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23/05/2021 12:48 pm
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Express the following in the form p/q, where p and q are integers and q 0.
(i) \(0.\bar{6}\)
(ii) \(0.4\bar{7}\)
(iii) \(0.\overline{001}\)
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23/05/2021 12:53 pm
(i) \(0.\bar{6}\) = 0.666 .....
Assume that x = 0.666…
Then,10x = 6.666…
10x = 6 + x
9x = 6
x = \(\frac{2}{3}\)
(ii) \(0.4\bar{7}\) = 0.4777...
= \(\frac{4}{10}\) + \(\frac{0.777}{10}\)
Assume that x = 0.777…
Then, 10x = 7.777…
10x = 7 + x
x = \(\frac{7}{9}\)
(4/10)+(0.777../10) = (4/10)+(7/90) ( x = 7/9 and x = 0.777…0.777…/10 = 7/(9×10) = 7/90 )
= (36/90)+(7/90) = 43/90
(iii) \(0.\overline{001}\) = 0.001001....
Assume that x = 0.001001…
Then, 1000x = 1.001001…
1000x = 1 + x
999x = 1
x = \(\frac{1}{999}\)