OPERATIONS WITH RECURRING DECIMALS

Problem 1 :

Work out the following addition. Give your answer as a simplified fraction.

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

0.555... = 5/9

Let x = 0.212121... --> (3)

Since two digit is repeating, we will multiply it by 100.

100x = 100 × 0.212121...

100x = 21.2121... --> (4)

From (4) - (3)

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

99x = 21

x = 21/99

0.2121... = 7/33

Take LCM of 9 and 33. 

Problem 2 :

Work out the following 

Give your answer as a simplified fraction.

Solution:

Let x = 0.272727... --> (1)

Since two digit is repeating, we will multiply it by 100.

100x = 100 × 0.272727...

100x = 27.2727... --> (2)

From (2) - (1)

100x - x = 27.2727... - 0.2727...

99x = 27

x = 27/99

0.2727... = 3/11

Let x = 0.646464... --> (3)

Since two digit is repeating, we will multiply it by 100.

100x = 100 × 0.646464...

100x = 64.6464... --> (4)

From (4) - (3)

100x - x = 64.6464... - 0.6464...

99x = 64

x = 64/99

0.6464... = 64/99

Let x = 0.5333... --> (5)

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

10x = 10 × 0.5333...

10x = 5.333... --> (6)

From (6) - (5)

10x - x = 5.333... - 0.5333...

9x = 4.8

x = 4.8/9

Problem 3 :

Arrange in order from smallest to largest.

Solution:

Let x = 0.17878...--->(1)

Since two digit is repeating, we will multiply it by 100.

100x = 100 × 0.17878...

100x = 17.878... --> (2)

From (2) - (1)

100x - x = 17.878... - 0.17878...

99x = 17.7

Arrange from smallest to largest,

Problem 4 :

Solution :

Let x = 0.545454... --> (1)

Since two digit is repeating, we will multiply it by 100.

100x = 100 × 0.545454...

100x = 54.5454... --> (2)

From (2) - (1)

100x - x = 54.5454... - 0.5454...

99x = 54

x = 54/99

0.5454... = 18/33

Let x = 0.555... --> (3)

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

10x = 10 × 0.555...

10x = 5.55... --> (4)

From (4) - (3)

10x - x = 5.55... - 0.555...

9x = 5

x = 5/9

0.555... = 5/9

Problem 5 :

Solution:

Let x = 0.3939...--->(1)

Since two digit is repeating, we will multiply it by 100.

100x = 100 × 0.3939...

100x = 39.3939... --> (2)

From (2) - (1)

100x - x = 39.3939... - 0.3939...

99x = 39

x = 39/99

x = 13/33

Let x = 0.6363...--->(3)

Since two digit is repeating, we will multiply it by 100.

100x = 100 × 0.6363...

100x = 63.6363... --> (4)

From (4) - (3)

100x - x = 63.6363... - 0.6363...

99x = 63

x = 63/99

x = 21/33

Problem 6 :

Solution:

Let x = 0.077... --> (1)

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

10x = 10 × 0.077...

10x = 0.77... --> (2)

From (2) - (1)

10x - x = 0.77... - 0.077...

9x = 0.7

Let x = 0.185185....--->(3)

Since three digit is repeating, we will multiply it by 1000.

1000x = 1000 × 0.185185...

1000x = 185.185... --> (4)

From (4) - (3)

1000x - x = 185.185... - 0.185185...

999x = 185