# Conversion of Pure Recurring Decimal into Vulgar Fraction

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:

1. Express 0.4 as a vulgar fraction.

Solution:

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

2. Express 0.38 as a vulgar fraction.

Solution:

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

3. Express 0.532 as a vulgar fraction.

Solution:

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;

(a) 0.5

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

= 5/9

(b) 0.45

Numerator = period = 45

Denominator = as many nines as the number of digits in the denominator

= 45/99

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