How to turn a repeating decimal into a fraction

A repeating decimal is a decimal number where one or more digits repeat forever—like 0.333... (which is 1/3) or 0.142857142857... (which is 1/7). You can convert any repeating decimal to a fraction using algebra. The method depends on whether the entire decimal repeats or only part of it repeats.

The core idea is to multiply the decimal by a power of 10 to shift the repeating part, then subtract the original equation to cancel out the repeating digits. What remains is a fraction you can simplify. This works because the repeating pattern, when subtracted from itself at a different position, leaves only whole numbers behind.

Key Takeaways

  • Write the repeating decimal as an equation set equal to a variable like x, then multiply both sides by 10 (or a higher power of 10) to shift the decimal point.
  • Subtract the original equation from the new one to eliminate the repeating part, leaving you with a whole number on one side.
  • Solve for x by dividing both sides, then reduce the fraction to lowest terms by finding the greatest common divisor.
  • If only part of the decimal repeats (like 0.16666...), multiply by different powers of 10 to isolate the repeating section before subtracting.
  • Always check your work by dividing the numerator by the denominator to confirm it produces the original repeating decimal.

Converting a fully repeating decimal

Start with a decimal where the repeating part begins immediately after the decimal point. An example is 0.555... (written as 0.5̄, where the bar shows which digit repeats).

Set the decimal equal to x:

x = 0.555...

Since one digit repeats, multiply both sides by 10:

10x = 5.555...

Now subtract the first equation from the second:

10x − x = 5.555... − 0.555...9x = 5x = 5/9

The repeating decimals cancel out, leaving only whole numbers. Reduce if needed—in this case, 5/9 is already in lowest terms. You can verify this by dividing 5 by 9 on a calculator or by hand: you will get 0.555... again.

Converting a partially repeating decimal

Some decimals have non-repeating digits after the decimal point before the repeating part starts. An example is 0.1666... (written as 0.16̄), which equals 1/6. The non-repeating part is the 1, and the repeating part is the 6.

Set the decimal equal to x:

x = 0.1666...

Multiply by 10 to move the decimal one place (this puts you just before the repeating part):

10x = 1.666...

Now multiply the original equation by 100 (one power of 10 higher) to shift past one full cycle of the repeating digit:

100x = 16.666...

Subtract the 10x equation from the 100x equation:

100x − 10x = 16.666... − 1.666...90x = 15x = 15/90

Reduce by dividing both numerator and denominator by their greatest common divisor (15):

x = 1/6

Handling decimals with multiple repeating digits

When two or more digits repeat, the same method works—you just multiply by a higher power of 10. For 0.142857142857... (which repeats every 6 digits), multiply by 1,000,000 instead of 10.

Set x = 0.142857142857...

Multiply by 1,000,000 (since 6 digits repeat):

1,000,000x = 142857.142857...

Subtract:

1,000,000x − x = 142857.142857... − 0.142857...999,999x = 142857x = 142857/999999

Reduce by finding the greatest common divisor. In this case, it simplifies to 1/7. The pattern holds: if n digits repeat, multiply by 10^n (10 to the power of n).

Reducing fractions to lowest terms

After you have your fraction, divide both the numerator and denominator by their greatest common divisor (GCD). The GCD is the largest number that divides evenly into both.

For 15/90: The factors of 15 are 1, 3, 5, 15. The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90. The greatest common factor is 15. Divide both by 15 to get 1/6.

If you are unsure of the GCD, try dividing by small primes (2, 3, 5, 7) until the numerator and denominator share no common factors. This ensures your final fraction is in simplest form.

Checking your answer

Divide the numerator by the denominator using long division to verify your fraction matches the original decimal. For 1/6, divide 1 by 6: you get 0.1666..., which matches.

This check catches arithmetic errors and confirms the repeating pattern is correct. If the decimal does not match, retrace your steps—usually the error is in the subtraction, in choosing the wrong power of 10, or in reducing the fraction incorrectly. Working backward from your answer is one of the most reliable ways to spot mistakes.

Frequently Asked Questions

What if the decimal does not repeat?

A non-repeating decimal (called a terminating decimal) like 0.25 is already simple to convert: it has a finite number of digits. Write 0.25 as 25/100, then reduce to 1/4. You do not need the multiplication-and-subtraction method for these.

How do I know what power of 10 to multiply by?

Count how many digits repeat. If one digit repeats, multiply by 10. If two digits repeat, multiply by 100. If three digits repeat, multiply by 1000. For partially repeating decimals, use one power of 10 to isolate the non-repeating part, then a higher power to include one full cycle of the repeating part.

Can I convert 0.999... to a fraction?

Yes. Set x = 0.999..., multiply by 10 to get 10x = 9.999..., then subtract: 10x − x = 9, so 9x = 9, and x = 1. This shows that 0.999... equals exactly 1, which surprises many people but is mathematically correct.

What if my fraction does not reduce?

Some fractions are already in lowest terms. For example, 5/9 cannot be reduced further because 5 and 9 share no common factors other than 1. Check by testing whether small primes divide both the numerator and denominator evenly.

What power of 10 do I use for 0.12̄3̄ (where 23 repeats)?

The non-repeating part is 1, and the repeating part is 23 (two digits). Multiply the original by 10 to get past the non-repeating digit, then multiply by 1000 (10^3) to shift past one cycle of the two-digit repeat. Subtract the 10x equation from the 1000x equation to isolate the repeating part.