Fraction Calculator
Add, subtract, multiply, and divide fractions with full step-by-step work. Automatic simplification, mixed-number conversion, and decimal equivalent.
How to Use
- Enter the first fraction's numerator (top) and denominator (bottom).
- Pick the operation: +, −, ×, or ÷.
- Enter the second fraction's numerator and denominator.
- The result is shown in three forms: simplified fraction, decimal, and mixed number (when applicable).
- For mixed numbers like 2½, convert to improper first: 2½ = 5/2.
- Negative fractions: place the minus sign on the numerator (e.g., -3/4).
Formulas & Rules
A Brief History of Fractions
Egyptian mathematicians (c. 2000 BCE) used "unit fractions" — sums of 1/n terms — almost exclusively, with the Eye of Horus symbols representing 1/2, 1/4, 1/8, 1/16, 1/32, 1/64. The Rhind Mathematical Papyrus (c. 1550 BCE) demonstrates clever arithmetic on these limited tools. Babylonian mathematicians used a base-60 sexagesimal system that handled fractions naturally: any number divisible by 2, 3, 4, 5, or 6 has a clean representation. We still see this legacy today in 60 minutes per hour and 360 degrees per circle.
The fraction notation we use — numerator over denominator with a horizontal bar — comes from Arabic mathematicians of the 12th century. Al-Hassar's Kitab al-Bayan (c. 1192) is among the earliest works to use this layout. European mathematicians adopted it after Fibonacci's Liber Abaci (1202) and it has been standard ever since. The slash-fraction (1/2) is a typographic shorthand that emerged with the printing press, where stacked fractions were mechanically inconvenient.
Decimal fractions (representing 0.5 instead of 1/2) were popularized by Simon Stevin in 1585 and have steadily displaced fractions in scientific and engineering work. Two domains where fractions remain dominant: U.S. customary measurements (inch fractions for woodworking, cooking, fabric) and pure mathematics (where exact rational values matter). Both will keep fractions alive indefinitely — decimal 1/3 = 0.333… can never be exact, but 1/3 always is.
About This Calculator
This calculator handles the four basic fraction operations using exact integer arithmetic — no floating-point rounding errors. Inputs accept negative integers; the result is automatically simplified using the Euclidean algorithm to find the greatest common divisor of numerator and denominator. The decimal equivalent is shown for readability, and improper fractions are also shown in mixed-number form when applicable.
Everything runs entirely in your browser; no values are transmitted or stored. For very large fractions (numerators or denominators with more than ~15 digits), browser integer precision can become a factor — the calculator stays in BigInt-safe ranges for typical inputs but degrades gracefully for extreme values.
Frequently Asked Questions
Why do I need a common denominator to add fractions?
Fractions only add directly when they're parts of the same-sized whole. 1/2 + 1/3 doesn't work as 2/5 because the halves and thirds aren't the same units. Convert both to sixths (3/6 + 2/6 = 5/6) and the sum makes sense. Multiplication and division don't need this — they operate on numerators and denominators independently.
What's an improper fraction vs. a mixed number?
An improper fraction has a numerator larger than its denominator (e.g., 7/3). A mixed number combines a whole number and a proper fraction (e.g., 2⅓). They're equivalent — 7/3 = 2⅓. For arithmetic, convert mixed numbers to improper form first (multiply whole by denom, add numerator). For final answers, convert back if mixed-number form is more readable.
How do I simplify a fraction?
Find the greatest common divisor (GCD) of numerator and denominator and divide both by it. For 18/24: GCD(18, 24) = 6, so 18/24 = 3/4. The Euclidean algorithm finds the GCD efficiently: GCD(a, b) = GCD(b, a mod b), continuing until the second argument hits zero. The calculator does this automatically.
Can I divide by a fraction?
Yes — "flip and multiply." a/b ÷ c/d = a/b × d/c. For example, 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1⅞. The reciprocal trick works because dividing by a number is the same as multiplying by its inverse.
Why does multiplying two proper fractions give a smaller result?
Because you're taking a fraction <em>of</em> a fraction. Half of one-third is one-sixth (1/2 × 1/3 = 1/6). Each multiplication by something less than 1 shrinks the result. This is the opposite of integer multiplication and a common source of confusion when learning fractions.
How do fractions, decimals, and percentages relate?
All three express the same kind of quantity. 3/4 = 0.75 = 75%. To convert: fraction → decimal, divide; decimal → percent, multiply by 100; percent → fraction, divide by 100 then simplify. Some fractions like 1/3 produce repeating decimals (0.333…) — keep them in fraction form for exact arithmetic.
Common Use Cases
Cooking and recipe scaling
Halve a recipe (multiply each fraction by 1/2) or convert ⅓ cup to tablespoons by dividing by 1/16.
Construction and woodworking
Add lengths in fractional inches (e.g., 2 3/8 + 1 1/4) without converting to decimal first.
Sewing and pattern making
Combine seam allowances (1/4" + 5/8" + 3/8") and convert the total to a measurement on the ruler.
Math homework and tutoring
Verify worked solutions with the step-by-step output and use mixed-number conversion to match textbook style.
Music theory
Compute composite note durations (e.g., quarter + sixteenth = 5/16 of a measure).
Finance and probabilities
Combine odds expressed as fractions (1/3 chance × 1/2 chance = 1/6 joint probability).
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