Question

Express each of the following as a fraction in simplest form.

$\left(i\right)0.\overline{8}\left(ii\right)2.\overline{4}\left(iii\right)0.\overline{24}\left(iv\right)0.1\overline{2}\left(v\right)2.2\overline{4}\left(vi\right)0.\overline{365}$

Answer 1

(v) 2.24¯¯¯2.24¯

(vi) 0.365¯¯¯¯¯¯¯¯0.365¯

Sol:

(i) Let x = 0.8¯

Therefore, x = 0.888 . . . . . . . . (1)

10x = 8.888 . . . . . . (2)

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

9x = 8⇒⇒=x=8/9

(ii) Let x = 2.4¯2.4¯

Therefore, x =2.444 … (1)

10x = 24.444 . . . . . . . . (2)

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

9x = 22 ⇒⇒ x= 22/9

(iii) Let x = 0.24¯

Therefore, x = 0.2424 …(1)

100x = 24.2424 …(2)

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

99x = 24 ⇒⇒ x= 833833

Therefore, 0.24 = 833¯833¯

(iv) Let x = 0.12¯

Therefore, x = 0.1212 …….(1)

100x = 12.1212 …….(2)

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

99x=12 ⇒⇒ x=4/33

(v) Let x = 2.24¯

Therefore, x = 2.2444 –(1)

10x = 22.444 …(2)

100x = 224.444 …(3)

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

90x = 202 ⇒⇒ x = 202/90

(vi) Let x = 0.365¯

Therefore, x = 0.3656565 …(1)

10x = 3.656565 …(2)

1000x = 365.656565 ….(3)

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

990x = 362 = x = 362/990

Hence, 0.365¯ = 181/495