About this calculator
Fraction arithmetic done exactly — with whole numbers only, so there is no rounding at all. Enter plain fractions (3/4) or mixed numbers (1 ½), add, subtract, multiply or divide them, and get the answer in lowest terms, as a mixed number and as an exact decimal with any repeating digits marked. You can also simplify one fraction, or turn a decimal — including a repeating decimal such as 0.1666… — into an exact fraction.
For inch fractions and millimetres, use the Inch Fraction ↔ Decimal ↔ mm Converter.
Formula
Two fractions (+ − × ÷)
a/b ± c/d = (ad ± bc) / bd a/b × c/d = ac / bd a/b ÷ c/d = ad / bc
A mixed number w n/d is first written as (w·d + n)/d. The result is reduced by the greatest common divisor (Euclid’s algorithm).
Simplify one fraction
n/d = (n ÷ g) / (d ÷ g), g = gcd(n, d)
Decimal → fraction
0.N(R) = ( NR − N ) ÷ ( 10k × (10r − 1) )
N is the non-repeating part with k digits and R the repeating part with r digits. A terminating decimal is simply its digits over a power of 10. The closest fraction within a denominator limit is found with continued fractions (best rational approximation).
Worked example
- 1 ½ + ¾ = 3/2 + 3/4 = (3·4 + 3·2) / 8 = 18/8 = 9/4 = 2 ¼ = 2.25
- Simplify 1 ½ → 3/2
- 0.1(6) = (16 − 1) ÷ (10 × 9) = 15/90 = 1/6; 0.(3) = 3/9 = 1/3
- 7/12 = 0.58(3)
Assumptions and limitations
- Whole numbers, numerators and denominators must be integers up to 10¹² in size; the arithmetic is exact (BigInt), so results are never rounded.
- A minus sign goes on the whole number (for mixed numbers) or on the numerator; denominators must be positive.
- The exact decimal shows up to 60 digits after the point; a longer repeating part is cut off with “…”. The decimal value is rounded to 12 significant figures.
References
- Euclid’s algorithm for the greatest common divisor.
- Khinchin, A. Ya. Continued Fractions — best rational approximations.