Recurring Decimals. How to covert a recurring decimal into a fraction?


The method for converting a recurring decimal into a fraction is a little bit unusual. To do this you need to make some equations which eliminate the decimals. This is much easier to see if we carry out some examples.

Note: In the examples that follow the r means recurring, instead of a decimal point above the number which is the normal way to express a recurring decimal)

Example 1

Convert 0.7r to a fraction (The r means that the 7 is recurring)

First make this recurring decimal equal to x:

x = 0.777... (Equation 1)

Next multiply both sides of the equation by 10:

10x = 7.777... (Equation 2)

The two equations can now be taken away so that the decimal parts are eliminated:

9x = 7

Next solve this equation which will give the values of x as a fraction instead of a decimal:

x = 7/9

Example 2

Convert 0.38r to a fraction (The r means that the 8 is recurring)

First make this recurring decimal equal to x:

x = 0.3888... (Equation 1)

Next multiply both sides of the equation by 10:

10x = 3.888... (Equation 2)

If you take the two equations away the decimal part won’t be eliminated so you need to multiply by 10 again:

100x = 38.888... (Equation 3)

Now if you take the 3rd and 2nd equations away the decimal part of the numbers will be eliminated.

The two equations can now be taken away so that the decimal parts are eliminated:

90x = 35

Next solve this equation which will give the values of x as a fraction instead of a decimal:

x = 35/90 which simplifies to 7/18

Let’s take a look at 1 last example.

Example 3

Change 2.36r1r into a fraction

The r means the 6 is repeating and similarly for the 1. So you have:

x = 2.3616161...

Convert 0.38r to a fraction (The r means that the 8 is recurring)

Next multiply both sides of the equation by 10:

10x = 23.616161... (Equation 2)

If you take the two equations away the decimal part won’t be eliminated so you need to multiply by 10 again:

100x = 236.16161... (Equation 3)

And still the decimal parts cannot be eliminated by subtracting any of the above equations so you need to multiply by 10 again:

1000x = 2361.616161...(Equation 4)

Now the decimals match in equations 2 and 4. So these can be subtracted to give:

990x = 2338

Next solve this equation which will give the values of x as a fraction instead of a decimal:

x = 2338/990 which simplifies to 1169/495

To summarise, all you need to remember is to make x equal the recurring decimal and keep multiplying this by 10 until you can subtract any of the 2 equations so that the decimals will eliminate.

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Patty Kenyon profile image

Patty Kenyon 4 years ago from Ledyard, Connecticut

Interesting and Useful!!! I love math and I think this article will help people that do not know how to do this. Good Job!!

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