CONVERTING RECURRING DECIMALS TO FRACTIONS

About "Converting Recurring Decimals to Fractions"

Converting Recurring Decimals to Fractions :

In this section, we are going to learn, how to convert recurring decimals into fractions. 

How to convert recurring decimals into fractions ?

Step 1 : 

Let  x = Given decimal number 

For example,

If the given decimal number is 2.0343434.........

then, let x = 2.0343434...........

Step 2 : 

Identify the repeated pattern

For example,

In 2.0343434..........., the repeated pattern is 34

(Because 34 is being repeated)

Step 3 :

Identify the first repeated pattern and second repeated pattern as as explained in the example given below. 

Step 4 :

Count the number of digits between the decimal point and first repeated pattern as given in the picture below. 

Step 5 :

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

Step 7 :

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

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

Converting recurring decimals to fractions - 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

So, multiply the given decimal by 100. Then, we have

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

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

So, multiply the given decimal by 100000. Then, we have

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

Subtracting (1) from (2), we get

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

x  =  3200053/99900

Hence, 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)

So, multiply the given decimal by 10000. Then, we have

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

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

So, multiply the given decimal by 100000. Then, we have

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

Subtracting (1) from (2), we get

(2) - (1) --------> 90000x  =  1109

x  =  1109/90000

Hence, 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)

So, multiply the given decimal by 100. Then, we have

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

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

So, multiply the given decimal by 10000. Then, we have

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

Subtracting (1) from (2), we get

(2) - (1) --------> 9900x  =  20129

x  =  9900/20129

Hence, 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 

So, multiply the given decimal by 1. Then, we have

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

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

So, multiply the given decimal by 100. Then, we have

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

Subtracting (1) from (2), we get

(2) - (1) --------> 99x  =  25

x  =  25/99

Hence, 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)

So, multiply the given decimal by 1. Then, we have

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)

So, multiply the given decimal by 10. Then, we have

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

Subtracting (1) from (2), we get

(2) - (1) --------> 9x  =  30

x  =  30/9  =  10/3

Hence, 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 

So, multiply the given decimal by 1000. Then, we have

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

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

So, multiply the given decimal by 1000000. Then, we have

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

Subtracting (1) from (2), we get

(2) - (1) --------> 999000x  =  1022538

x  =  1022539/999000 

Hence, 1.023562562562.......... =  1022539/999000

After having gone through the stuff and examples, we hope that the students would have understood, "Converting recurring decimals to fractions"

Apart from the stuff and examples, if you want to know more about, "Converting recurring decimals to fractions", please click here.

Apart from the stuff "Converting recurring decimals to fractions", if you need any other stuff in math, please use our google custom search here.