Compound rule of three calculator

A compound rule of three problem asks for one unknown when several quantities change at once and each quantity is either directly or inversely proportional to that unknown. Students meet this model in middle-school arithmetic, vocational exams, and shop-floor planning: more workers produce more pieces (direct), but more workers finish a job in fewer days (inverse). A simple cross-multiply handles one pair; the compound form multiplies every direct ratio of sought over given and every inverse ratio of given over sought, then scales the known magnitude. This compound rule of three calculator accepts that known magnitude as value plus two or more labeled factors, each with a given value, a sought value, and a relation of direct or inverse, and returns the unknown together with the product of ratios and a filled-in setup string.

What the compound rule of three means and how mixed ratios combine

The simple rule of three is a single proportion: if four is to six as ten is to x, then x equals fifteen. The compound rule of three is the same idea with several quantities moving at once. The unknown scales with every quantity that is directly proportional to it, and it scales against every quantity that is inversely proportional to it.

Write the known magnitude of the unknown as value. For each other quantity, record its given value in the first situation, its sought value in the second situation, and whether the relation is direct or inverse. Direct means more of that quantity produces more of the unknown, so the factor is sought divided by given.

Inverse means more of that quantity produces less of the unknown, so the factor is given divided by sought. A classroom classic is eight workers in six days working five hours a day producing two hundred forty pieces; twelve workers in four days working six hours a day then produce two hundred eighty-eight pieces, because all three factors are direct: two hundred forty times twelve over eight times four over six times six over five.

Flip a relation to inverse when the story says the opposite: more machines finish the same job in fewer days, so days is inverse to machines. Zero or negative magnitudes are rejected because a ratio with a zero denominator is undefined and a negative head-count or duration is outside the usual domain of these word problems.

What to enter and what you get back

Each object needs given and sought as strictly positive finite numbers and relation as the closed enum direct or inverse so the form can render a select instead of a free-text trap. An optional name labels the row for display (workers, days, hours) but never changes the arithmetic. Optional decimals is an integer from zero to twelve and defaults to six; it rounds only the unknown, while each ratio and the product of ratios stay at twelve places for a stable contract.

Sending fewer than two factors is rejected because a single pair is a simple proportion, not a compound one. Example: value two hundred forty with workers eight to twelve direct, days six to four direct, and hours five to six direct yields unknown two hundred eighty-eight and the setup unknown equals two hundred forty times twelve over eight times four over six times six over five.