**Recurring decimals to fractions :**

"Converting recurring decimals to fractions" is a much required stuff for the students who study high school math.

Here, we are going to see, how to convert recurring decimals to fractions step by step.

**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 :**

Write the following recurring decimal as a 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)

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

X = 3200053 / 99900

Hence, 32.03256256256.......... = 3200053 / 99900

**Problem 2 :**

Write the following recurring decimal as a 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)

(2) - (1) --------> 9900X = 20129

X = 9900 / 20129

Hence, 2.03323232.......... = 9900 / 20129

**Problem 3 :**

Write the following recurring decimal as a 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)

(2) - (1) --------> 99X = 25

X = 25 / 99

Hence, 0.252525.......... = 25 / 99

**Problem 4 :**

Write the following recurring decimal as a 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)

(2) - (1) --------> 9X = 30

X = 30 / 9 = 10 / 3

Hence, 3.3333.............. = 10 / 9

After having gone through the stuff given above, we hope that the students would have understood "Recurring decimals to fractions".

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