HOW TO CONVERT RECURRING DECIMAL INTO FRACTION

Convert the following recurring decimals to fractions :

Example 1 :

0.555…

Solution :     

Let x = 0.555… --->(1)

Since one digit is repeating, we will multiply it by 10.

10x = 10 × 0.555….

10x = 5.555… --->(2)

From (2) - (1)

10x - x = 5.555… - 0.555…

9x = 5

x = 5/9

So, 0.555… = 5/9

Example 2 : 

0.777….

Solution :

Let x = 0.777… --->(1)

Since one digit is repeating, we will multiply it by 10.

10x = 10 × 0.777….

10x = 7.777… --- > (2)

From (2) - (1)

10x - x = 7.777… - 0.777…

9x = 7

x = 7/9

So, 0.777… = 7/9

Example 3 :

 0.888….

Solution :

Let x = 0.888… --->(1)

Since one digit is repeating, we will multiply by 10.

10x = 10 × 0.888….

10x = 8.888… --->(2)

From (2) - (1)

10x - x = 8.888… - 0.888…

9x = 8

`x = 8/9

So, 0.888… = 8/9

Convert the following recurring decimals to fractions:

Example 4 :

1.262626….

Solution :

Let x = 1.262626… --->(1)

Since two digits are repeating, we multiply by 100.

100x = 100 × 1.262626….

100x = 126.2626… --->(2)

From (2) - (1)

100x - x = 126.2626… - 1.262626…

99x = 125

`x = 125/99

So, 1.262626… = 125/99

Example 5 :

1.0202….

Solution :

Let x = 1.0202… --->(1)

Since two digits are repeating, we multiply by 100.

100x = 100 × 1.0202….

100x = 102.02… --->(2)

From (2) - (1)

100x - x = 102.02… - 1.0202…

99x = 101

`x = 101/99

So, 1.0202… = 101/99

Example 6 :

0.5333….

Solution :

Let x = 0.5333… ---> (1)

Since one digit is repeating, we will multiply by 10.

10x = 10 × 0.5333….

10x = 5.333… ---> (2)

From (2) - (1)

10x - x = 5.333… - 0.5333…

9x = 4.8

`x = 4.8/9

x = (4.8 × 10) / (9 × 10)

x = 48/90

x = 8/15

So, 0.5333… = 8/15.

Example 7 :

1.7333….

Solution :

Let x = 1.7333… --- > (1)

Since one digit is repeating, we will multiply by 10.

10x = 10 × 1.7333….

10x = 17.333… --- > (2)

From (2) - (1)

10x - x = 17.333… - 1.7333…

9x = 15.6

`x = 15.6/9

x = (15.6 × 10) / (9 × 10)

x = 156/90

Doing simplification, we get

x = 26/15

So, 1.7333… = 26/15

Example 8 :

0.9444….

Solution :

Let x = 0.9444… --->(1)

Since one digit is repeating, we will multiply by 10.

10x = 10 × 0.9444….

10x = 9.444… --->(2)

From (2) - (1)

10x - x = 9.444… - 0.9444…

9x = 8.5

`x = 8.5/9

x = (8.5 × 10) / (9 × 10)

x = 85/90

Doing simplification, we get

= 17/18

So, 0.9444… = 17/18

Example 9 :

2.0555…

Solution :

Let x = 2.0555… ---> (1)

Since one digit is repeating, we multiply it by 10.

10x = 10 × 2.0555….

10x = 20.555… --->(2)

From (2) - (1)

10x - x = 20.555… - 2.0555…

9x = 18.5

`x = 18.5/9

x = (18.5 × 10) / (9 × 10)

x = 185/90

Doing simplification, we get

x = 37/18

So, 2.0555… = 37/18

Example 10 :

0.5555............+ 0.212121............

Solution :

Let x = 0.55555............. ----(1)

Since we have one digit which is repeating, then we have to multiply by 10 on both sides.

10x = 5.55555............. ----(2)

(2) - (1)

10x - x = 5.55555...... - 0.5555..........

9x = 5

x = 5/9

Let x = 0.2121................. ----(1)

Since we have two digits which is repeating, then we have to multiply by 100 on both sides.

100x = 21.2121.................. ----(2)

(2) - (1)

100x - x = 21.2121.................. - 0.2121.................

99x = 21

x = 21/99

x = 7/33

0.5555............+ 0.212121............ = 5/9 + 7/33

Least common multiple of 9 and 33 is 99

= 5/9 x (11/11) + 7/33 x (3/3)

= 55/99 + 21/99

= (55 + 21) / 99

= 76/99

Example 10 :

8.6797979...........

Solution :

Let x = 8.6797979...........----(1)

Since we have two digits are repeating, then we have to multiply by 10 on both sides.

100x = 867.97979............. ----(2)

(2) - (1)

100x - x = 867.97979............ - 8.6797979.........

99x = 859.3

x = 859.3/99

x = 8593/990