#### Question

Express the following in the form p/q, where p and q are integers and q ≠ 0.

`(i) 0.bar6`

`(ii) 0.4bar7`

`(iii) 0.bar001`

#### Solution

`(i) 0.bar6`

Let x = 0.666 ....................(1)

∴10x = 6.666 ......................(2)

(2) - (1)

⇒9x = 6

`x = 6/9 = 2/3`

`(ii) 0.4bar7`

`"Let "x = 0.4bar7 = 4.7777` ....................(1)

Multiplying both sides by 10 (since one digit is repeating),

we get 10x = 4.7777 ..........................(2)

(2) - (1)

⇒ 10x - x = 4.3

⇒ 9x = 4.3

`⇒ x = (4.3/9) = 43/90`

`"Thus, " 0.4bar7=43/90`

Here p = 43, q = 90(≠0)

(iii) 0.bar001

`"Let "x = 0.bar001 = 0.001001001` .........................(1)

Multiplying both sides by 1000(since three digits are repeating),

we get 1000x = 1.001001 ..........................(2)

(2) - (1)

⇒ 100x - x = 1

⇒ 999x = 1

`⇒ x = 1/999`

`"Thus, "0.bar001 = 1/999`

Here P = 1, q=999(≠0)