# How to Convert Repeating Decimals to Fractions

Whenever a fraction is represented as a decimal, it can either be a terminating decimal, such as

1/2 = 0.5 and

7/50 = 0.14,

or a repeating decimal, such as

17/60 = 0.283 and

2/7 = 0.285714.

The bar lies over the numerical string that repeats, called the *repetend*. Repeating decimals are frequently encountered in practical math applications when fractions are converted to percents or decimals. One of the main reasons for converting fractions to decimals is for ease of adding and subtracting. However, when repeating decimals are truncated or rounded off, some accuracy may be lost.

You can easily convert a repeating decimal back to its original fraction form following the mathematical steps below, or you can use the convenient calculator on the right. To use the calculator, enter the non-repeating part and the repetend. You can leave blank any fields that do not apply. For example, if converting 0.3 to 1/3, leave the first field blank. If converting 0.5 to 1/2, leave the last field blank.

**(Step 1)** First separate the non-repeating part and the repetend after the decimal point. For example, if we wish to convert

0.7110506

to a fraction, then we separate 711 and 0506.

**(Step 2)** For the non-repeating part, write that number over a power of 10 that has as many zeroes as there are digits in the non-repeating part (including all zeroes in the non-repeating part!) For example, since 711 has three digits, we write the fraction 711/1000.

**(Step 3)** For the repetend, write that number over as many nines as there are digits in the repetend (again, including all zeroes in the repetend.) For example, since 0506 has four digits, we write 0506/9999. Now divide this fraction by the power of 10 used in Step 2. For example, since we used 1000 in Step 2, we write (0506/9999)/1000 = 0506/9999000 = 506/9999000.

**(Step 4)** Add the two fractions you obtained in Steps 2 and 3. (Remember the rules for adding fractions; you must give them a common denominator) For example, we add

711/1000 + 506/9999000

= 7109289/9999000 + 506/9999000

= 7109795/9999000

**(Step 5)** Now reduce the fraction you obtained in Step 4. For example, 55 divides evenly into both 7109749 and 9999000. Thus, if we divide the numerator and denominator by 55, we get a final answer of

129269/181800.

This is the fraction equivalent of 0.7110506.

### Why does this method work?

You can use algebra to prove that all repeating decimals are rational numbers. For example, let us set x = 0.7110506. Then the following algebraic steps show that x can be rewritten as a fraction:x = 0.7110506

x = 711/1000 + 0.0000506

x - 711/1000 = 0.0000506

**1000(x - 711/1000) = 0.0506**

10000(1000(x - 711/1000)) = 506.0506

10000(1000(x - 711/1000)) = 506 +

**0.0506**

10000(1000(x - 711/1000)) = 506 +

**1000(x - 711/1000)**

10000000x - 7110000 = 506 + 1000x - 711

9999000x = 7109795

x = 7109795/9999000 = 129269/181800

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