Fraction Calculator

How fraction arithmetic works

A fraction a/b means "a parts out of b equal parts". The four operations follow simple rules, and the result panel lists every step it takes: converting mixed numbers to improper fractions, finding the least common denominator for addition and subtraction, combining, reducing by the GCD and rewriting as a mixed number:

a/b + c/d = (a·d + c·b) / (b·d)
a/b − c/d = (a·d − c·b) / (b·d)
a/b × c/d = (a·c) / (b·d)
a/b ÷ c/d = (a·d) / (b·c)

After the raw result is formed, it is simplified by dividing the numerator and denominator by their greatest common divisor (GCD), found with Euclid's algorithm: repeatedly replace the larger number with the remainder of dividing it by the smaller until the remainder is zero. Any minus sign is moved to the numerator so the denominator is always positive.

Worked examples

Example 1: 1/2 + 3/4

Cross-multiply: (1×4 + 3×2) / (2×4) = (4 + 6)/8 = 10/8. The GCD of 10 and 8 is 2, so the answer is 5/4, which is the mixed number 1 1/4 and the decimal 1.25.

Example 2: 5/6 − 1/4

(5×4 − 1×6) / (6×4) = (20 − 6)/24 = 14/24. Dividing by the GCD of 2 gives 7/12 ≈ 0.583333.

Example 3: 2/3 ÷ 4/9

Dividing is multiplying by the reciprocal: 2/3 × 9/4 = 18/12. The GCD is 6, giving 3/2 = 1 1/2 = 1.5.

Example 4: −3/5 × 10/7

(−3 × 10)/(5 × 7) = −30/35, which simplifies by 5 to −6/7 ≈ −0.857143.

How to read the result

The large figure is the simplified fraction. If the numerator is larger than the denominator (an improper fraction), the mixed-number form is shown too: 7/3 becomes 2 1/3, meaning two wholes and one third. The decimal is rounded to six places; recurring decimals like 1/3 = 0.333333… are cut off rather than shown with a bar. The circle shades the fractional part of the answer — for 5/4 it shades one quarter, the part left after the whole one. When the answer is a whole number, the fraction is shown without a denominator.

Mixed numbers and whole numbers as input

Each fraction has an optional Whole box. To enter 2 3/8, type 2 in Whole, 3 in Numerator and 8 in Denominator; the calculator converts it with n′ = sign(w) × (|w| × d + n), so 2 3/8 becomes 19/8. For a negative mixed number such as −2 3/8, put the minus sign on the whole number only: −2, 3 and 8 give −19/8, because the minus applies to the entire quantity, not just the whole part. If you leave Whole empty or at 0, the numerator keeps its own sign, so 0 with −3/4 is simply −3/4. To enter a whole number on its own, such as 5, use 5 in Whole with 0/1, or 5/1 with Whole left blank.

Each value you type must be smaller than one billion in size. Behind the scenes the arithmetic uses exact big-integer math, so even when two nine-digit denominators are multiplied together the answer is not rounded or corrupted.

Limits of the calculator

Denominators cannot be zero — a fraction with zero parts in the whole is undefined — and you cannot divide by a fraction whose numerator is zero, since that is dividing by zero. In both cases the calculator shows a friendly error instead of a result. Inputs must be whole numbers; if you have a decimal like 0.5 in a numerator, scale both parts (0.5/3 = 1/6).

To turn a fraction into a percentage, multiply the decimal by 100 or use the percentage calculator in "X is what % of Y" mode.

Where fractions show up

Fractions are the natural language of recipes (scaling 3/4 cup by 2/3), woodworking and construction (adding 5/8 inch to 1 3/16 inch), music (note lengths), probability (a 1/6 chance on a die) and ratios in science. In these settings decimals can be awkward: 1/3 never terminates, and tape measures are marked in sixteenths, not tenths. Working in fractions keeps answers exact, and simplifying to lowest terms makes them easy to read and measure.

For example, doubling a recipe that calls for 3/8 teaspoon gives 3/8 × 2/1 = 6/8 = 3/4 teaspoon — a measure you actually own.

Common mistakes

The classic error is adding tops and bottoms separately: 1/2 + 1/3 is not 2/5. Both fractions must be expressed over the same denominator first (3/6 + 2/6 = 5/6), which the cross-multiplication rule does automatically. Another is forgetting to flip the second fraction when dividing, and a third is leaving answers unsimplified, such as 12/16 instead of 3/4. Use the calculator to check homework steps, not just final answers.