Why every result is reduced with GCD#
Raw cross-multiplication often leaves answers like 4/8 or 10/15. This calculator always divides numerator and denominator by their greatest common divisor using the Euclidean algorithm, so you get lowest terms such as 1/2 or 2/3. The sign is kept on the numerator, which matches how fractions are usually written in algebra and most programming libraries.
Reducing first also keeps intermediate products smaller when you chain mental checks, and it makes equality obvious: 6/9 and 2/3 are the same value once both are simplified. If a denominator would be zero, the tool stops with an error instead of inventing a result.
Improper fractions vs mixed numbers#
An improper fraction has a numerator larger than or equal to its denominator (7/3). A mixed number splits that into a whole part plus a proper fraction (2 1/3). Both represent the same value; this tool always returns the simplified improper form and the mixed form side by side so you can use whichever your homework, recipe, blueprint, or code expects.
Negative mixed numbers put the minus on the whole value (−2 1/3 means −7/3), not on the fractional piece alone. Whole numbers appear as n/1 in improper form and as a bare integer in the mixed panel when there is no leftover fraction.
How add, subtract, multiply, and divide work#
Addition and subtraction use a common denominator: (a×d ± c×b) / (b×d), then reduce. Multiplication is (a×c)/(b×d). Division multiplies by the reciprocal: (a×d)/(b×c), and fails cleanly if the second numerator is zero — “cannot divide by zero” is distinct from a zero denominator on either input fraction.
All of this stays in integer arithmetic until the optional decimal display, which avoids most floating-point drift for everyday classroom and DIY sizes. Very large numerators can still hit JavaScript number limits; for those edge cases, reduce inputs first or split the problem.
Converting decimals and mixed numbers#
Convert mode turns a terminating decimal into a fraction by expanding powers of ten and simplifying (0.75 → 75/100 → 3/4). Mixed input combines whole and fractional parts into one improper fraction first, then runs the same GCD pass. Going the other way, any fraction is shown as a decimal so you can check calculator or spreadsheet output against exact rational form.
Repeating decimals that only exist as float approximations (for example typing 0.3333333 hoping for 1/3) use a continued-fraction best approximation within a large max denominator. Prefer entering 1/3 directly in fraction mode when you need an exact third. Nothing leaves your browser: conversions and arithmetic run entirely on-device.