Fraction ↔ Decimal Converter

Common fractions chart

Fraction	Decimal	Percent
1/8	0.125	12.5%
1/4	0.25	25%
1/3	0.333	33.3%
3/8	0.375	37.5%
1/2	0.5	50%
5/8	0.625	62.5%
2/3	0.667	66.7%
3/4	0.75	75%
7/8	0.875	87.5%

How this is calculated

decimal = numerator ÷ denominator · reverse: search the simplest fraction within rounding distance

Worked example: 3/8 = 0.375 = 37.5%. Going backwards, 0.375 recovers 3/8 exactly, and a truncated 0.333 sensibly recovers 1/3 — the simplifier prefers the fraction you meant over the one the rounding literally spells.

Why some fractions terminate and others repeat forever

A fraction in lowest terms becomes a terminating decimal when its denominator has no prime factors other than 2 and 5 — the primes that divide ten. So 1/8 is 0.125 exactly, because 8 = 2³, and 3/20 is 0.15 exactly, because 20 = 2²×5. Every other denominator repeats: 1/3 = 0.333…, 1/7 = 0.142857142857…, 1/6 = 0.1666…

The length of the repeat is also predictable. For 1/7 the cycle is six digits long, and it is the longest possible for a denominator of seven — such primes are called full reptend primes. For 1/3 it is one digit. The repeating block can never be longer than one less than the denominator, which is a useful sanity check when a calculator display runs out of room.

Converting by hand, both directions

Fraction to decimal is long division: numerator divided by denominator. 5/8 is 5 ÷ 8 = 0.625. Decimal to fraction is place value: 0.625 is 625/1000, which simplifies by dividing both by 125 to give 5/8.

Repeating decimals convert with a trick worth seeing once. For 0.777…, let x = 0.777…, then 10x = 7.777…, so 10x − x = 7, giving 9x = 7 and x = 7/9. The same method with two repeating digits multiplies by 100. It also produces the notorious result that 0.999… = 1 exactly — not approximately, but as an identity, since 10x − x = 9 gives x = 1. That is a fact about how decimal notation works, not a rounding error.

The fractions worth memorising

Fraction	Decimal	Percent
1/2	0.5	50%
1/3	0.333…	33.3%
1/4	0.25	25%
1/5	0.2	20%
1/8	0.125	12.5%
1/16	0.0625	6.25%
2/3	0.666…	66.7%
3/4	0.75	75%

The eighths and sixteenths matter in practice because imperial measurement is built on them: a drill bit marked 3/8" is 0.375", and a tape measure divided into sixteenths reads 0.0625" per mark. Anyone working between imperial drawings and metric tools is converting these constantly.

Rounding, and where the error goes

A repeating decimal has to be cut somewhere, and where you cut determines the error. Rounding 1/3 to 0.33 loses about 0.1% of the value; across a hundred such operations that becomes visible. In money this is why financial systems either carry extra decimal places internally or work in whole cents throughout — and why an invoice rounded line by line can differ from the same invoice rounded once at the end.