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.

** **

Worked-out examples for the conversion of pure recurring decimal into vulgar fraction:

Let n = 0.4

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**

Let n = 0.38

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**

Let n = 0.532

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;**

Here numerator is the period (5) and the denominator is 9 because there is one digit in the period.

= 5/9

Numerator = 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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