Solves the rule of three in both directions: direct proportion, where doubling one quantity doubles the other, and inverse proportion, where doubling one halves the other.
Direct proportion divides to find the value of one unit and then multiplies: if A gives B, then C gives B divided by A times C. Inverse proportion holds the product constant instead, so the answer is A times B divided by C. Choosing the wrong one is the mistake the tool cannot catch for you, because both produce a plausible-looking number. The test is what happens at the extremes: if more of the first quantity should mean more of the second, the relationship is direct; if more should mean less, it is inverse.
Direct or inverse
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Direct proportion: more gives more:
three kilos cost nine euros, so five kilos cost fifteen. Price, distance at constant speed and quantities in a recipe all behave this way.
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Inverse proportion: more gives less:
four workers take six hours, so eight workers take three. Speed against time, and workers against duration, behave this way.
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Test at an extreme before you choose:
ask what happens if the first quantity doubles. If the answer should grow, the relationship is direct; if it should shrink, inverse.
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Not everything is proportional:
twelve workers rarely finish in half the time six would take. Coordination, space and diminishing returns break the model, and the arithmetic will not warn you.
The one decision the arithmetic cannot make for you
The [msm_tool slug="dreisatzrechner"] solves the rule of three in both directions — direct proportion, where doubling one quantity doubles the other, and inverse proportion, where doubling one halves the other — and choosing between them is the actual judgement call, since both produce an equally plausible-looking number for the wrong relationship. The tool cannot catch that mistake, because nothing about a single set of three values reveals on its own which kind of relationship they belong to.
Test at an extreme before choosing
Asking what should happen if the first quantity doubled settles the question reliably: if the answer should grow along with it, the relationship is direct; if the answer should shrink instead, it is inverse. Three kilograms costing nine euros and five costing fifteen is a direct relationship — more weight, more cost. Four workers taking six hours and eight workers taking three is inverse — more workers, less time.
How direct proportion actually works
The calculation divides to find the value of a single unit and then multiplies back up: if A gives B, then C gives B divided by A, times C. Price against quantity, distance at a constant speed, and ingredient amounts in a recipe all follow this same shape.
How inverse proportion actually works
Here the product of the two quantities stays constant instead, so the answer is A times B, divided by C. Workers against the time a job takes, and speed against the time a journey takes, both follow this shape rather than the direct one.
Not everything that looks proportional actually is
Twelve workers rarely finish a job in exactly half the time six would take, because coordination overhead, limited space and diminishing returns all interfere with what a strictly proportional model assumes. The arithmetic itself stays exact throughout; it simply has no way to warn when the underlying assumption of proportionality has quietly stopped holding.