Fraction Calculator

A fraction represents a part of a whole, written as a numerator (the top number, how many parts you have) over a denominator (the bottom number, how many equal parts make up the whole). Fractions show up everywhere: cooking measurements, construction cuts, financial splits, and the building blocks of algebra. Being able to combine two fractions correctly — rather than just approximating with a decimal — matters because exact fractional answers are often required in recipes, woodworking, and schoolwork, where "0.333..." is a less useful answer than "1/3". The reason fraction arithmetic needs its own rules, instead of just adding numerators and denominators directly, is that fractions with different denominators represent different-sized "slices". You cannot add 1/2 and 1/3 by adding tops and bottoms, which would incorrectly give 2/5, because a half and a third are not divided into equal-sized pieces. Instead, both fractions are rewritten over a shared common denominator first. This calculator uses the simplest reliable method: cross-multiplication. For addition and subtraction, a/b op c/d becomes (a×d op c×b) / (b×d) — effectively giving both fractions the denominator b×d before combining their numerators. Worked example: adding 1/2 and 1/3. Using the formula, the numerator becomes (1×3 + 1×2) = 5, and the denominator becomes (2×3) = 6, giving 5/6 — already in simplest form, since 5 and 6 share no common factor besides 1. As a decimal, that's approximately 0.833333. Multiplication is simpler: multiply the numerators together and the denominators together, so 1/2 × 1/3 = (1×1)/(2×3) = 1/6. Division flips the second fraction and multiplies: 1/2 ÷ 1/3 = 1/2 × 3/1 = 3/2, which is 1.5 as a decimal. Every result is automatically reduced using the greatest common factor (GCF) — for instance, a raw result of 4/8 is immediately shown as 1/2, because 4 is the largest number that divides evenly into both 4 and 8. This is calculated with the Euclidean algorithm, the same method taught in most number-theory classes: repeatedly divide the larger number by the smaller and keep the remainder until it reaches zero, at which point the last non-zero remainder is the GCF. A common misconception, especially for anyone new to fractions, is assuming you can add or subtract numerators and denominators independently the way you would whole numbers — "1/2 + 1/3 = 2/5" is a frequent and completely incorrect shortcut. Another common mix-up is with division: many people try to divide numerator-by-numerator and denominator-by-denominator instead of flipping the second fraction and multiplying. This tool sidesteps both mistakes by applying the correct cross-multiplication and flip-and-multiply formulas automatically, so you can check your own manual work or just get a fast, reliable answer for a real-world measurement problem, without re-deriving the rules from scratch every time.