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 = \(\frac{1}{999}\)