How to convert recurring decimal to fraction?

We need to deal with recurring or repeating decimals in school, in our competitive exams and even later.  Today we’ll discuss a shortcut trick to convert recurring decimals to fractions. However, to understand it’s effectiveness, we need to first understand the method taught in schools.

Just follow the steps below carefully. Say you need to find the value of 0.44444……

Step 1: Let x be the value of the repeating decimal which you are converting to fraction
x = 0.44444444… 

We know the repeating digit is 4

Step 2: Multiply x by a power of 10, such that the resultant has same repeating digits on the right side of decimal. In this case if we multiply 10 both side, we get –

10x = 4.44444444…….

Step 3: Subtract output of step 2 from step 1

10x – x = 4.44444444……. -  0.44444444…….

9x = 4

x = 4/9

Hence we have converted decimal into fraction.

Another example,

X = 1.3454545

Multiply x by 10, we get 10x = 13.45454545……

Multiplying x by 1000 we get, 1000 = 1345.454545….

Subtracting 10x from 1000x we get,

1000x – 10x = 1345.454545…. – 13.45454545……

990x = 1332

X = 1332/990 =1 (342/990)

So again we have converted recurring decimal to fraction.

 

Shortcut trick to convert recurring decimals into fractions

The same thing which we have done above can be done just by looking at the original number. The shortcut trick is explained below step by step with example for easy understanding -

Say if the number is 2.13636363…

First ignore the digit to the left of decimal. We’ll deal with it later.

For the remaining part i.e. right hand side part of the decimal, we need to distinguish repeating and non-repeating (fixed) digits. To get the denominator of the fraction, we need to use 9s for each digit of the repeating part. The fixed digit(s) is offset by putting a trailing zero (990). To get the numerator, we need to subtract the fixed part from the number (without repetition)

2.13636363…… conversion in fraction:

First ignore 2 initially.

Right side of the decimal have 1 fixed digit and 2 digits in the repeating portion.

Hence the denominator will be 990

For the numerator, we subtract the fixed part from the number (without repetition). Hence Numerator = 136 -1 = 135

2 which was ignored earlier will simply be added to the left of the resultant fraction

Hence the answer becomes, 2 (135/990)

Example 2

Converting the easy one using the above shortcut:  0.44444444…. into fraction

0.44444444…. =

Since there is only one repeating digit we take denominator as 9 and for numerator we take the repeating digit itself as there is no fixed digit, hence nothing to subtract. Hence it is equal to 4/9

Example 3

1.3454545….

Numerator = the number (without repeatation) – the fixed digit =  345 – 3

Denominator = two 9s followed by 0 as there are 2 recurring digits and 1 fixed digit.

1 is written as it is separately before the fraction

=1 (345-3/990)  = 1 (342/990)

By posting a comment below, let me know if you found this method useful. You can even ask your questions here.

4 Comments
  1. jason
  2. Manju R
    • Vineet Patawari
      • Ankur

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