Repeating Decimal to Fraction Conversion Calculator

You can use this repeating decimal to fraction conversion calculator to revert a repeating decimal to its original fraction form.

Simply input the repeating part of the decimal (the repetend) and its non-repeating part (where applicable). For instance, if you are converting 0.6 to 2/3, leave the non-repeating field blank.

Repeating Decimal to Fraction Calculator
 

                   

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How to Convert Repeating Decimals to Fractions

When a fraction is represented as a decimal, it can take the form of a terminating decimal; for example:

3/5 = 0.6 and 1/8 = 0.125,

or a repeating decimal; for example,

19/70 = 0.2714285 and 1/6 = 0.16

The bar depicted above is presented above the repeating element of the numerical string. This is known as the repetend. You may wish to convert a fraction to a decimal to make adding and subtracting quantities more straightforward. However, it is common to encounter a repeating decimal in practical math when you convert fractions to percentages or decimals, and this reduces the accuracy of the calculation.

You can revert a decimal to its original fraction by following the steps described below. However, if you want to make life a little easier, use our decimal to fraction conversion calculator instead.

Step 1: Separate the non-repeating part of the decimal from the repeating part. For instance, let's say you wanted to convert the following to a fraction:

0.3210708

The bar is positioned above the non-repeating part of the decimal. As such, you should separate 321 from 0708.

Step 2: Record the non-repeating part of the decimal over a power of 10 that incorporates as many zeros as there are numbers in the non-repeating part of the decimal (including any zeros). For instance, as 321 consists of three numbers, we represent the fraction as 321/1000.

Step 3: Record the repetend over as many nines as there are numbers in that repetend (again, including any zeros). For instance, as 0708 consists of four numbers, it is represented as 0708/9999. Next, divide this fraction by the power of 10 applied in Step 2. For instance, since we applied 1000 in step two, we calculate the following: (0708/9999)/1000 = 0708/9999000 = 708/9999000.

Step 4: Sum the two fractions generated in Step 2 and 3 respectively (as per the rules for adding fractions, make sure you give them a common denominator). For instance:

321/1000 + 708/9999000

= 3209679/9999000 + 708/9999000

= 3210387/9999000

Step 5: Reduce the fraction generated in Step 4. For instance, both 3210387 and 9999000 can be divided by 3. As such, we divide the numerator and denominator by 3 to produce the following:

1070129/3333000.

This is the fraction equivalent of 0.3210708.

Why Does This Method Work?

Algebra can be used to demonstrate that all repeating decimals are rational numbers. For instance, let's say we have x = 0.3210708. The following algebraic steps can be applied to demonstrate that x can be represented as a fraction:

x = 0.3210708

x = 321/1000 + 0.0000708

x − 321/1000 = 0.0000708

1000 (x − 321/1000) = 0.0708

10000 (1000 (x − 321/1000)) = 708.0708

10000 (1000 (x − 321/1000)) = 708 + 0.0708

10000 (1000 (x − 321/1000)) = 708 + 1000 (x − 321/1000)

10000000x − 3210000 = 708 + 1000x - 321

9999000x = 3210387

x = 3210387/9999000 = 1070129/3333000

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