Convert a Pure Recurring Decimal into Vulgar Fraction

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1. Convert a Pure Recurring Decimal into Vulgar FractionTo convert a pure recurring decimal into a vulgar fraction,
(1) Remove the number left to the decimal point, if any.
(2) Write the repeated figures only once in the numerator without the decimal point.
(3) Write as many nines in the denominator as the number of repeating figures.
(4) Add the number removed in step 1(if any) with the fraction obtained in the above steps.

Examples

(a)  $0.\overline{3}=\frac{3}{9}=\frac{1}{3}$

(b)  $0.\overline{7}=\frac{7}{9}$

(c)  $0.\overline{36}=\frac{36}{99}=\frac{4}{11}$

(d)  $0.\overline{067}=\frac{067}{999}=\frac{67}{999}$

(f)  $1.\overline{3}=1+\frac{3}{9}=1+\frac{1}{3}=\frac{4}{3}$

(g)  $5.\overline{36}=5+\frac{36}{99}=5+\frac{4}{11}=\frac{59}{11}$

2. Convert a Mixed Recurring Decimal into Vulgar FractionTo convert a mixed recurring decimal into a vulgar fraction,
(1) Remove the number left to the decimal point, if any.
(2) Numerator is the difference between the number formed by all the digits (taking repeated digits only once) and that formed by the digits which are not repeated.
(3) Denominator is the number formed by taking as many nines as the number of repeating figures followed by as many zeros as the number of non-repeating digits.
(4) Add the number removed in step 1(if any) with the fraction obtained in the above steps.

Examples

(a)  $0.54\overline{29}=\frac{(5429-54)}{9900}=\frac{5375}{9900}=\frac{215}{396}$

(b)  $0.1\overline{6}=\frac{(16-1)}{90}=\frac{15}{90}=\frac{1}{6}$

(c)  $0.58\overline{3}=\frac{(583-58)}{900}=\frac{525}{900}=\frac{21}{36}=\frac{7}{12}$

(d)  $1.3\overline{18}=1+\frac{(318-3)}{990}=1+\frac{315}{990}=1+\frac{63}{198}=1+\frac{21}{66}=1+\frac{7}{22}=\frac{29}{22}$