CONVERTING RECURRING DECIMALS TO FRACTIONS

The following steps will much useful to convert recurring decimals to fractions.

Step 1 : 

Assume that the given decimal number is equal to the variable x. 

For example, if the given decimal number is  2.7343434....,

then, we have

x  =  2.7343434..............

Step 2 : 

Identify the repeated pattern

In 2.7343434..........., the repeated pattern is 34

(Because 34 is being repeated)

Step 3 :

Identify the first repeated pattern and second repeated pattern as shown below.  

Step 4 :

Count the number of digits between the decimal point and first repeated pattern as shown below.

Step 5 :

Because there is 1 digit between the decimal point and the first repeated pattern, we have to multiply the given decimal by 10 as shown below. 

(If there are two digits -----------> multiply by 100,

three digits -----------> multiply by 1000  and  so on )

Note :

In (1), we have only repeated patterns after the decimal.

Step 6 : 

Count the number of digits between the decimal point and second repeated pattern as shown below. 

Step 7 :

Because there are 3 digits between the decimal point and the second repeated pattern, we have to multiply the given decimal by 1000 as shown below. 

Note :

In (2), we have only repeated patterns after the decimal.

Step 8 :

Now, we have to subtract the result of step 5 from step 7 as shown below. 

Now we got the fraction which is equal to the given decimal.

That is, 

Practice Problems

Problem 1 :

Covert the given repeating decimal into fraction

32.03256256256..........

Solution : 

Let x  =  32.03256256256.............

Here, the repeated pattern is 256

No. of digits between the 1st repeated pattern and decimal  =  2

Then, multiply the given decimal by 100.

100x  =  3203.256256256...............-----(1) 

No. of digits between the 2nd repeated pattern and decimal  =  5

Then, multiply the given decimal by 100000.

100000x  =  3203256.256256256...............-----(2)

Subtracting (1) from (2), we get

(2) - (1) --------> 99900x  =  3200053

x  =  3200053/99900

So,

32.03256256256.......... =  3200053/99900

Problem 2 :

Covert the given repeating decimal into fraction

0.01232222........

Solution : 

Let x  =  0.01232222.............

Here, the repeated pattern is 2

No. of digits between the 1st repeated pattern and decimal  =  4

(Here, the first repeated pattern starts after four digits of the decimal)

Then, multiply the given decimal by 10000. 

10000X  =  123.2222...............-----(1) 

No. of digits between the 2nd repeated pattern and decimal  =  5

Then, multiply the given decimal by 100000.

100000x  =  1232.2222...............-----(2)

Subtracting (1) from (2), we get

(2) - (1) -----> 90000x  =  1109

x  =  1109/90000

So,

0.01232222........... =  1109/90000

Problem 3 :

Covert the given repeating decimal into fraction

2.03323232..........

Solution : 

Let x  =  2.03323232.............

Here, the repeated pattern is 32

No. of digits between the 1st repeated pattern and decimal  =  2

(Here, the first repeated pattern starts after two digits of the decimal)

Then, multiply the given decimal by 100. 

100x  =  203.323232...............-----(1) 

No. of digits between the 2nd repeated pattern and decimal  =  4

Then, multiply the given decimal by 10000. 

10000x  =  20332.323232...............-----(2)

Subtracting (1) from (2), we get

(2) - (1) -----> 9900x  =  20129

x  =  9900/20129

So,

2.03323232.......... =  9900/20129

Problem 4 :

Covert the given repeating decimal into fraction

0.252525..........

Solution : 

Let x  =  0.252525.............

Here, the repeated pattern is 25

No. of digits between the 1st repeated pattern and decimal  =  0 

Then, multiply the given decimal by 1. 

x  =  0.252525...............-----(1) 

No. of digits between the 2nd repeated pattern and decimal  =  2

Then, multiply the given decimal by 100.

100x  =  25.252525...............-----(2)

Subtracting (1) from (2), we get

(2) - (1) --------> 99x  =  25

x  =  25/99

So,

0.252525.......... =  25/99

Problem 5 :

Covert the given repeating decimal into fraction

3.3333..........

Solution : 

Let x  =  3.3333.............

Here, the repeated pattern is 3

No. of digits between the 1st repeated pattern and decimal  =  0

(Here, the first repeated pattern is "3" which comes right after the decimal point)

Then, multiply the given decimal by 1. 

x  =  3.3333...............-----(1) 

No. of digits between the 2nd repeated pattern and decimal  =  1

(Here, the second repeated pattern is "3" which comes one digit  after the decimal point)

Then, multiply the given decimal by 10. 

10x  =  33.3333...............-----(2)

Subtracting (1) from (2), we get

(2) - (1) -----> 9x  =  30

x  =  30/9  =  10/3

So,

3.3333.............. =  10/3.

Problem 6 :

Covert the given repeating decimal into fraction

1.023562562562..........

Solution : 

Let x  =  1.023562562562.............

Here, the repeated pattern is 562

No. of digits between the 1st repeated pattern and decimal  =  3 

Then, multiply the given decimal by 1000.

1000x  =  1023.562562562...............-----(1) 

No. of digits between the 2nd repeated pattern and decimal  =  6

Then, multiply the given decimal by 1000000. 

1000000x  =  1023562.562562562...............-----(2)

Subtracting (1) from (2), we get

(2) - (1) -----> 999000x  =  1022538

x  =  1022539/999000 

So,

1.023562562562.......... =  1022539/999000

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