3. Express the following in the form 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)\( 0\overline{.6}\)

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…)

i.e., 9x = 6

i.e., x = 6/9

Hence, x = 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)

i.e., 9x = 4.3

i.e., x = 4.3/9

Hence, x = 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....)

i.e., 999x = 1

Hence, x = 1/999