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 10(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 10(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:**

**Solution:**

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

10x = 1.810x = 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**

**Solution: **

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

10x = 2.310x = 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 =x = 7/30

**Therefore, the vulgar fraction = 7/30**

**Solution: **

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

100x = 43.213100x = 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 = 4317x = 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;**

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**

● **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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