Word Problem Parser Math Calculator

Each of the five templates here is a single relation that can be rearranged for whichever quantity is unknown. Uniform motion uses distance equals rate times time, which holds only at constant speed; for a multi-leg trip the average speed is total distance over total time, never the mean of the leg speeds. The percentage family is one relation, part equals whole times rate, read three ways: find the part, find the rate as part over whole times 100, or recover the whole as part divided by the rate. Work-rate problems rest on the fact that rates add while times do not - if one worker finishes in T1 and another in T2, the combined rate is 1/T1 plus 1/T2 and the joint time is the reciprocal of that sum, so 6 hours and 4 hours give 2.4 hours, not 5. Mixture problems apply conservation of solute, C1V1 plus C2V2 equals the final concentration times the combined volume, with dilution as the special case where one concentration is zero. The recurring error across all five is unit inconsistency - mixing minutes with hours, or percentages with decimals - which is why dimensional analysis is the standard check: if the units of the answer do not resolve to miles, hours or units of solution, the setup is wrong before the arithmetic starts. These templates also assume strict linearity, so they do not extend without modification to accelerating motion, compound interest, or reaction kinetics where rates depend on concentration.

Word problems are among the oldest surviving mathematics. The Rhind Mathematical Papyrus, copied by the scribe Ahmes in the mid-second millennium BCE from an older Egyptian source, contains division-of-loaves and proportional-share problems solved by the method of false position. Babylonian clay tablets from a comparable era pose excavation and labour problems in the same rhetorical form, stated in words with no symbolic notation at all. The Chinese Nine Chapters on the Mathematical Art, compiled by roughly the first century CE, devotes whole chapters to proportional distribution and to excess-and-deficit methods, which are mixture and rate problems in all but name. The cistern problem - two pipes filling a tank at different rates - appears in Greek sources associated with Heron of Alexandria and recurs in the collection Propositiones ad Acuendos Juvenes attributed to Alcuin of York in the eighth century, which also preserves the familiar river-crossing puzzles. Systematic symbolic method arrived with al-Khwarizmi, whose ninth-century treatise on al-jabr gave algebra its name and codified the reduction of a stated problem to a small number of canonical equation forms. Renaissance commercial arithmetics then popularised percentage, interest and alligation, the merchants' term for mixture calculations, and those chapters are the direct ancestors of the problem types that fill modern school algebra textbooks.