Answer: Enter your values and the Fraction Calculator returns the exact result instantly — formula, worked example, and a plain-English explanation are included below the tool.
Add, subtract, multiply, and divide fractions.
A fraction represents a part of a whole, written as one integer over another: the numerator (top) counts how many parts you have, and the denominator (bottom) says how many equal parts the whole was divided into. Fractions are fundamental in arithmetic, measurement, carpentry, cooking, finance, and every branch of mathematics. This calculator adds, subtracts, multiplies, and divides fractions and simplifies the result using exact integer arithmetic, so answers are precise rather than decimal approximations.
Fractions have been used for millennia: the Egyptian Rhind Mathematical Papyrus (circa 1550 BC) works extensively with unit fractions, and modern numerator-over-denominator notation stabilized in medieval Europe, with the horizontal fraction bar popularized by Fibonacci in his 1228 Liber Abaci. Decimals, percentages, and fractions are interchangeable representations: 1/2 = 0.5 = 50%.
Adding and subtracting fractions requires a common denominator. For a/b + c/d you compute (a·d + c·b)/(b·d), or use the least common multiple of the denominators to keep numbers small. Multiplication is direct: (a/b)·(c/d) = (a·c)/(b·d). Division multiplies by the reciprocal: (a/b) ÷ (c/d) = (a/b)·(d/c). Finally, the result is simplified by dividing numerator and denominator by their greatest common divisor (GCD), a method formalized in Euclid's Elements (Book VII, Proposition 2, circa 300 BC).
For example, 1/6 + 1/4: the LCM of 6 and 4 is 12, so 2/12 + 3/12 = 5/12, which is already in lowest terms because gcd(5,12)=1.
The table below lists benchmark equivalents worth memorizing. Each decimal is exact where it terminates; the fraction column gives the value in lowest terms.
None
| 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% |
| 1/10 | 0.1 | 10% |
| 3/4 | 0.75 | 75% |
| 2/3 | 0.666… | 66.6…% |
A proper fraction has a numerator smaller than its denominator (3/4); an improper fraction has a numerator greater than or equal to it (7/4); and a mixed number combines a whole number with a proper fraction (1 3/4 = 7/4). Any improper fraction converts to a mixed number by dividing: 7 ÷ 4 = 1 remainder 3, giving 1 3/4. This calculator reports exact simplified fractions so you can convert as needed.
Fractions appear wherever quantities do not divide evenly. In cooking, US customary measures are binary fractions: 1/2 cup, 1/4 teaspoon, 1/3 cup, and scaling a recipe by 3/2 or 2/3 is pure fraction multiplication. In carpentry and metalworking, imperial tape measures are graduated in halves, quarters, eighths, and sixteenths of an inch, so adding measurements like 3 5/16" and 1 7/8" is routine fraction arithmetic.
In music, note durations are fractions of a whole note: a half note is 1/2, a quarter note 1/4, an eighth 1/8. Time signatures such as 3/4 specify fractional structure, and dotted notes extend a duration by half again (a dotted quarter equals 3/8 of a whole note). Musicians read fractions constantly, usually without thinking of them as mathematics.
In finance, fractional shares let investors buy dollar amounts rather than whole shares, and interest computations often produce fractional quantities of a year. Odds and probability are naturally fractional: a probability of 1/6 for each face of a fair die, or 13/52 = 1/4 for drawing a heart from a standard deck.
Two cautions worth knowing. First, never simplify by "cancelling" digits (64/16 is not 4/1, even though striking the sixes suggests it); only common factors of the whole numerator and whole denominator may be divided out. Second, comparing fractions requires a common basis: 3/7 is greater than 2/5 because cross-multiplying gives 15 versus 14 — a technique equivalent to rewriting both over the common denominator 35.
Historically, fraction notation evolved for practical commerce. Medieval merchants in the Fibonacci tradition computed with fractions for currency and weights, and the sexagesimal (base-60) fractions of Babylonian astronomy — the reason we still divide hours into 60 minutes and minutes into 60 seconds — survive in every clock. Understanding fractions as division (3/4 means 3 divided by 4) unifies the notation with decimals and makes conversions mechanical.
How do I add fractions with different denominators?
Rewrite both fractions over a common denominator — the least common multiple of the two denominators — then add the numerators. For example, 1/6 + 1/4 = 2/12 + 3/12 = 5/12.
How are fractions simplified?
The numerator and denominator are divided by their greatest common divisor (GCD), the algorithm for which dates back to Euclid's Elements (Book VII). For example, 8/12 simplifies to 2/3 because gcd(8,12)=4.
How do I divide by a fraction?
Multiply by its reciprocal: dividing by c/d is the same as multiplying by d/c. For example, (1/2) ÷ (1/4) = (1/2)·(4/1) = 2.
Why is my answer shown as a fraction instead of a decimal?
Fraction arithmetic is exact, while many fractions such as 1/3 have no terminating decimal. Reporting the simplified fraction avoids rounding error; you can convert to a decimal yourself whenever an approximation is acceptable.
Can this calculator handle improper fractions and mixed numbers?
Yes. Enter improper fractions directly; results are returned as exact simplified fractions, and any improper result can be converted to a mixed number by dividing numerator by denominator.