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

(i) 0.͞6 = 0.666…

Let x = 0.666666…

Multiplying both sides by 10

10x = 6.666

10x = 6 + x 9x = 6

x = 2/3

(ii) 0.47͞ = 0.4777… = 4/10 + 0.777/10

Let x = 0.777…

Multiplying both sides by 10

10x = 7.777…

10x = 7 + x x = 7/9

4/10 + 0.777/10 = 4/10 + 7/90

= 36/90 + 7/90

= 43/90

(iii) 0.0͞0͞1 = 0.001001…

Let x = 0.001001…

Multiplying both sides by 1000

1000x = 1.001001…

1000x = 1 + x 999x = 1

x = 1/999