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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)\( 0\overline{.001}\)
Answer :

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