Repeating Decimal to Fraction

Introduction
Fractions  can be written in the form p/q where p and q are rational numbers.
p and q can be integers also and q ≠ 0.
Decimals can be converted into fractions.
0.25 = 1/4  and 0.5= 1/2
But when the decimal has repeating digits, it becomes necessary to convert them into fractions by using variable x
Repeating decimals to fractions:-
Repeating decimals have a digit or a block of digits repeating itself again and again.
0.333….., 0.416666…….., 0.57145714………… are repeating decimals.
If the digits after the decimals are repeated , it is called pure recurring decimals.
Mixed recurring decimals have  mixed numbers repeating such as 0.242424……

Repeating Decimals to Fractions – Conversion

The repeating decimals can be converted into fractions.

# Convert 0.7777……. into fraction

Solution:-

Let  x = 0.7777

step1 multiply x by 10

Then 10x = 7.777

step 2 subtract x from 10 x

Then 10x – x = 7.777 – 0.777

               9x = 7.000

Step 3 Dividing both sides by 9  we get  x = 7/9

Answer  0.7777…… = 7/9

 

#Convert 0.2424… to fractions.

Let  x = 0.2424

Step1 multiply  x by 100

Then 100x = 24.2424

Step2 subtract x from 100x

100x – x = 24.2424 – 0.2424

        99x = 24

Step3 Divide both sides by 99

x = 24/99 which on simplification gives  8/33

Hence 0.2424…. = 8/33

 

#Convert 0.27777,,,,, into fractions

Let x = 0.27777

step1 multiply x by 10

then 10x = 2.7777

step 2 multiply x by 100

and 100x = 27.777

step 3 subtract 10x from 100x

100x  – 10 x  =  27.777 – 2.777

 90x  = 25

Step 4 divide  both sides by 90

 x = `25/90 = 5/18`

Answer 0.27777 = `5/18`

 

# Convert 0.35656…. into fractions

Let x = 0.35656

Then 10x = 3.5656

1000x = 356.5656

1000x – 10x = 356.5656 – 3.5656

           990x = 353

                 x = `353/990`

 

# Convert 0.3333 t fractions

Let x = 0.333

10x   = 3.33

10x – x= 3.33 – 0.33

   9x    =  3

     x   = `3/9 = 1/3`

Hence  o,3333 = `1/3`

# Convert 0.416666… to fractions

Let x = 0.41666

Then 100x = 41.666 and 1000x = 416.666

1000x – 100x = 416.666 – 41.666

            900x = 375

                 x  =`375/900 = 5/12`

Answer 0.416666….. = `5/12`

 

#Convert 0.888888…. into fractions.

Let x = 0.88

Then 10 x =  8.88

10x – x = 8.88 – 0.88

       9x = 8

 x = 8/9

# Convert 0.4646…. as fraction

Let x = 0.4646

100x = 46.4646

100x – x = 46.4646 – 0.4646

       99x =  46

           x = `46/99`

Answer 0.4646 =`46/99`

# Convert 0.717171 into fraction

Let x = 0.717171

100x= 71.7171

100x – x = 71.7171  – 0.7171

99x = 71

x =`71/99`

Repeating Decimal to Fraction-practice Problems

#Convert 0.5555…. into fraction (Ans:`5/9` )

#Convert 0.232323…into fraction  (Ans:`23/99` )

#Convert 0.757575..into fraction   (Ans:`25/33` )

#Convert 0.522522..into fraction   (Answer:`58/111` )

#Convert 0.057057..into fraction   (Answer:`19/333` )

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