Question

Question 7
$\text{Express the following in the form}$$\frac{p}{q}$, $\text{where p and q are integers and}q\ne 0$
(i) 0.2
(ii) 0.888…
(iii) $5.\overline{2}$
(iv) $0\overline{.001}$
(v)0.2555…
(vi)$0.1\overline{34}$
(vii) 0.00323232...
(viii) 0.404040…

Answer 1

$\text{(i) Let x}=0.2=\frac{2}{10}=\frac{1}{5}$
$\therefore x=\frac{1}{5}$

(ii) Let x = 0.888...
On multiplying the above by 10 , we get
10x=8.888...
We have, 10x – x = ( 8.88...) – ( 0.888...)
$\Rightarrow 9x=8$
$\therefore x=\frac{8}{9}$

$\text{(iii) Let x =5.}\overline{2}=\mathrm{5.222...}$
On multiplying the above by 10, we get
10x=52.222...
We have, 10x – x = ( 52.22...) – ( 5.22…)
$\Rightarrow 9x=47$
$\therefore x=\frac{47}{9}$

{(iv) $\Rightarrow Letx=0.\overline{001}=\mathrm{0.001001...}$

On multiplying the above by 1000, we get
1000x=001.001...
We have, 1000x – x = (001.001...) – ( 0.001001...)
$\Rightarrow 999x=001=1$
$\therefore x=\frac{1}{999}$

(v) Let x =0.2555...
On multiplying the above equation by 10, we get
10x=2.555...
On multiplying the above equation by 10, we get
100x=25.55...
We have, 100x – 10x = (25.55…) – ( 2.555…)
$\Rightarrow 90x=23$
$\therefore x=\frac{23}{90}$

$\text{(vi) Let x}=0.1\overline{34}$
$\Rightarrow x=0.1\overline{34}=\mathrm{0.13434...}$
On multiplying the above equation by 10, we get
10x=1.3434…
On multiplying the above equation by 100, we get
1000x=134.3434...
We have, 1000x – 10x = (134.34...) – (1.3434…)
$\Rightarrow$990x = 133
$\therefore x=\frac{133}{990}$

(vii) Let x = 0.00323232...
On multiplying the above equation by 100 we get
100x = 0.3232...
On multiplying the above equation by 100 we get
10000x = 32.3232...
$\therefore$ 10000x – 100x = (32.32...) - (0.3232…)
$\Rightarrow 9900x=32$
$\therefore x=\frac{32}{9900}=\frac{8}{2475}$ [ Dividing numerator and denominator by 4]

(viii) Let x=0.404040...
On Multiplying the above equation by 100, we get
100x=40.4040...
$\Rightarrow$100x-x=(40.4040...) - (0.404040...)
$\Rightarrow 99x=40$
$\therefore x=\frac{40}{99}$