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 into 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, subtract the original decimal, and solve for the fraction. This works because multiplication shifts the repeating part, and subtraction cancels it out. The process takes about two minutes once you know the steps.

Key Takeaways

  • Write the repeating decimal as a variable (like x = 0.333...), then multiply both sides by 10, 100, or 1000 depending on how many digits repeat.
  • Subtract the original equation from the new one to eliminate the repeating part and get a whole number.
  • 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 0.1666...), multiply by different powers of 10 to shift both the non-repeating and repeating parts into position before subtracting.
  • Check your answer by dividing the numerator by the denominator on a calculator to confirm it matches the original decimal.

Converting a decimal where all digits repeat

Start with a decimal like 0.555... (written as 0.5̄, where the bar means the digit repeats). Set it equal to a variable: x = 0.555...

Count how many digits repeat. Here, only one digit (5) repeats, so multiply both sides by 10:

x = 0.555... 10x = 5.555...

Now subtract the first equation from the second:

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

The repeating part cancels out, leaving you with a simple fraction. Reduce if needed — in this case, 5/9 is already in lowest terms. You can verify: 5 ÷ 9 = 0.555... on a calculator.

Converting a decimal where two or more digits repeat

If two digits repeat, like 0.454545... (written 0.4̄5̄), multiply by 100 instead of 10, because you need to shift two decimal places:

x = 0.454545... 100x = 45.454545...

Subtract:

100x − x = 45.454545... − 0.454545... 99x = 45 x = 45/99

Now reduce by finding the greatest common divisor. Both 45 and 99 divide by 9:

45 ÷ 9 = 5 99 ÷ 9 = 11 x = 5/11

The rule is simple: if n digits repeat, multiply by 10^n. Three repeating digits means multiply by 1000; four means 10,000.

Converting a decimal where only some digits repeat

This is trickier. Take 0.1666... (written 0.1̄6̄ or sometimes 0.16̄). The 1 does not repeat, but the 6 does. You need two equations to shift both parts into the right position.

First, multiply by 10 to move past the non-repeating digit:

x = 0.1666... 10x = 1.666...

Then multiply by 100 to shift the repeating part:

100x = 16.666...

Subtract the first from the second:

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

Reduce by dividing both by 15:

x = 1/6

Verify: 1 ÷ 6 = 0.1666... ✓

Reducing fractions to lowest terms

After you get your fraction, always check if it can be reduced. Find the greatest common divisor (GCD) — the largest number that divides evenly into both the numerator and denominator.

For 45/99: list the factors of 45 (1, 3, 5, 9, 15, 45) and factors of 99 (1, 3, 9, 11, 33, 99). The largest one they share is 9. Divide both by 9 to get 5/11.

If you are unsure, try dividing by small primes: 2, 3, 5, 7. Keep dividing until no prime divides both numbers evenly. A fraction is fully reduced when the only common factor is 1.

Checking your work

Use a calculator to divide the numerator by the denominator. If your fraction is correct, the result should match the original repeating decimal.

Example: you converted 0.777... to 7/9. Divide 7 ÷ 9 on a calculator. You get 0.777..., which matches. If your result does not match, recheck your subtraction step — that is where most errors happen.

Frequently Asked Questions

What if the decimal does not repeat — like 0.5 or 0.25?

Those are terminating decimals, not repeating ones. You can still convert them: 0.5 = 5/10 = 1/2, and 0.25 = 25/100 = 1/4. Just count decimal places, put the digits over a power of 10, and reduce. No algebra needed.

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

Count the number of digits that repeat. If one digit repeats, multiply by 10. If two repeat, multiply by 100. If three repeat, multiply by 1000. The exponent on 10 matches the count of repeating digits.

What if part of the decimal repeats and part does not?

Multiply by two different powers of 10 to create two equations. Multiply by 10^a to shift past the non-repeating part, then by 10^(a+b) to shift past both. Subtract the first equation from the second. The non-repeating part cancels, leaving only the repeating part to solve.

Can every repeating decimal be written as a fraction?

Yes. Any repeating decimal is a rational number, meaning it can always be expressed as a fraction of two whole numbers. Non-repeating, non-terminating decimals (like pi or the square root of 2) cannot be written as fractions.

Do I have to reduce the fraction all the way?

Technically no, but it is standard practice. A reduced fraction is simpler and easier to work with. Most teachers and textbooks expect the final answer in lowest terms, so reduce it.