Represent the following as rational numbers :
`(i) 0.bar43` (ii) `0.4bar3` (iii) `24.28bar(137)`

Explanation

(i) `(i) 0.bar43 = 0.434343... = 0.43 + 0.000043 + ...`
`=(43)/(100) + (43)/(10000) + (43)/(1000000) + ...`
= Sum of infinite G.P. with ` a = (43)/(100) and r = (1)/(100)`
Here ` r = (1)/(100)`, Clearly `|(1)/(r)| lt 1`
`therefore 0.bar43 = S_(oo) = ((43)/(100))/(1-(1)/(100))= ((43)/(100))/((99)/(100) )=(43)/(99)`
(ii) `0.bar43 = 0.4333... = 0.4 + 0.03 + 0.003 +0.0003 + ...`
`= (4)/(10) + ((3)/(100) + (3)/(1000)+(3)/(10000) + ...)`
` (4)/(10) + ` (Sum of infinite G.P. with a ` = (3)/(100) and r = (1)/(10))`
`= (4)/(10) + ((3)/(100))/(1-(1)/(10))`
`=(4)/(10) + (3)/(100) xx(10)/(9)`
`=(4)/(10) + (1)/(30) = (13)/(30)`
(iii) `24.28bar137 = 24.28137137137...`
24.28 +0.00137 + 0.00000137 + ...`
` 24.28 + (Sum of infinite G.P. with a ` = (137)/(100000) and r = (1)/(1000))`
`24.28 + ((137)/(100000))/(1-(1)/(1000))`
` 24.28 + (137)/(100000)xx(1000)/(999)`
`(2428)/(100) + (137)/(99900)`
` = (2425709)/(99900)`