$ $\times$ $\frac{100}{100}$
= $\left(\frac{100}{11}\right)$ $\times$ $\frac{1}{100}$
= $\left(9\frac{1}{11}\right)$ $\times$ $\frac{1}{100}$
= $(9+\frac{1}{11})$ $\times$ $\frac{1}{100}$
= $\frac{9}{100}$ + $\frac{1}{1100}$.......(2)

Now, $\frac{1}{1100}$ = $\frac{1}{11\times 100}$
= $\left(\frac{1}{11}\right)$ $\times$ $\frac{1}{100}$.......(3)
From (2), $\frac{1}{11}$ = $\frac{9}{100}$ + $\frac{1}{1100}$; substitute it in eq (3).

$\Rightarrow$ $\frac{1}{1100}$ = $(\frac{9}{100}$ + $\frac{1}{1100})$ $\times$ $\frac{1}{100}$

= $\frac{9}{10000}$ + $\frac{1}{110000}$
Now substituting this value in eq(1), we get

$\frac{1}{11}$ = $\frac{9}{100}$ + $(\frac{9}{10000}$ + $\frac{1}{110000})$

= $(\frac{9}{100}$ + $\frac{9}{10000})$ + $\frac{1}{110000}$

Similarly finding the value of $\frac{1}{110000}=(\frac{1}{11}\times \frac{1}{10000})$
and on solving that will give $\frac{9}{1000000}$ + $\frac{1}{11000000}$

$\Rightarrow$ $\frac{1}{11}$ = $\frac{9}{100}$ + $\frac{9}{10000}$ + $(\frac{9}{1000000}$ + $\frac{1}{11000000})$

This cycle will continue and leads to $\frac{1}{11}$ = $\frac{9}{100}$ + $\frac{9}{10000}$ + $\frac{9}{1000000}$ + ........
= 0.09 + 0.0009 + 0.000009 + .....

$\therefore$ $\frac{1}{11}$ = 0.090909.....
Now, substituting $\frac{1}{11}$ in (1) gives
$\frac{10}{11}=10\times \mathrm{0.090909...}$
= 0.909090....