Express the following in the form \(\frac{p}{q} \) where p and q are integers and \(q \ne 0\) .
i)\( 0\overline{.6}\)
ii)\( 0.4\overline{7}\)
iii)\( 0\overline{.001}\)

i) Let x = 0.66666... .......(i)


multiplying by eq. (i) by 10 , we get,

10 x = 6.666...... .....(ii)

On subtracting Eq. (ii) from Eq. (i), we get ,

(10 x – x) = (6.666…) – (0.666…)
\(\Rightarrow \) 9x = 6
\(\Rightarrow \)x = \(\frac{6}{9} \)
Hence, x = \(\frac{2}{3} \)


ii)Let x = 0.477777... .......(i)


multiplying by eq. (i) by 10 , we get,

10 x = 4.777....... ......(ii)

On subtracting Eq. (ii) from Eq. (i), we get,

(10 x – x) = (4.777) - (0.4777)
\(\Rightarrow \) 9x = 4.3
\(\Rightarrow \) x = \(\frac{4.3}{9} \)
Hence, x = \(\frac{43}{90} \)


iii)Let x = 0.001001001...... ...... (i)


multiplying by eq. (i) by 1000 , we get,

1000 x = 1.001001001.... .....(ii)

On subtracting Eq. (ii) from Eq. (i), we get

(1000 x – x) = (1.001001001....) – (0.001001001....)
\(\Rightarrow \) 999x = 1
Hence, x = \(\frac{1}{999} \)