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Fraction and Decimal Converter

Turn a decimal, including a repeating one, into an exact fraction, or turn a fraction into its exact decimal with the repeating block marked.

Level: Grades 7–8

  • Common Core standard 7.NS.A.2.dConvert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
  • Common Core standard 8.NS.A.1Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
Printable worksheet

Fraction and Decimal Converter

Inputs
Convert

Put a repeating block in parentheses: 0.(3) means 0.333…, 0.1(6) means 0.1666…

Result

Fraction (lowest terms)

3/8

0.375 = 3/8

Mixed number
3/8
Decimal value
0.375
Type of decimal
Terminating

See the step-by-step solution ↓

Step-by-step solution

  1. There are 3 digits after the decimal point, so write the digits over 10 to the power 3 = 1,000

    0.375=3751,000

  2. Find the biggest number that divides both the top and bottom evenly, called the GCD. Divide, keep what's left over, and repeat until nothing is left

    GCD(375, 1,000)

  3. 1,000 ÷ 375 leaves 250 left over

    1,000=2×375+250

  4. 375 ÷ 250 leaves 125 left over

    375=1×250+125

  5. Nothing is left over, so the last number you divided by, 125, is the GCD

    250=2×125+0

  6. Divide the top and bottom by the GCD, 125, to get the simplest form

    375÷1251,000÷125=38

  7. That's the decimal as a fraction in lowest terms

    0.375=38

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Turn any decimal into an exact fraction, or any fraction into its exact decimal. A fraction of whole numbers always becomes a decimal that either stops (3/8 = 0.375) or repeats. A repeating decimal settles into a block of digits that goes on forever (5/12 = 0.41666…). Going the other way, every decimal that stops or repeats can be written as an exact fraction. This converter works in both directions without rounding along the way.

Decimals are read as text, digit by digit. So 0.3 becomes exactly 3/10, not the long, slightly-off value a computer stores in binary. For repeating decimals, put the repeating block in parentheses: 0.(3) is 0.333…, and 0.1(6) is 0.1666…. In the other direction, the calculator does long division. It watches the remainders and marks the block that repeats.

Where it's used

Why this math matters outside the classroom:

  • Tape measures. A carpenter converts 0.375 inches to 3/8 inch to find the mark on a tape measure.
  • Recipes. A cook reads 0.333… cups as exactly 1/3 cup, because a repeating 3 is equal to one third.
  • Drill bits. A machinist converts a 5/16 inch drill bit to 0.3125 inches to compare it with other bit sizes.
  • Batting averages. A player with 7 hits in 22 at-bats has 7/22 = 0.31818…, with the 18 repeating, shown as .318.
  • Money. Shoppers see that 3/4 of a dollar is 0.75, or 75 cents, by dividing 3 by 4.

Grade level and standards: Grades 7–8; Common Core State Standards for Mathematics 7.NS.A.2.d, 8.NS.A.1. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Choose the direction: decimal to fraction, or fraction to decimal.
  2. For a decimal, type it as written. Wrap any repeating digits in parentheses, such as 2.(45) or 0.08(3).
  3. For a fraction, enter a whole-number numerator (top number) and a denominator (bottom number) that isn't zero.
  4. Read the exact answer, then follow the steps to see how the calculator got there.

What the inputs mean

Decimal
Up to 14 significant digits (digits not counting leading zeros). Only the digits inside the parentheses repeat: 0.1(6) is 0.16666…, while 0.(16) is 0.161616…. A minus sign in front is fine. Don't type '…' or extra digits to show repetition. Use the parentheses.
Numerator
Any whole number, positive, negative or zero.
Denominator
Any whole number other than zero, up to 1,000,000 in size. A negative denominator just makes the whole value negative.

Formula and method

Terminating: 0.d₁d₂…dₖ = (digits) / 10^k
Repeating: x = I.N(R), with k digits in N and r digits in R
10^(k+r) · x − 10^k · x = INR − IN
x = (INR − IN) / (10^(k+r) − 10^k)
a/b terminates ⇔ the reduced b has no prime factors other than 2 and 5

x is the decimal you're converting. I is its whole-number part, N is the digits after the point that don't repeat and R is the repeating block. k counts the digits in N, and r counts the digits in R. In the last line, a/b is a fraction with top a and bottom b.

A terminating decimal (one that stops) is its digits over a power of ten: 0.375 is 375/1000. Divide top and bottom by their greatest common divisor (the biggest number that divides both), 125, to get 3/8.

A repeating decimal is handled by shifting it twice. Multiplying by 10^(k+r) moves the non-repeating digits and one full repeating block in front of the point. Multiplying by 10^k moves only the non-repeating digits. Both results have the same endless tail. So subtracting them leaves a whole number, and x is that whole number over 10^(k+r) − 10^k. Here INR means all those digits read as one whole number, and IN means the same without the repeating block.

For fraction to decimal, each step of long division multiplies the remainder by 10 and divides again. There are only b − 1 possible remainders other than zero. So either a remainder hits 0 and the decimal stops, or one comes back. From that point on, the same digits repeat.

Worked example

A measurement reads 0.1666… with the 6 repeating forever. What fraction is it?

  1. Write it as 0.1(6) and call it x. There is k = 1 non-repeating digit (1) and r = 1 repeating digit (6).
  2. Multiply by 10^2 = 100: 100x = 16.(6).
  3. Multiply by 10^1 = 10: 10x = 1.(6).
  4. Subtract: 90x = 16 − 1 = 15, so x = 15/90.
  5. The GCD of 15 and 90 is 15, so 15/90 reduces to 1/6.

Result: 0.1(6) = 1/6. In the other direction, entering 5/12 gives 0.41(6). After the third decimal place, the remainder 8 comes back, so the 6 repeats forever.

Understanding the result

The fraction is always exact and in lowest terms. For values larger than 1, you'll also see it as a mixed number, like 2 1/2. The rounded decimal follows your precision setting.

In fraction-to-decimal mode, the exact decimal puts the repeating block in parentheses, and you'll see how long that block is. A block longer than 60 digits is shortened with '…', but its full length is still given.

Assumptions and limitations

  • Parentheses mark the repeating block. Everything else in the decimal is taken exactly as typed.
  • A repeating block of zeros, such as 0.5(0), is treated as a decimal that stops.
  • Fractions are reduced before converting, which doesn't change the decimal.
  • Decimals are limited to 14 significant digits, and denominators to 1,000,000. That way every number in the working stays exact.
  • Irrational numbers such as π or √2 never stop or repeat, so they have no exact fraction. Entering 3.14159 gives the fraction for that rounded value, not for π.
  • Thousands separators (like the comma in 1,000) and scientific notation (like 1e5) aren't accepted in the decimal field.

Frequently asked questions

Is 0.999… really equal to 1?

Yes. Use the subtraction method. x = 0.(9) gives 10x = 9.(9). Subtracting leaves 9x = 9, so x = 1. Two different ways of writing a decimal can name the same number, just as 1/2 and 2/4 do.

Why does 1/3 never end as a decimal?

Our decimal system is built on tens, and 10 = 2 × 5. A reduced fraction can only end if its denominator divides evenly into some power of ten. That means its only prime factors are 2 and 5. Since 3 is neither, the remainders in the long division keep cycling, and the digits repeat.

How long can the repeating block be?

For a reduced fraction a/b, it's at most b − 1 digits. Denominators such as 7, 17 and 97 reach that limit. 1/7 has a 6-digit block, and 1/97 has a 96-digit block.

What is the difference between 0.1(6) and 0.(16)?

The parentheses mark exactly which digits repeat. 0.1(6) is 0.16666… and equals 1/6. 0.(16) is 0.161616… and equals 16/99.

How this calculator is checked

Method: Standard place-value conversion for terminating decimals, the shift-and-subtract method for repeating decimals, and long division with remainder tracking for fractions. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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