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# 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 = $$\frac{1}{999}$$