Decimal to Fraction Calculator

Terminating or repeating, reduced to lowest terms.

Terminating decimals

Count the digits after the point. That count is the power of ten you need. 0.625 has three digits, so it is 625/1000, and dividing both by their GCF of 125 gives 5/8.

Repeating decimals

These need algebra rather than counting. For 0.333… let x = 0.333…, multiply by 10 to get 10x = 3.333…, and subtract: 9x = 3, so x = 1/3. The subtraction is what removes the infinite tail, and multiplying by 10 to the power of the block length is what lines the tails up.

For a decimal with a non-repeating start such as 0.1666…, the same idea applies with two powers of ten. Enter 0.16 with 1 repeating digit above and the calculator shows that arithmetic.

Which decimals repeat

A fraction in lowest terms terminates if its denominator's only prime factors are 2 and 5, and repeats otherwise. That is a fact about base ten: ten is 2 × 5, so only those primes divide a power of ten cleanly. In base three, 1/3 would terminate.

Common mistakes

• Forgetting to reduce. 625/1000 is correct but not in lowest terms.
• Rounding first. Converting 0.33 rather than 0.333… gives 33/100, not 1/3.
• Miscounting the repeating block — it is the length of the pattern, not the number of digits shown.