The Basic Method for Converting Repeating Decimals

To convert a repeating decimal to a fraction, you multiply the decimal by a power of 10 to shift the repeating part, then subtract the original decimal from the result. This works because repeating decimals follow a predictable pattern, and subtraction cancels out the infinite tail.

The exact power of 10 you use depends on how many digits repeat. If one digit repeats (like 0.333...), you multiply by 10. If two digits repeat (like 0.454545...), you multiply by 100. If three digits repeat, you multiply by 1000, and so on. The key is counting the repeating digits accurately before you start.

Key Takeaways

  • Count how many digits repeat in the decimal, then multiply both sides of an equation by the matching power of 10 (10 for one digit, 100 for two digits, 1000 for three digits).
  • Subtract the original decimal from the multiplied version to eliminate the repeating part and leave only a whole number.
  • Divide both sides by the coefficient of the original decimal to isolate the fraction.
  • Reduce the fraction to lowest terms by dividing the numerator and denominator by their greatest common factor.

Converting a Single Repeating Digit

When one digit repeats forever, multiply by 10. Let's use 0.333... (which is 0.3 with the 3 repeating).

Set up the equation: x = 0.333...

Multiply both sides by 10: 10x = 3.333...

Now subtract the first equation from the second:

10x − x = 3.333... − 0.333... 9x = 3 x = 3/9

Reduce by dividing both numerator and denominator by 3: x = 1/3

You can check this by dividing 1 by 3 on a calculator — you get 0.333... forever. This same process works for any single repeating digit: 0.777... becomes 7/9, and 0.999... becomes 9/9 or 1.

Converting Two or More Repeating Digits

When two digits repeat, multiply by 100. When three digits repeat, multiply by 1000. The rule is: multiply by 10 raised to the power of however many digits repeat.

Let's convert 0.454545... (where 45 repeats):

Set up: x = 0.454545...

Multiply by 100 (because two digits repeat): 100x = 45.454545...

Subtract:

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

Reduce by dividing both by 9: x = 5/11

For 0.142857142857... (where 142857 repeats — six digits), you would multiply by 1,000,000 and subtract to get the fraction. The denominator would be 999,999, which you would then reduce by finding common factors.

Handling Decimals That Start Non-Repeating, Then Repeat

Some decimals have non-repeating digits before the repeating part starts. For example, 0.1666... has a 1 that doesn't repeat, then 6 repeating forever.

For these, you need two equations. Multiply by 10 to move past the non-repeating part, then multiply by a higher power of 10 to move past one cycle of the repeating part.

Let's convert 0.1666...:

Set up: x = 0.1666...

Multiply by 10: 10x = 1.666...

Multiply by 100: 100x = 16.666...

Subtract the first from the second:

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

Reduce by dividing by 15: x = 1/6

The denominator (90) comes from subtracting the powers of 10: 100 − 10 = 90. This method handles any mix of non-repeating and repeating digits.

Reducing Your Fraction to Lowest Terms

After subtraction, 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 45/99: both divide by 9, giving you 5/11. For 15/90: both divide by 15, giving you 1/6. For 3/9: both divide by 3, giving you 1/3.

If you're unsure of the GCF, try dividing both numbers by small primes (2, 3, 5, 7) until no number divides both evenly. You can also list all factors of each number and pick the largest one they share. A fraction is fully reduced when the numerator and denominator have no common factors except 1.

Checking Your Work

The easiest check is to divide the numerator by the denominator using long division or a calculator. If you get back the original repeating decimal, your fraction is correct.

For example, 1 ÷ 3 = 0.333..., and 5 ÷ 11 = 0.454545..., and 1 ÷ 6 = 0.1666... If your result doesn't match, go back and check that you counted the repeating digits correctly and subtracted the equations in the right order.

Frequently Asked Questions

What if the entire decimal repeats versus just part of it?

If the entire decimal repeats (like 0.454545...), the repeating part starts when ready after the decimal point. If only part repeats (like 0.1666...), you have non-repeating digits first. The method is the same — count only the repeating digits to decide which power of 10 to use.

Can I convert a terminating decimal the same way?

Terminating decimals (like 0.5 or 0.25) don't repeat, so this method doesn't explore. Instead, count the decimal places: 0.5 is 5/10, and 0.25 is 25/100. Then reduce. A terminating decimal can always be written as a fraction without using the multiplication-and-subtraction technique.

What if I get a fraction that doesn't reduce?

That's fine — it means the fraction is already in lowest terms. For example, 1/3 cannot be reduced further because 1 and 3 share no common factors other than 1.

Do I always multiply by 10, 100, or 1000?

Yes, always a power of 10. The exponent matches the number of repeating digits. One repeating digit = 10¹. Two repeating digits = 10². Three repeating digits = 10³. And so on.

Why does subtracting the equations make the repeating part disappear?

Because both sides have the same infinite repeating tail. When you subtract 0.333... from 3.333..., the repeating parts cancel out, leaving only 3. That's why the method works — the infinity cancels itself.