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Repeating Decimal to Fraction Calculator

This repeating decimal to fraction calculator turns a decimal such as 0.1666… or 1.272727… into an exact fraction in lowest terms, and shows the algebra behind it. Put the digits that repeat in square brackets, so 0.1[6] gives 1/6 and 1.[27] gives 14/11. It uses whole-number arithmetic, not floating point, so even a long repeating block comes out exact. It is worked out on your device and nothing is uploaded or stored.

How to convert a repeating decimal to a fraction

Decimal number - type the number. Use a dot or a comma for the decimal point.
Put the digits that repeat in square brackets straight after the point and any non-repeating digits: 0.1[6] means 0.1666…, with only the 6 repeating.
Simplified fraction - read the answer, shown stacked and again on one line. A minus sign in front of the number carries to the fraction.
Copy fraction - copies it as plain text, such as 1/6, ready to paste.
Steps - read the working underneath to see how the fraction was found.

There is no Convert button: the result changes on every keystroke. Round brackets also work, so 0.1(6) is read the same as 0.1[6]. Input can be up to 50 characters.

How to type a repeating decimal with brackets

Brackets mark exactly which digits repeat, which is why one decimal can have different answers: 0.[27] repeats 27, while 0.2[7] repeats only the 7. Only one pair of brackets is allowed, and it must come after the decimal point.

Type this	Meaning	Fraction
0.[3]	0.333…	1/3
0.1[6]	0.1666…	1/6
0.[142857]	0.142857142857…	1/7
1.[27]	1.272727…	14/11
-0.[3]	-0.333…	-1/3
0.75	0.75, no repeat	3/4
5	a whole number	5/1

The calculator rejects a thousands separator such as 1,000.5, a second decimal point, and brackets with no decimal point before them, and tells you what format it expects. Overline, dot and “…” notation are not read, so rewrite those with brackets.

How the algebra works: 0.1666… as a fraction

The trick is to make two copies of the number that have the same endless tail, then subtract so the tail disappears. This is the working shown under Steps for 0.1[6]:

Example: 0.1[6]
x = 0.1666…
100x = 16.666… (move past the 1 and one full repeat)
10x = 1.666… (move past the 1 only)
Subtract: 90x = 15
Divide both sides by 90: x = 15/90
Divide top and bottom by 15: x = 1/6

In general, if n digits come after the point before the repeat and r digits repeat, subtract 10ⁿx from 10^n+rx. The number you divide by is 10^n+r − 10ⁿ, which is why a single repeating digit gives a denominator built from 9s and a two-digit block gives one built from 99s.

Repeating decimals as fractions: common values

Decimal	Type	Fraction
0.111…	0.[1]	1/9
0.666…	0.[6]	2/3
0.0909…	0.[09]	1/11
0.2727…	0.[27]	3/11
0.0833…	0.08[3]	1/12
0.8333…	0.8[3]	5/6
0.12333…	0.12[3]	37/300
0.999…	0.[9]	1/1

Why 0.9 repeating equals 1

Type 0.[9] and the calculator returns 1/1. The steps show why: 10x = 9.999… and x = 0.999…, so 9x = 9 and x = 1. There is no number between 0.999… and 1, so they are the same number written two ways.

Terminating decimals, negatives and mixed numbers

A decimal with no brackets is treated as terminating: 0.75 is written over 100 as 75/100, then reduced to 3/4. A whole number is returned over 1. A leading minus or plus sign is accepted. The answer is always an improper fraction in lowest terms; to turn 14/11 into a mixed number, divide: 11 goes into 14 once with 3 left over, so 1 3/11.

Frequently asked questions

How do you convert a repeating decimal to a fraction?

Let x be the decimal, multiply x by a power of 10 so the repeating digits line up with a second copy, subtract the two lines so the endless tail cancels, then divide to solve for x and reduce. For 0.1666... that gives 100x - 10x = 16.666... - 1.666..., so 90x = 15 and x = 1/6. The calculator does this as you type and shows each line under Steps.

How to convert a repeating decimal to a fraction without a calculator?

Write x = the decimal. If n digits come after the point before the repeat and r digits repeat, subtract 10^n x from 10^(n+r) x. The repeating tail cancels and you get a whole number equal to (10^(n+r) - 10^n) x. Divide both sides by that number, then divide the top and bottom by their greatest common divisor. The Steps list shows the same working for your own number so you can check yours.

How do you convert a recurring decimal to a fraction?

Recurring and repeating mean the same thing here. Put the recurring digits in square brackets after the decimal point, for example 0.[3] for 0.333... or 1.[27] for 1.272727..., and the calculator returns the fraction in lowest terms: 1/3 and 14/11.

What is 0.1666 repeating as a fraction?

1/6. Type 0.1[6], because only the 6 repeats: 100x = 16.666..., 10x = 1.666..., subtracting gives 90x = 15, so x = 15/90, which reduces to 1/6. If you meant the whole block 1666 to repeat, type 0.[1666] instead and you get 1666/9999.

What is 0.333 repeating as a fraction?

1/3. Type 0.[3]. Subtracting x from 10x gives 9x = 3, so x = 3/9 = 1/3. Likewise 0.[6] is 2/3 and 0.[1] is 1/9.

What fraction is 0.9 repeating?

0.9 repeating is exactly 1, so the calculator shows 1/1. Subtracting x from 10x gives 9x = 9, so x = 1. The two numbers are not just very close: there is no number between them, which means they are the same number written two ways.

What is 0.27 repeating as a fraction?

It depends on what repeats. If both digits repeat (0.272727...), type 0.[27] to get 3/11. If only the 7 repeats (0.2777...), type 0.2[7] to get 5/18. The brackets tell the calculator which digits repeat.

How do you turn a repeating decimal into a mixed number?

The calculator always shows an improper fraction in lowest terms and has no mixed-number output. To get a mixed number, divide the top by the bottom: the quotient is the whole part and the remainder is the new top. For 1.[27] the answer is 14/11, and 14 divided by 11 is 1 remainder 3, so 1 3/11.