Conversion of Mixed Recurring Decimals into Vulgar Fractions

Follow the steps for the conversion of mixed recurring decimals into vulgar fractions:

(i) First write the decimal form by removing the bar from the top and put it equal to x (any variable).

(ii) Now, find the number of digits having no bar after the decimal point.

(iii) Suppose there are n-digits without bar, multiply both sides by 10n, so that only the repeating decimal digit is on the right side of the decimal point.

(iv) Now write the repeating digits at least twice.

(v) Now, find the number of digits having bar after the decimal point.

(vi) Suppose there are n-digits having bar, multiply both sides by 10n.

(vii) Then subtract the number obtained in step (i) from the number obtained in step (ii).

(viii) Then divide both the sides of the equation by the coefficient of x.

(ix) Therefore, we get the required vulgar fraction in the lowest form.

Worked-out examples for the conversion of mixed recurring decimals into vulgar fractions:

1. Express 0.18 as vulgar fraction.

Solution:

x = 0.18

Multiply both sides by 10 (Since number of digits without bar is 1)

10x = 1.8

10x = 1.88…… ---------- (i)

10 × 10x = 1.88…… × 10 (Since number of digits having bars is 1)

100x = 18.8….. ------------- (ii)

Subtracting (i) from (ii)

100x - 10x = 18.8 - 1.8

90x = 17

x = 17/90

Therefore, the vulgar fraction = 17/90

2. Express 0.23 as vulgar fraction.

Solution:

x = 0.23

Multiply both sides by 10 (Since number of digits without bar is 1)

10x = 2.3

10x = 2.33…… ---------- (i)

10 × 10x = 2.33…… × 10 (Since number of digits having bars is 1)

100x = 23.3….. ------------- (ii)

Subtracting (i) from (ii)

100x - 10x = 23.3 - 2.3

90x = 21

x = 21/90

x = 7/30

Therefore, the vulgar fraction = 7/30

3. Express 0.43213 as vulgar fraction.

Solution:

x = 0.43213

Multiply both sides by 100 (Since number of digits without bar is 2)

100x = 43.213

100x = 43.213213…… ---------- (i)

100 × 1000x = 43.213…… × 1000 (Since number of digits having bars is 3)

100000x = 43213.213….. ------------- (ii)

Subtracting (i) from (ii)

100000x - 100x = 43213.213 - 43.213213

99900x = 43170

x = 43170/99900

x = 4317/9990

Therefore, the vulgar fraction = 4317/9990

Shortcut method for solving the problems on conversion of mixed recurring decimals into vulgar fractions:

The difference between the number formed by all the digits in decimal part and the number formed by digits that are not repeated, gives the numerator of the vulgar fraction and for its denominator the number formed by as many nines as there are recurring digits which are repeated followed by as many zeros as the number of non-repeating or non-recurring digits.

For example;

Express 0.123 as vulgar fraction.
Numerator = 123 - 12 = 111

Denominator = one nine (as there are one recurring digit) followed by two zeros (as there are two non-recurring digits) = 900

Required fraction = 111/900 (reduce to its simplest form)

Therefore, the vulgar fraction = 37/300

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