The Basic Method: Set Up an Equation

To change a repeating decimal to a fraction, you multiply the decimal by a power of 10, subtract the original decimal, and solve for the unknown. This works because the repeating part cancels out, leaving you with a whole number you can use to build a fraction.

Start by writing your repeating decimal as a variable. For example, if your decimal is 0.333... (the 3 repeats forever), write it as x = 0.333... Then multiply both sides by 10 to shift the decimal point one place to the right: 10x = 3.333...

Now subtract the first equation from the second. The repeating parts line up and cancel out, leaving you with a straightforward equation: 10x − x = 3.333... − 0.333... which simplifies to 9x = 3. Divide both sides by 9 to get x = 3/9, which reduces to 1/3.

Key Takeaways

  • Write the repeating decimal as a variable (x), then multiply by 10, 100, or 1000 depending on how many digits repeat.
  • Subtract the original equation from the new one so the repeating part cancels and you are left with a whole number.
  • Divide both sides by the coefficient of x to isolate the fraction.
  • Always reduce the fraction to lowest terms by dividing the numerator and denominator by their greatest common factor.
  • Check your answer by dividing the numerator by the denominator on a calculator to confirm it matches the original decimal.

When More Than One Digit Repeats

If two or more digits repeat, multiply by a higher power of 10. For the decimal 0.454545... (where 45 repeats), multiply by 100 instead of 10, because two digits repeat.

Write x = 0.454545... and 100x = 45.454545... Subtract to get 100x − x = 45.454545... − 0.454545..., which gives you 99x = 45. Divide by 99 to get x = 45/99. Both 45 and 99 are divisible by 9, so the reduced fraction is 5/11.

The rule is straightforward: multiply by 10 for every digit that repeats. One repeating digit means multiply by 10. Two repeating digits means multiply by 100. Three repeating digits means multiply by 1000.

Handling Decimals That Have a Non-Repeating Part

Some decimals have digits that do not repeat, followed by digits that do. For example, 0.1666... has a 1 that appears once, then a 6 that repeats forever.

For these, you need two equations. Write x = 0.1666... Since one digit does not repeat before the 6 starts, multiply by 10 to get 10x = 1.666... Now multiply by 100 (not 10) to shift past both the non-repeating and repeating parts: 100x = 16.666...

Subtract the second equation from the third: 100x − 10x = 16.666... − 1.666..., which gives 90x = 15. Divide by 90 to get x = 15/90. Reduce by dividing both by 15 to get 1/6. You can verify: 1 ÷ 6 = 0.1666...

Reducing Your Fraction to Lowest Terms

After you solve for x, your fraction may not be in simplest form. To reduce it, find the greatest common factor (GCF) — the largest number that divides evenly into both the numerator and denominator.

For the fraction 45/99, list the factors of each: 45 has factors 1, 3, 5, 9, 15, 45. The number 99 has factors 1, 3, 9, 11, 33, 99. The greatest factor they share is 9. Divide both the numerator and denominator by 9: 45 ÷ 9 = 5 and 99 ÷ 9 = 11, so the reduced fraction is 5/11.

If you are unsure whether a fraction is fully reduced, try dividing both parts by small primes like 2, 3, 5, and 7. If none of them divide evenly into both the numerator and denominator, the fraction is in lowest terms.

Checking Your Work With Division

The fastest way to verify your answer is to divide the numerator by the denominator using a calculator. If you converted 0.333... to 1/3, divide 1 by 3 and you should get 0.333... (or 0.3333333 on the calculator display, which represents the repeating 3).

If your decimal has a non-repeating part, the match will be exact. For 0.1666..., dividing 1 by 6 gives 0.16666..., which matches. If your calculator result does not match the original decimal, go back and check your subtraction step — that is where most errors happen.

Common Mistakes to Avoid

The most frequent error is multiplying by the wrong power of 10. Remember: multiply by 10 for each digit that repeats. If you multiply by 10 when you should multiply by 100, your repeating part will not cancel out and you will get the wrong answer.

Another common mistake is forgetting to reduce the fraction. The fraction 45/99 is technically correct, but 5/11 is the answer in lowest terms, which is what you should report. Always check whether the numerator and denominator share any common factors before you finish.

Finally, be careful with subtraction when the repeating parts have different numbers of decimal places. Line them up carefully so that the repeating digits align, or you will subtract incorrectly and get a wrong numerator.

Frequently Asked Questions

What if the entire decimal is just one digit repeating, like 0.777...?

Use the basic method: x = 0.777... and 10x = 7.777... Subtract to get 9x = 7, so x = 7/9. Since 7 and 9 share no common factors, 7/9 is already in lowest terms.

Can I use this method for decimals that do not repeat?

No. Non-repeating decimals like 0.5 or 0.25 are already terminating, meaning they end. You can convert them to fractions directly: 0.5 = 5/10 = 1/2, and 0.25 = 25/100 = 1/4. The equation method only works when the decimal repeats forever.

What if I have a whole number with a repeating decimal, like 2.333...?

Treat the whole number and decimal part separately. The decimal 0.333... converts to 1/3 using the method above. Then add the whole number: 2 + 1/3 = 6/3 + 1/3 = 7/3. You can verify by dividing 7 by 3, which gives 2.333...

How do I know if a fraction is fully reduced?

A fraction is fully reduced when the numerator and denominator have no common factors other than 1. If you can divide both by the same number, they are not fully reduced yet. Use the GCF method or try dividing by small primes until nothing works.