Question

Express each of the following as a rational number in its simplest form:

(i) \(0.\bar8\) 

(ii) \(2.\bar4\) 

(iii) \(0.\overline{24}\) 

(iv) \(0.\overline{12}\) 

(v) \(2.\overline{24}\) 

(vi) \(0.\overline{365}\) 

Answer 1

(i) Let x = \(\overline{0.8}\) 

∴ x = 0.888 …(1) 

10x = 8.888 …(2) 

On subtracting equation (1) from (2), we get 

9x = 8 ⇒ x = \(\frac{8}9\) 

∴0.8 = \(\frac{\bar8}9\) 

(ii) Let x = \(\overline{2.4}\) 

∴ x = 2.444 …(1) 

10x = 24.444 …(2) 

On subtracting equation (1) from (2), we get 

9x = 22 ⇒ x = \(\frac{22}9\)

∴ 2.4 = \(\frac{\overline{22}}9\)

(iii) Let x = \(\overline{0.24}\) 

∴ x = 0.2424 …(1) 

100x = 24.2424 …(2)

On subtracting equation (1) from (2), we get 

99x = 24 ⇒ x = \(\frac{8}{33}\) 

∴ 0.24 = \(\frac{\bar8}{33}\) 

(iv) Let x = \(\overline{0.12}\) 

∴ x = 0.1212 …(1) 

100x = 12.1212 …(2) 

On subtracting equation (1) from (2), we get 

99x = 12 ⇒ x = \(\frac{4}{33}\) 

∴ 0.12 = \(\frac{\bar4}{33}\) 

(v) Let x = \(\overline{2.24}\) 

∴ x = 2.2444 …(1) 

10x = 22.444 …(2) 

100x = 224.444 …(3) 

On subtracting equation (2) from (3), we get 

90x = 202 ⇒ x = \(\frac{202}{90}\) = \(\frac{101}{45}\)

∴ \(\overline{2.24}\) = \(\frac{101}{45}\) 

(vi) Let x = \(\overline{0.365}\) 

∴ x = 0.3656565 …(1) 

10x = 3.656565 …(2) 

1000x = 365.656565 …(3) 

On subtracting equation (2) from (3), we get 

990x = 362 ⇒ x = \(\frac{362}{990}\) = \(\frac{181}{495}\) 

∴ \(\overline{0.365}\) = \(\frac{181}{495}\)