About this tool
This fraction calculator does the arithmetic students and cooks and woodworkers actually need: add, subtract, multiply, or divide two fractions or mixed numbers and see the answer simplified, as a mixed number, as a decimal, and as a percentage, with the working written out line by line (common denominator, converted numerators, GCD used to simplify). Separate panels convert a decimal such as 0.375 or a repeating decimal such as 0.1(6) into the closest fraction with a denominator you choose, expand any fraction into its decimal with the repeating block marked, simplify any fraction or mixed number, and compare two fractions to say which is larger. It is built for homework checking, recipe scaling, and reading tape measures, and every result can be copied.
How it works
Fractions are stored as an integer numerator and denominator and always reduced by their greatest common divisor (Euclid's algorithm), with the sign carried on the numerator. Addition and subtraction rewrite both fractions over the least common denominator, LCD = a x b / gcd(a, b), then combine numerators. Multiplication multiplies straight across; division multiplies by the reciprocal of the divisor. Mixed numbers are converted to improper fractions first (2 1/2 = 5/2). Decimal-to-fraction conversion uses the continued-fraction algorithm: repeatedly take the integer part and invert the remainder, building convergents until the next denominator would exceed your limit, then check the final semiconvergent so the result is the best rational approximation with that bound (pi with limit 1,000 gives 355/113). A repeating decimal 0.ab(c) becomes (abc - ab) / (10^2 x (10^1 - 1)). Fraction-to-decimal uses long division and detects the repeating block when a remainder recurs.
Frequently asked questions
How do you add fractions with different denominators?
Rewrite both over a common denominator, ideally the least common denominator (LCD). For 1/2 + 1/3 the LCD of 2 and 3 is 6, so 1/2 = 3/6 and 1/3 = 2/6, and the sum is 5/6. The LCD is the least common multiple of the two denominators: multiply them and divide by their greatest common divisor. Using the plain product also works (2 x 3 = 6 here) but often leaves a fraction that needs simplifying afterwards.
How do you divide fractions?
Flip the second fraction and multiply. 3/4 divided by 1/2 becomes 3/4 x 2/1 = 6/4 = 3/2 = 1 1/2. The reason: dividing by a number is the same as multiplying by its reciprocal, and the reciprocal of 1/2 is 2. With mixed numbers convert to improper fractions first: 2 1/2 divided by 1/4 is 5/2 x 4/1 = 20/2 = 10. Dividing by zero, including by a fraction that reduces to 0, has no answer.
How do I turn a decimal into a fraction?
For a terminating decimal, write the digits over the matching power of ten and simplify: 0.375 = 375/1000 = 3/8. For a repeating decimal, let x equal the number, multiply by 10 raised to the length of the repeating block, and subtract: x = 0.333..., 10x = 3.333..., so 9x = 3 and x = 1/3. For a measured or rounded value like 0.3333, pick a maximum denominator (such as 16 for a tape measure or 100 for percentages) and the calculator finds the closest fraction with the continued-fraction method.
Why does the calculator show a mixed number and an improper fraction?
They are the same value written two ways. An improper fraction such as 7/4 has a numerator larger than its denominator; the mixed number 1 3/4 separates the whole part. Improper fractions are easier to compute with, which is why the working uses them, while mixed numbers are easier to read on a ruler or a recipe. Simplified means the numerator and denominator share no factor other than 1, so 6/4 becomes 3/2 (or 1 1/2).