Decimal to Fraction and Repeating Decimal Converter

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repeating decimal to fraction

Repeating decimal to fraction

Type the repeating decimal with its repeating part in parentheses, such as 0.1(6), or with an ellipsis, such as 0.1666..., and the tool returns the fraction in lowest terms with the full working in both methods. This page shows how the fraction comes out of each kind of repeating decimal and where students slip most.

How the fraction comes out of each kind of repeating decimal

In a purely repeating decimal the repeating block starts right after the decimal point. The fraction has the block as its numerator and, as its denominator, one 9 for each digit of the block. So 0.(45) is 45/99, which divided by 9 top and bottom gives 5/11, and 0.(142857) is 142857/999999, which is exactly 1/7 because 999999 equals 7 times 142857. The possible denominators are always divisors of 9, 99, 999 and so on, which is why 3, 9, 11, 27 and 37 show up so often in these answers.
In a mixed repeating decimal some digits sit between the decimal point and the repeating block. In 0.41(6) the non-repeating part is 41 and the repeating block is 6. The numerator is the number made of both parts minus the number made of the non-repeating part alone: 416 − 41 = 375. The denominator has a 9 for the block and a 0 for each digit of the non-repeating part: 900. The fraction 375/900 reduces by 75 to 5/12. With algebra, x = 0.41666..., 1000x = 416.666... and 100x = 41.666..., and the subtraction 1000x − 100x = 375 removes the infinite tail, leaving 900x = 375.
When there is an integer part it goes into the number that forms the numerator: in 2.4(54) you use 2454 − 24 = 2430 over 990, which reduces by 90 to 27/11, or 2 5/11 as a mixed number. The same result comes from adding 2 to the fraction of 0.4(54), which is 5/11: 2 + 5/11 = 27/11. So there is nothing extra to memorize: apply the rule to the whole number with the decimal point removed. A negative sign is handled separately: −0.(3) is −1/3. Repeating decimals with a block of 9s, such as 0.(9) or 0.3(9), give fractions that match a terminating decimal: 9/9 = 1 and 36/90 = 2/5.
The most common mistake is mixing up what repeats. 0.16 (terminating) is 16/100 = 4/25; 0.1(6) is 15/90 = 1/6; and 0.(16) is 16/99. They are three different numbers whose first decimal places look almost the same. That is why the tool asks for the repeating part in parentheses and, when you type only an ellipsis, looks for the repetition itself and lets you correct how many final digits form the block. An ellipsis after only a few digits, as in 0.5..., proves no repetition at all.

Checked examples

Pure repeating, one digit

Input: 0,(3)
Expected output: 1/3

3/9 divided by 3.

Pure repeating, two digits

Input: 0,(45)
Expected output: 5/11

45/99 divided by 9.

Six-digit block

Input: 0,(142857)
Expected output: 1/7

142857/999999, gcd 142857.

Mixed repeating

Input: 0,41(6)
Expected output: 5/12

(416 − 41)/900 = 375/900.

With an integer part

Input: 2,4(54)
Expected output: 27/11

2430/990, or 2 5/11 as a mixed number.

Block of nines

Input: 0,3(9)
Expected output: 2/5

36/90: equal to 0.4.

Full tool FAQ

How do you turn a decimal into a fraction?

Count the decimal places, write the digits without the point as the numerator and use 1 followed by that many zeros as the denominator. Then reduce by the gcd. Example: 0.375 has 3 places, becomes 375/1000 and, dividing by 125, 3/8. With an integer part, 2.5 = 25/10 = 5/2 = 2 1/2.

What is the fraction of a repeating decimal?

How do you convert a purely repeating decimal?

What about a repeating decimal with non-repeating digits first?

Is 0.999... really equal to 1?

How can I tell if a fraction terminates or repeats?

How do I type a repeating decimal?

Does it handle negatives, percentages and mixed numbers?

Why do π and √2 have no fraction?

Is there a size limit?

Frequently asked questions

What separates a pure from a mixed repeating decimal?

In a pure one the block starts right after the decimal point (0.454545...). In a mixed one a non-repeating part comes first (0.41666...). The difference only changes the denominator: nines alone in the pure case, nines followed by zeros in the mixed case.

How do I find the repeating block?

Look for the shortest block that repeats through the end of the digits written. In 0.1666... the block is 6 and the 1 is the non-repeating part. In 0.121212... the block is 12. With at least two repetitions the tool detects it by itself; with fewer, choose it by hand.

Why is the final denominator sometimes smaller than the rule gives?

Because the rule produces a fraction that can still be reduced. 0.(6) gives 6/9, but the gcd of 6 and 9 is 3 and the fraction in lowest terms is 2/3. The tool always shows the raw fraction and then the reduction.

Can I use the fraction to add repeating decimals?

Yes, and it is the exact way. 0.(3) + 0.(6) becomes 1/3 + 2/3 = 1, while adding truncated decimals gives 0.999.... Convert each repeating decimal to a fraction, add the fractions and, if you like, go back to a decimal.

Next steps

Decimal	Fraction
0,25	1/4
0,(3)	1/3
1,2(5)	113/90
-0,8	-4/5

Decimal

Fraction

0,25

1/4

0,(3)

1/3

1,2(5)

113/90

-0,8

-4/5

Repeating decimal to fraction

How the fraction comes out of each kind of repeating decimal

In a purely repeating decimal the repeating block starts right after the decimal point. The fraction has the block as its numerator and, as its denominator, one 9 for each digit of the block. So 0.(45) is 45/99, which divided by 9 top and bottom gives 5/11, and 0.(142857) is 142857/999999, which is exactly 1/7 because 999999 equals 7 times 142857. The possible denominators are always divisors of 9, 99, 999 and so on, which is why 3, 9, 11, 27 and 37 show up so often in these answers.

In a mixed repeating decimal some digits sit between the decimal point and the repeating block. In 0.41(6) the non-repeating part is 41 and the repeating block is 6. The numerator is the number made of both parts minus the number made of the non-repeating part alone: 416 − 41 = 375. The denominator has a 9 for the block and a 0 for each digit of the non-repeating part: 900. The fraction 375/900 reduces by 75 to 5/12. With algebra, x = 0.41666..., 1000x = 416.666... and 100x = 41.666..., and the subtraction 1000x − 100x = 375 removes the infinite tail, leaving 900x = 375.

When there is an integer part it goes into the number that forms the numerator: in 2.4(54) you use 2454 − 24 = 2430 over 990, which reduces by 90 to 27/11, or 2 5/11 as a mixed number. The same result comes from adding 2 to the fraction of 0.4(54), which is 5/11: 2 + 5/11 = 27/11. So there is nothing extra to memorize: apply the rule to the whole number with the decimal point removed. A negative sign is handled separately: −0.(3) is −1/3. Repeating decimals with a block of 9s, such as 0.(9) or 0.3(9), give fractions that match a terminating decimal: 9/9 = 1 and 36/90 = 2/5.

The most common mistake is mixing up what repeats. 0.16 (terminating) is 16/100 = 4/25; 0.1(6) is 15/90 = 1/6; and 0.(16) is 16/99. They are three different numbers whose first decimal places look almost the same. That is why the tool asks for the repeating part in parentheses and, when you type only an ellipsis, looks for the repetition itself and lets you correct how many final digits form the block. An ellipsis after only a few digits, as in 0.5..., proves no repetition at all.

Checked examples

Pure repeating, one digit

Input: 0,(3)
Expected output: 1/3

3/9 divided by 3.

Pure repeating, two digits

Input: 0,(45)
Expected output: 5/11

45/99 divided by 9.

Six-digit block

Input: 0,(142857)
Expected output: 1/7

142857/999999, gcd 142857.

Mixed repeating

Input: 0,41(6)
Expected output: 5/12

(416 − 41)/900 = 375/900.

With an integer part

Input: 2,4(54)
Expected output: 27/11

2430/990, or 2 5/11 as a mixed number.

Block of nines

Input: 0,3(9)
Expected output: 2/5

36/90: equal to 0.4.

Full tool FAQ

How do you turn a decimal into a fraction?

What is the fraction of a repeating decimal?

How do you convert a purely repeating decimal?

What about a repeating decimal with non-repeating digits first?

Is 0.999... really equal to 1?

How can I tell if a fraction terminates or repeats?

How do I type a repeating decimal?

Does it handle negatives, percentages and mixed numbers?

Why do π and √2 have no fraction?

Is there a size limit?

Frequently asked questions