Question

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

(i) 0.6(ii) 0.47 (iii) 0.001.

Answer 1

(i) $\overline{0.6}$ = 0.666....

One digit 6 is repeating. We multiply it with 10 on both sides.

10x = $\overline{6.6}$ ⇒ 10x = 6 + x

⇒ 10x - x = 6 ⇒ 9x = 6 ⇒ x = $\frac{6}{9}$ = $\frac{2}{3}$

(ii) $\overline{0.47}$= 0.4777....

One digit is repeating. We multiply it with 10 on both sides.

∴ 10x = 4.7= 4.3 + .47 = 4.3 + x

⇒ 9x = 4.3 ⇒ x = $\frac{4.3}{9}$ =$\frac{43}{90}$

(iii) 0.001= x = 0.001

Here three digits repeats; we multiply with 1000.

∴ 1000x = $\overline{1.001}$= 1000x = 1 + x

⇒ 1000x - x = 1 ⇒ 999x = 1

⇒ x = $\frac{1}{999}$