Follow the steps for the conversion of pure recurring decimal into vulgar fraction:
(i) First write the decimal form by removing the bar from the top and put it equal to n (any variable).
(ii) Then write the repeating digits at least twice.
(iii) Now find the number of digits having bars on their heads.
● If the repeating decimal has 1 place repetition, then multiply both sides by 10.
● If the repeating decimal has 2 place repetitions, then multiply both sides by 100.
● If the repeating decimal has 3 place repetitions, then multiply both sides by 1000 and so on.
(iv) Then subtract the number obtained in step (i) from the number obtained in step (ii).
(v) Then divide both the sides of the equation by the coefficient of n.
(vi) Therefore, we get the required vulgar fraction in the lowest form.
Workedout examples for the conversion of pure recurring decimal into vulgar fraction:
1. Express 0.4 as a vulgar fraction.n = 0.444  (i)
Since, one digit is repeated after the decimal point, so we multiply both sides by 10.
Therefore, 10n = 4.44  (ii)
Subtracting (i) from (ii) we get;
10n  n = 4.44  0.44
9n = 4
n = 4/9 [dividing both the sides of the equation by 9]
Therefore, the vulgar fraction = 4/9
n = 0.3838  (i)
Since, two digits are repeated after the decimal point, so we multiply both sides by 100.
Therefore, 100n = 38.38  (ii)
Subtracting (i) from (ii) we get;
100n  n = 38.38  0.38
99n = 38
n = 38/99
Therefore, the vulgar fraction = 38/99
n = 0.532532  (i)
Since, three digits are repeated after the decimal point, so we multiply both sides by 1000.
Therefore, 1000n = 532.532  (ii)
Subtracting (i) from (ii) we get;
1000n  n = 532.532  0.532
999n = 532
n = 532/999
Therefore, the vulgar fraction = 532/999
Shortcut method for solving the problems on conversion of pure recurring decimal into vulgar fraction:
Write the recurring digits only once in the numerator and write as many nines in the denominator as is the number of digits repeated.
For example;
(a) 0.5Here numerator is the period (5) and the denominator is 9 because there is one digit in the period.
= 5/9
(b) 0.45Numerator = period = 45
Denominator = as many nines as the number of digits in the denominator
= 45/99
● Related Concept
● Decimals
● Conversion of Unlike Decimals to Like Decimals
● Decimal and Fractional Expansion
● Converting Decimals to Fractions
● Converting Fractions to Decimals
● H.C.F. and L.C.M. of Decimals
● Repeating or Recurring Decimal
● BODMAS/PEMDAS Rules  Involving Decimals
● PEMDAS Rules  Involving Integers
● PEMDAS Rules  Involving Decimals
● BODMAS Rules  Involving Integers
● Conversion of Pure Recurring Decimal into Vulgar Fraction
● Conversion of Mixed Recurring Decimals into Vulgar Fractions
● Rounding Decimals to the Nearest Whole Number
● Rounding Decimals to the Nearest Tenths
● Rounding Decimals to the Nearest Hundredths
● Simplify Decimals Involving Addition and Subtraction Decimals
● Multiplying Decimal by a Decimal Number
● Multiplying Decimal by a Whole Number
● Dividing Decimal by a Whole Number
● Dividing Decimal by a Decimal Number
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