The rule
terminating: digits ÷ a power of ten · repeating: digits ÷ a string of nines
Why it works
A terminating decimal is already a fraction: 0.75 is 75 hundredths. A repeating one needs the trick of nines — 0.333… is 3/9, because a single repeating digit sits over a single nine. Both then simplify.
How to do it by hand
- For a terminating decimal, put the digits over the matching power of ten
- For a repeating one, put the repeating digits over as many nines
- Combine the two parts if the decimal has both
- Simplify by the greatest common divisor
What is worth knowing
The nines rule looks like a trick and is a theorem. Multiply 0.333… by 10, subtract the original, and the repeating tail cancels exactly, leaving 9x = 3. The same manipulation proves that 0.999… equals 1 — which is not an approximation or a rounding convention but an identity, and the most reliably contested fact in elementary mathematics. The practical value of converting back to fractions is exactness: 1/3 is exact where 0.333 is not, which matters whenever the result feeds into further calculation.