 # How to convert a recurring decimal into fraction

Firstly, we need to define what a recurring decimal and a fraction are.

A fraction is an expression in which a number is being divided by another number. The number at the top is called the numerator and the one at the bottom is called the denominator, for example ⅔.

A recurring decimal is a decimal representation of a number whose digits are repeating its values at regular intervals infinitely.

For example:

0.444444 is a recurring decimal.

Now there are also types of recurring decimal, thus making the conversion to fraction slightly different for each type.

Type 1:

0.5555555….

Step 1: Let x = recurring decimal in expanded form.

x = 0.5555555….

Step 2: Let the number of recurring digits = n.

Number of recurring digits in our case equals 1.

Step 3: Multiply recurring decimal by 10^n.

10^n = 10 in our case because n=1.

Hence, 10x= 10 × 0.55555555…

= 5.55555….

Step 4: Subtract  Step 1 from  Step 3 to eliminate the recurring part and solve for x.

10x - x = 5.5555 - 0.5555

9x = 5

x = 5/9

Type 2:

0.1515151515….

Step 1: Let x = recurring decimal in expanded form.

x = 0.15151515….

Step 2: Let the number of recurring digits = n.

Number of recurring digits in our case equals 2.

Step 3: Multiply recurring decimal by 10^n.

10^n = 100 in our case because n=2.

Hence, 100x= 100 × 0.15151515…

= 15.151515….

Step 4: Subtract  Step 1 from  Step 3 to eliminate the recurring part and solve for x.

100x - x = 15.151515 - 0.151515

98x = 15

x = 15/99

As the number of recurring numbers would increase the value of n would increase simultaneously.