OPERATIONS WITH RECURRING DECIMALS

What is recurring decimals ?

A decimal fraction in which a figure or group of figures is repeated indefinitely.

For example,

0.7777777.......(One digit is repeating)

2.090909..........(Two digits are repeating)

1.73333.....(One digit is repeating)

To covert the repeating decimals or recurring decimal into fraction, we follow the steps given below.

Step 1 :

Let x be the given decimal keep it as (1) and count the number of digits repeating.

Step 2 :

Multiply both sides by 10ⁿ.

Here n is number of digits repeating. For example,

0.73333......

Since one digit is repeating, we have to multiply it by 10 and keep it as (2).

Step 3 :

Subtract (2) and (1), we will get the value of x and that required fraction of the repeating decimal.

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

OPERATIONS WITH RECURRING DECIMALS

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