How to convert recurring decimals to fractions :

1. Make sure that there are only recurring digits after the decimal point.

2. If there is only one digit in the recurring pattern, subtract 1 from 10, the result is 9.

3. Take the recurring digit in the numerator and 9 in the denominator.

4. Simplify the fraction, as needed.

Note :

If there are two digits in the recurring pattern, subtract 1 from 100, for three digits, subtract 1 from 1000 and so on. Continue the rest of the process as explained above.

Covert the 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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