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,
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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