# Repeating decimal to fraction

We have already learnt how to convert decimal into fraction.
In this post we will understand advance topic of converting repeating decimal into fraction.

All the concept explained are as per the syllabus of Saxon Math Curriculum generally preferred in USA and Canada schools.

## What is Repeating Decimal?

In Repeating decimal, one or more digits after decimal points repeat infinite times.

For Example

### Representation of Repeating Decimal

In Math, the repeating digit is represented by line over that digit.

For Example
In below decimal, digit 4 is continuously repeated
0.44444 . . . .

This decimal is represented as :

Note the red line above repeating digit 4.

## Steps to convert repeating decimal into fraction

The process of converting decimal into fraction is easy and straight forward.

We will understand the process with the help of example.

Example 01
Let the recurring decimal is:
0.3333 . . . .

Step 01
Express the decimal into the form of equation
Let x = 0.3333 – – – -eq (1)

Step 02
Note the digit which is being repeated
Here only 1 digit is repeated (i.e 3)

Step 03
Take one set of repeated digit to front of decimal

To do this we have to multiply the equation with 10

⟹ x = 0.3333…
⟹ 10x = 3.3333 – – – – -eq(2)

Step 04
Subtract eq (2) with eq(1)

Simplify the expression further:
⟹ 9x = 3
⟹ x = 3/9
⟹ x = 1/3

Hence the fraction of decimal 0.3333… is 1/3

I hope the concept is clear.
Let us move on to solve some other examples.

Example 02
Convert 1.454545…. into fraction

Solution

Step 01
Express the decimal into form of expression
x = 1.454545…… eq(1)

Step 02
Note the repeated digit
Here two digits are repeated (i.e 45)

Step 03
Take set of repeated digit to front of decimal

To do this we have to multiply the equation with 100

⟹ x = 1.454545
⟹ 100 x = 145.4545 —eq(2)

Step 04
Subtract eq(2) with eq(1)

Simplify the expression further

⟹ 99x = 144
⟹ x = 144/99
Dividing numerator and denominator by 9
⟹ x = 16/11

Hence 16/11 is the required fraction

Example 03
2.0787878 . . .

Step 01
Express the decimal into the form of expression
x = 2.0787878 . . . eq(1)

Step 02
Find the repeated digits
Here two digits are repeated (i.e. 78)

Step 03
Take one set of repeated digit to front of decimal

To do this we have to multiply equation with 1000
1000x = 2078.7878 —- eq(2)

Step 04
Subtract eq(2) with eq(1)

PROBLEM: You can see from above image that decimal part are not getting subtracted fully.

In order to solve the problem, multiply eq(1) with 10 so that the zero in decimal can come forward

Now subtract eq (2) with (3)

Simplify the equation further
⟹ 90x = 2058
⟹ x = 2058/90

Divide Numerator and Denominator by 6
⟹ x = 343/15

Hence 343/15 is the required fraction

Example 04
12.007777. . . . . .

Step 01
Express the decimal in the form of Math Expression
x = 12.007777. . . . . . — eq(1)

Step 02
Find the repeated digits
Here only one digit is being repeated (i.e 7)

Step 03
Take one set of the repeated digit in the front of decimal

To do this we have to multiply the eq(1) with 1000
⟹ x = 12.007777. . . . .
⟹ 1000x = 12007.7777. . . . eq(2)

Step 04
Subtract eq(2) with eq(1)

You can see that because of zeros the subtraction is getting complicated

You have to multiply eq(1) with 100 so that all zero come before decimals
⟹ x = 12.007777. . . . .
⟹ 100x = 1200.7777. . . . eq(3)

Now subtract eq(2) with eq(3)

Further solving the equation:
⟹ 900 x = 10807
⟹ x = 10807/900

Hence 10807/900 is the required raction

## Questions on Converting Recurring Decimal to Fraction

(01) Convert the decimal into fraction
0.242424…….

(a) 1/66
(b) 4/33
(c) 8/33
(d) 8/66

(i) Math Expression
x = 0.242424 . . . . . eq(1)

(ii) Count number of repeating digits
There are two repeating digits (i.e. 24)

(iii) Take one set on repeating digits before decimal
Multiply eq(1) with 100
100x = 24.2424 . . . . . eq(2)

(iv) Any digits between decimal and repeating numbers?
NO

(v) Subtract eq(2) with eq(1)

Simplify the expression
⟹ 99x = 24
⟹ x = 24/99
Divide numerator and denominator by 3
⟹ x = 8/33

Hence 8/33 is the required fraction
Option (c) is the right answer

(02) Find the fraction for decimal, 1.122222. . . .

(a) 101/90
(b) 102/25
(c) 97/90
(d) 103/90

(i) Write Math Expression
⟹ x = 1.12222 . . . .

(ii) Count the digits which are repeated
Only one digit is being repeated (i.e digit 2)

(iii) Take one set of repeating digit before decimal
For that you have to multiply eq(1) with 100

⟹ x = 1.12222 . . . .
⟹ 100x = 112.2222 . . . . -eq(2)

(iv) Any digits between decimal and repeating number

Multiply eq(1) with 10 to create equation (3)
⟹ 10x = 11.22222 . . . -eq(3)

(v) Subtract eq(2) with eq(1)

Simplifying the above equation
⟹ 90x = 101
⟹ x = 101/90

Hence 101/90 is the required fraction
Option (a) is the right answer

(03) Find the fraction of given recurring decimal
⟹ 0.94777777 . . .

(a) 513/900
(b) 713/900
(c) 753/900
(d) 853/900

(i) Write Math Expression
⟹ x = 0.94777777 . . . . eq(1)

(ii) Count the digits which are repeated
Only one digit is being repeated (i.e digit 7)

(iii) Take one set of repeating digit before decimal
For that you have to multiply eq(1) with 1000

⟹ x = 0.94777777 . . . .
⟹ 1000x = 947.77777. . . -eq(2)

(iv) Any digits between decimal and repeating number
Yes, There are two digits between decimal and repeating numbers

Multiply the equation (1) with 100
⟹ x = 0.94777777 . .
⟹ 100x = 94.777777 . . . . eq(3)

(v) Subtract eq(2) with eq(3)

Simplifying the equation, we get:
⟹ 900x = 853
⟹ x = 853/900

Hence 853/900 is the required fraction
Option (d) is the right answer

(04) Convert the decimal into fraction
0.73737373…….

(a) 79/99
(b) 45/99
(c) 73/99
(d) 81/66

(a) Math Expression
x = 0.737373 . . . . . eq(1)

(b) Count number of repeating digits
There are two repeating digits (i.e. 73)

(c) Take one set on repeating digits before decimal
Multiply eq(1) with 100
100x = 73.737373 . . . . . eq(2)

(d) Any digits between decimal and repeating numbers?
NO

(e) Subtract eq(2) with eq(1)

Simplifying the equation
⟹ 99x = 73
⟹ x = 73/99

Option (c) is the right answer

(05) Convert the recurring decimal into fraction
⟹ 7.05639639639…..

(a) 704929/99900
(b) 704934/99900
(c) 704931/99900
(d) 704930/99900

(i) Write Math Expression
⟹ x = 7.05639639639….. eq(1)

(ii) Count the digits which are repeated
Three digits are being repeated (i.e digit 639)

(iii) Take one set of repeating digit before decimal
For that you have to multiply eq(1) with 100000

⟹ x = 7.05639639639 . . . .
⟹ 100000x = 705639.639639. . . -eq(2)

(iv) Any digits between decimal and repeating number
Yes, There are two digits between decimal and repeating numbers

Multiply equation (1) with 100
⟹ x = 7.05639639639 . . . .
⟹ 100x = 705.639639639 . . . . eq(3)

(v) Subtract eq(2) with eq(3)

⟹ 99900x = 704934
⟹ x = 704934/99900

Hence 704934/99900 is the required fraction
option (b) is th eright answer

(06) Convert the decimal into fraction
⟹ 1.49494949……..

(a) 148/99
(b) 150/99
(c) 149/99
(d) 161/99

(i) Write Math Expression
⟹ x = 1.49494949….. eq(1)

(ii) Count the digits which are repeated
Two digits are being repeated (i.e digit 49)

(iii) Take one set of repeating digit before decimal
For that you have to multiply eq(1) with 100

⟹ x =1.494949 . . . .
⟹ 100x =149.494949. . . -eq(2)

(iv) Any digits between decimal and repeating numbers?
NO

(v) Subtract eq(2) with eq(1)

⟹ 99x = 148
⟹ x = 148/99

Option (a) is the solution

(07) Convert the repeated decimal into fraction
⟹ 6.6666666……..

(a) 6/5
(b) 20/3
(c) 40/3
(d) 12/5

(i) Write Math Expression
⟹ x = 6.66666….. eq(1)

(ii) Count the digits which are repeated
One digits is repeated (i.e digit 6)

(iii) Take one set of repeating digit before decimal
For that you have to multiply eq(1) with 10

⟹ x = 6.66666…..
⟹ 10x =66.66666……. -eq(2)

(iv) Any digits between decimal and repeating numbers?
NO

(v) Subtract eq(2) with eq(1)

Simplifying the equation further
⟹ 9x = 60
⟹ x = 60/9

Divide numerator and denominator by 3
⟹ x = 20/3

Hence 20/3 is the solution
Option (b) is the right answer

(08) Given below is recurring decimal. Convert the decimal into fraction
⟹ 0.12366666 . . . .

(a) 371/3000
(b) 370/3000
(c) 372/3000
(d) 373/3000

(i) Write Math Expression
⟹ x = 0.12366666….. eq(1)

(ii) Count the digits which are repeated
One digits is getting repeated (i.e digit 6)

(iii) Take one set of repeating digit before decimal
For that you have to multiply eq(1) with 10000

⟹ x = 0.12366666…..
⟹ 10000x = 1236.66666. . . -eq(2)

(iv) Any digits between decimal and repeating number
Yes, There are three digits between decimal and repeating numbers

Multiply 1000 to eq(1)
⟹ x = 0.12366666…..
⟹1000x = 123.66666….. eq(3)

(v) Subtract eq(2) with eq(3)

Simplify the equation
⟹ 9000x = 1113
⟹ x = 1113/9000

Divide numerator and denominator by 3
⟹ x = 371/3000

Option (a) is the right answer