# WRITE A REPEATING DECIMAL AS A FRACTION

## About "Write a repeating decimal as a fraction"

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

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