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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…)
\(\Rightarrow \) 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 = \(\frac{1}{999} \)