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. 

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