(i) 0.666…
Let x = 0.666…
⇒ x = 0.6666… …(i)
Multiplying equation (i) by 10 both sides
10x = 6.6666..
⇒ 10x = 6 + 0.6666…
⇒ 10x = 6 + x [From equation (i)]
⇒ 10x x = 6
⇒ 9x = 6
⇒ x = 6/9 = 2/3
(ii) 0.4777…
Let x = 0.4777…
⇒ x = 0.47777… …(i)
Multiplying equation (i) by 10 both sides
⇒ 10x = 4.7777… …(ii)
Multiplying equation (ii) by 10 both sides
⇒ 100x = 47.7777…
⇒ 100x = 43 + 4.7777…
⇒ 100x = 43 + 10x [From equation (ii)]
⇒ 100x = 10x = 43
⇒ 90x = 43
⇒ x = 43/90
(iii) 0.001001001…
Let x = 0.001001001…
⇒ x = 0.001001001…
Multiplying equation (i) by 1000 both sides
1000x = 1.001001001…
⇒ 1000x = 1 + 0.001001001…
⇒ 1000x = 1 = x [From equation (i)]
⇒ 1000x – x = 1
⇒ 999x = 1
⇒ x = 1/999

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