HOW TO CONVERT RECURRING DECIMALS TO FRACTIONS

Covert each of the following recurring decimals to fractions :

Example 1 :

0.33333..........

Solution :

There are only recurring digits after the decimal point in  

0.33333..........

There is only one digit in the recurring pattern, that is 3. 

Subtract 1 from 10, the result is 9.

Write a fraction with numerator 3 and denominator 9.

=  3/9

=  1/3

Example 2 :

0.77777..........

Solution :

There are only recurring digits after the decimal point in  

0.77777..........

There is only one digit in the recurring pattern, that is 7. 

Subtract 1 from 10, the result is 9.

Write a fraction with numerator 7 and denominator 9.

=  7/9

Example 3 :

0.36363636..........

Solution :

There are only recurring digits after the decimal point in  

0.36363636..........

There are two digits in the recurring pattern, that is 36. 

Subtract 1 from 100, the result is 99.

Write a fraction with numerator 36 and denominator 99.

=  36/99

=  4/11

Example 4 :

0.507507507507..........

Solution :

There are only recurring digits after the decimal point in  

0.507507507507..........

There are three digits in the recurring pattern, that is 507. 

Subtract 1 from 1000, the result is 999.

Write a fraction with numerator 507 and denominator 999.

=  507/999

=  169/333

Example 5 :

0.06666..........

Solution :

There is one digit between the decimal point and the recurring digits, that is 0.

0.06666..........  =  (0.6666..........) ÷ 10 ----(1)

There are only recurring digits after the decimal point in  

0.6666..........

There is only one digit in the recurring pattern, that is 6. 

Subtract 1 from 10, the result is 9.

Write a fraction with numerator 6 and denominator 9 for 0.6666.......... in (1). 

0.06666..........  =  (6/9) ÷ 10

=  (2/3) ÷ (10/1)

=  (2/3) ⋅ (1/10)

=  (2 ⋅ 1)/(3 ⋅ 10)

=  2/30

=  1/15

Example 6 :

0.00151515..........

Solution :

There are two digits between the decimal point and the recurring digits, that is 00.

0.00151515..........  =  (0.151515..........) ÷ 100 ----(1)

There are only recurring digits after the decimal point in  

0.151515..........

There are two digits in the recurring pattern, that is 15. 

Subtract 1 from 100, the result is 99.

Write a fraction with numerator 15 and denominator 99 for 0.151515.......... in (1). 

0.00151515..........  =  (15/99) ÷ 100

=  (5/33) ÷ (100/1)

=  (5/33) ⋅ (1/100)

=  (5 ⋅ 1)/(33 ⋅ 100)

=  5/3300

=  1/660

Example 7 :

0.37777..........

Solution :

There is one digit between the decimal point and the recurring digits, that is 3.

0.37777..........  =  0.3 + 0.07777..........

In 0.07777.........., there is one digit between the decimal point and the recurring digits, that is 0.

0.37777..........  =  0.3 + (0.7777..........) ÷ 10 ----(1)

There are only recurring digits after the decimal point in  

0.7777..........

There is only one digit in the recurring pattern, that is 7. 

Subtract 1 from 10, the result is 9.

Write a fraction with numerator 7 and denominator 9 for 0.7777.......... in (1). 

0.39999..........  =  0.3 + (7/9) ÷ 10

=  3/10 + 7/90

=  27/90 + 7/90

=  (27 + 7)/90

=  34/90

=  17/45

Example 8 :

1.21212121..........

Solution :

1.21212121..........  =  1 + 0.21212121.......... ----(1)

There are only recurring digits after the decimal point in  

0.21212121..........

There are two digits in the recurring pattern, that is 21. 

Subtract 1 from 100, the result is 99.

Write a fraction with numerator 21 and denominator 99 for 0.21212121.......... in (1). 

1.21212121..........  =  1 + 21/99

=  1 + 7/33

=  33/33 + 7/33

=  (33 + 7)/33

=  40/33

Example 9 :

2.342342342..........

Solution :

2.342342342..........  =  2 + 0.342342342.......... ----(1)

There are only recurring digits after the decimal point in  

0.342342342..........

There are three digits in the recurring pattern, that is 342. 

Subtract 1 from 1000, the result is 999.

Write a fraction with numerator 342 and denominator 999 for 0.342342342.......... in (1). 

2.342342342..........  =  2 + 342/999

=   2 + 38/111

=   222/111 + 38/111

=  (222 + 38)/111

=  260/111

Example 10 :

2.05555..........

Solution :

2.05555..........  =  2 + 0.05555.......... ----(1)

In 0.05555.........., there is one digit between the decimal point and the recurring digits, that is 0.

2.05555..........  =  2 + (0.5555..........) ÷ 10 ----(1)

There are only recurring digits after the decimal point in  

0.5555..........

There is only one digit in the recurring pattern, that is 5. 

Subtract 1 from 10, the result is 9.

Write a fraction with numerator 5 and denominator 9 for 0.5555.......... in (1). 

2.05555..........  =  2 + (5/9) ÷ 10

=  2 + 5/90

=  2 + 1/18

=  36/18 + 1/18

=  (36 + 1)/18

=  37/18

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