Q.3. Express the following in the form $\frac{\mathrm{p}}{\mathrm{q}}$ where p and q are integers and q ≠ 0.

(i) 0.$\stackrel{‾}{6}$ (ii) 0.4$\stackrel{‾}{7}$ (iii) 0.$\stackrel{‾}{001}$

Solution:

(i) 0.$\stackrel{‾}{6}$ = 0.666...

Let x = 0.666... (1)

⇒ 10x = 6.666... (2)

Subtracting (1) from (2), we get,

9x = 6

(ii) 0.4$\stackrel{‾}{7}$= 0.4777...

Let x = 0.4777… (1)

⇒ 10x = 4.777… (2)

Subtracting (1) from (2), we get,

9x = 4.3

(iii) 0.$\stackrel{‾}{001}$= 0.001001...

Let x = 0.001001... (1)

⇒ 1000x = 1.001001… (2)

Subtracting (1) from (2), we get,

999x = 1