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

● Related Concept

● Decimals

● Decimal Numbers

● Decimal Fractions

● Like and Unlike Decimals

● Comparing Decimals

● Decimal Places

● Conversion of Unlike Decimals to Like Decimals

● Decimal and Fractional Expansion

● Terminating Decimal

● Non-Terminating Decimal

● Converting Decimals to Fractions

● Converting Fractions to Decimals

● H.C.F. and L.C.M. of Decimals

● Repeating or Recurring Decimal

● Pure Recurring Decimal

● Mixed Recurring Decimal

● BODMAS Rule

● BODMAS/PEMDAS Rules - Involving Decimals

● PEMDAS Rules - Involving Integers

● PEMDAS Rules - Involving Decimals

● PEMDAS Rule

● BODMAS Rules - Involving Integers

● Conversion of Pure Recurring Decimal into Vulgar Fraction

● Conversion of Mixed Recurring Decimals into Vulgar Fractions

● Simplification of Decimal

● Rounding Decimals

● Rounding Decimals to the Nearest Whole Number

● Rounding Decimals to the Nearest Tenths

● Rounding Decimals to the Nearest Hundredths

● Round a Decimal

● Adding Decimals

● Subtracting Decimals

● 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

7th Grade Math Problems

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