Published · Updated

To work out a recurring decimal as a fraction, write the repeating block over the same number of nines. 0.3 recurring is 3/9, which simplifies to 1/3. 0.123 recurring is 123/999, which simplifies to 41/333. The shortcut holds whenever the repeat starts immediately after the decimal point, and needs one extra step when it does not.

This page is the fast route: the pattern itself, worked examples in both cases, and a reference list you can recall under time pressure. If you want the algebra behind the pattern, the method of setting x equal to the decimal, multiplying and subtracting is set out on our guide to converting recurring decimals to fractions.

Step 1: Count the digits in the repeating block

Write the decimal out with the repeat visible and mark the block that repeats. For 0.567567…, the repeating block is 567, so there are three recurring digits. Check that the repeat starts immediately after the decimal point, because that is what the shortcut depends on.

Step 2: Put the block over that many nines

The repeating block becomes the numerator, and the denominator is one nine for every digit in the block. We counted three digits and the block is 567, so the denominator is three nines. That gives 0.567 recurring = 567/999.

Step 3: Simplify and check the fraction

Cancel any common factors. 567 and 999 both divide by 27, so 567/999 becomes 21/37. Then check the result by dividing the numerator by the denominator, because 21 divided by 37 should return you to 0.567567… and it does.

Step 4: Shift first if the repeat starts late

If any digits sit between the decimal point and the repeat, the nines rule cannot be applied to the number as it stands. Split the decimal into the part that does not repeat and the part that does, convert the repeating part with the nines rule, divide it by 10 once for every digit you skipped, then add the two parts back together.

Worked examples: recurring decimals as fractions

Q) Express the following recurring decimals as fractions.

a. 0.123 recurring

Start by counting the digits in the block that repeats. The block is 123, so there are three of them.

Three digits means three nines on the denominator, and the block itself is the numerator:

0.123 recurring = 123/999

Now look for common factors. Both 123 and 999 divide by 3.

0.123 recurring = 41/333

Check it by dividing 41 by 333. That returns 0.123123…, which is the number we began with.

b. 0.5123 recurring

Here the repeat does not begin until after the 5, so the nines rule cannot be applied to the number as it stands. Split it into the part that does not repeat and the part that does:

The first part is straightforward: 0.5 = 5/10 = 1/2.

For the second part, 0.0123123… is 0.123123… moved one place to the right, so it is a tenth of the fraction from part a:

0.0123 recurring = (1/10) x (41/333) = 41/3330

Add the two parts over a common denominator of 3330:

1/2 + 41/3330 = 1665/3330 + 41/3330 = 1706/3330

Both 1706 and 3330 divide by 2, which gives the final answer:

0.5123 recurring = 853/1665

Divide 853 by 1665 to check, and you get 0.5123123…, as expected.

The slip to watch for in part b

The usual mistake in a question like this is to reuse the wrong fraction for the repeating part, writing 123/99 instead of 123/999, and so 41/33 instead of 41/333. Carried through, that produces the working below, which is wrong:

The steps look tidy, but the answer cannot be right: 103 divided by 165 is 0.6242…, while the number in the question starts 0.5123. One quick division catches it. Two nines belong to a two digit block and three nines to a three digit block, and the block here is 123.

When the nines rule applies, and when it does not

The rule applies to a purely recurring decimal, meaning one where the repeat starts immediately after the decimal point. 0.7 recurring, 0.45 recurring and 0.153 recurring all qualify, and each converts in a single step.

It does not apply directly when digits sit in front of the repeat, as in 0.5123 recurring or 0.0366 recurring. Those need the shift in step four above, or the algebraic method on our guide to converting recurring decimals to fractions, which handles both cases with identical working.

It also does not apply to a decimal that stops. A terminating 0.153 is 153/1000, not 153/999, and the two are easy to confuse because they are written almost identically. Read the notation before you pick a method.

How recurring decimals are written

A dot above a digit means that digit repeats forever, so 0.3 with a dot above the 3 is 0.3333…

When a block of digits repeats, dots sit above the first and last digit of the block, so 0.123 with dots above the 1 and the 3 is 0.123123123…

Some exam boards and textbooks draw a bar over the block instead, as the diagrams on this page do. Both notations mean exactly the same thing.

Quick reference: common recurring decimals

These are worth recognising on sight, because they come round again and again:

  • 0.1 recurring = 1/9
  • 0.2 recurring = 2/9
  • 0.3 recurring = 1/3
  • 0.4 recurring = 4/9
  • 0.5 recurring = 5/9
  • 0.6 recurring = 2/3
  • 0.7 recurring = 7/9
  • 0.8 recurring = 8/9
  • 0.16 recurring, with only the 6 repeating = 1/6
  • 0.83 recurring, with only the 3 repeating = 5/6
  • 0.142857 recurring = 1/7
  • 0.27 recurring = 3/11
  • 0.45 recurring = 5/11
  • 0.153 recurring = 17/111
  • 0.201 recurring = 67/333

Follow the pattern to its end and 0.9 recurring comes out as 9/9, which is 1. That is a genuine equality rather than a rounding artefact, and it is a favourite of interviewers who want to see whether you trust the method or your instinct.

Recurring decimals with a whole number in front

Leave the whole number where it is, convert only the decimal part, then put the two back together as a mixed number.

Take 8.3 recurring, with the 3 repeating. The decimal part is 0.3 recurring, which is 1/3, so the answer is 8 and 1/3, or 25/3 written as an improper fraction.

Take 94.83 recurring, with only the 3 repeating. The decimal part is 0.83 recurring, which is 5/6, so the answer is 94 and 5/6, or 569/6.

If you need to move between the two forms, our guide to expressing mixed fractions as improper fractions sets out the steps.

Practice recurring decimal questions

Conversions like these are quick marks once the pattern is automatic, and expensive ones when it is not. Put the method under a clock with our numerical reasoning tests, or shore up the arithmetic underneath it with our basic numeracy tests.