HOW TO CONVERT REPEATING DECIMALS TO FRACTIONS

Show that these are rational :

Problem 1 :

0∙444444 ⋯⋯

Solution :

Given, 0∙444444……

x  =  0∙444444…… -----(1)

Here 4 is repeating (1 digit)

Multiply by 10 on both sides.

10x  =  4.44444……-----(2)

(2) – (1)

10x  =  4.44444……

-x  =  - 0∙444444……

--------------------------

9x  =  4

x  =  4/9

So, 0∙44444……  =  4/9

Since we can convert the repeating decimal into fraction, it is rational number.

Problem 2 :

0.212121……

Solution :

Given, 0.212121……

x =  0.212121……  ------(1)

here , 2 digits are repeating so, we have to multiply by 100 on both sides.

100x  =  21.2121…… ------(2)

(2) – (1)

100  = 21.2121……

-x  =  - 0.212121……

-------------------------

99x  =  21

x  =  21/99 

x  =  7/33

So, 0.212121……  =  7/33

Since we can convert the repeating decimal into fraction, it is rational number.

Problem 3 :

0.7777777.............

Solution :

Given, 0.7777777..........

x  =   0.777777....... ------(1)

Here, 7 is  repeating (1 digit)                

Multiply by 10 on both sides

10x  =  7.777777.......  ------(2)

(2) – (1)

10x  =  7.777777.........

-x  =  - 0.7777777..........

-------------------------

9x  =  7

x  = 7/9

So, 0.7777777......  =  7/9

Since we can convert the repeating decimal into fraction, it is rational number.

Problem 4 :

0.363 636…… Are rational.

Solution :

Given, 0.363 636…… Are rational.

x  =  0.363 636……   ------(1)  

Here , 2 digits are repeating. So, we have to multiply by 100 on both sides.

100x  =  36. 3636……    ------(2)

(2) – (1)

100x  =  36. 3636……

 – x  =  - 0.363 636……..

----------------------------

99x  =  36

x   =  36/99

x  =  4/11

So,  0.363 636……. =  4/11

Since we can convert the repeating decimal into fraction, it is rational number.

Problem 5 :

0.325 325 325 .…..

Solution :

Given, 0.325 325 325 .…..

x  =  0.325 325 325……  ------(1)

Here, 3 digits are repeating. So we have to multiply by 1000 on both sides.

1000x  =  325.325325……  ------(2)

(2) – (1)

1000x  =  325.325325……

 – x  =  - 0.325 325 325……

---------------------------------

999x  =  325

x  =  325/999

So, 0.325 325 325 .….. =  325/999

Since we can convert the repeating decimal into fraction, it is rational number. 

Problem 6 :

2.360 360 360 ............

Solution :

Given, 2.360 360 360 ............

x  = 2.360 360 360 ............  ------(1)

Here, 3 digits are repeating. So we have to multiply by 1000 on both sides.

1000x  = 2360.360360......  ------(2)

(2) – (1)

1000x  =  2360.360360......

 – x  =  - 2.360 360 360 ...........

---------------------------------

999x  =  2358

x  =  2358/999

x = 262/111

Problem 7 :

Convert the recurring decimal 2.1363636....... as fraction and write down the answer as mixed number.

Solution :

Given, 2.1363636.......

x  = 2.1363636....... ------(1)

Here, 2 digits are repeating. So we have to multiply by 100 on both sides.

100x  = 213.63636.......   ------(2)

(2) – (1)

100x  = 213.63636....... 

 – x  = -2.1363636....... 

---------------------------------

99x  =  -211.5

x  =  211.5/99

Multiplying both numerator and denominator by 10, we get

= 2115/990

= 705/333

= 235/111

= 2  13/111

So, the answer is mixed form is 2  13/111.

Problem 8 :

convert the recurring decimal 2.0666....... as fraction and write down the answer as mixed number.

Solution :

Given, 2.0666.......

x  = 2.0666....... ------(1)

Here, 1 digit is repeating. So we have to multiply by 10 on both sides.

10x  = 20.666.......   ------(2)

(2) – (1)

10x  = 20.666....... 

 – x  = -2.0666.......

---------------------------------

9x  =  18.6

x  =  18.6/9

= 6.2/3

Multiplying both numerator and denominator by 10, we get

= 62/3

Converting as mixed number, we get

= 20  2/3

So, the answer is mixed form is 20  2/3.

Problem 9 :

Prove that recurring decimal 0.1717..... = 17/99

Solution :

Given, 0.1717.....

x  = 0.1717..... ------(1)

Here, two digits are repeating. So we have to multiply by 100 on both sides.

100x  = 17.1717..........   ------(2)

(2) – (1)

100x  = 17.1717..............

 – x  = - 0.1717.....

---------------------------------

99x  =  17

x  =  17/99

Problem 10 :

Work out 0.5454...... x 0.5555.............

Solution :

To multiply these two recurring decimals, we have to convert both recurring decimals as fraction and multiply.

Given, 0.5454......

x  = 0.5454...... ------(1)

Here, two digits are repeating. So we have to multiply by 100 on both sides.

100x  = 54.5454........------(2)

(2) – (1)

100x  = 54.5454........

-x  = -0.5454......

---------------------------------

99x  =  54

x  =  54/99

 0.5454...... = 54/99

Dividing both numerator and denominator by 9, 

= 6/11

x  = 0.5555.............------(1)

Here, two digits are repeating. So we have to multiply by 10 on both sides.

10x  = 5.5555.......------(2)

(2) – (1)

10x  = 5.555........

-x  = -0.555.............

---------------------------------

9x  =  5

x  =  5/9

0.5555............ = 5/9

 0.5454...... x 0.5555............ = (6/11) x (5/9)

= 30/99

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