**Write a repeating decimal as a fraction :**

In this topic we are going to learn how to change the repeating decimals as fraction.

Steps involved :

**Step 1 :**

Take the given number as "x"

**Step 2 :**

Count the number of digits which are repeating.

**Step 3 :**

Multiply the given number by 10, 100, 1000,..... on both sides according to the number of digits repeating.

**Step 4 :**

Subtract both equations.

**Example 1 :**

Convert the following repeating decimal as fraction

0.474747...........

**Solution :**

Let x = 0.474747............... ------(1)

Here repeating number of digits = 2. So we have to multiply 100 on both sides.

100 x = 47.4747.............. ------(2)

(2) - (1)

aaaaaaaaaaaaaaaaa100 x = 47.4747.........aaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaaaaax = 0.4747...............aaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaaaa(-)aaaaaa(-)aaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaa------------------------aaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaaa99 x = 47.0000aaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaa-----------------------aaaaaaaaaaaaaaaaa

x = 47/99

Hence the repeating decimal form of 0.474747........... is 47/99.

**Example 2 :**

Convert the following repeating decimal as fraction

0.57777..........

**Solution :**

Let x = 0.5777777............... ------(1)

Here repeating number of digits = 1. So we have to multiply 10 on both sides.

10 x = 0.57777777.............. ------(2)

(2) - (1)

aaaaaaaaaaaaaaaa10 x = 5.77777777.........aaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaax = 0.57777777...............aaaaaaaaaaaa

aaaaaaaaaaaaaaaaaa(-)aaaaaa(-)aaaaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaa------------------------aaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaaa9 x = 5.20000aaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaa-----------------------aaaaaaaaaaaaaaaaa

x = 5.2/9

Multiply the numerator and denominator by 10

x = 52/90 ==> x = 26/45

Hence the repeating decimal form of 0.57777.......... is 26/45.

**Example 3 :**

Convert the following repeating decimal as fraction

0.245245245.........

**Solution :**

Let x = 0.245245245............... ------(1)

Here repeating number of digits = 3. So we have to multiply 1000 on both sides.

1000 x = 245.245245.............. ------(2)

(2) - (1)

aaaaaaaaaaa1000 x = 245.245245.............. aaaaaaaaaaaaaa

aaaaaaaaaaaaaaaa x = aa0.245245.............. aaaaaaaaaaaaaa

aaaaaaaaaaaaaaaa(-)aaaaaa(-)aaaaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaa------------------------aaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaa999 x = 245aaaaaaaaaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaa----------------------aaaaaaaaaaaaaaaaaaaaaaaa

x = 245/999

Hence the repeating decimal form of 0.245245245......... is 245/999

After having gone through the stuff given above, we hope that the students would have understood "Write a repeating decimal as a fraction".

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