**Converting Recurring Decimals to Fractions :**

In this section, we are going to learn, 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.

**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"

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