Rule of Three Calculator (Direct & Inverse Proportion)
Solve direct and inverse proportionality problems with detailed intermediate step breakdown, proportional ratio tables, and clear explanations.
Note: You can use both comma (,) and dot (.) as decimal separators.
Mathematical Explanation
Der Dreisatz (auch Schlussrechnung genannt) ist ein mathematisches Lösungsverfahren, um aus drei gegebenen Werten eines Verhältnisses den unbekannten vierten Wert zu berechnen.
1. Proportionaler Dreisatz (Direkte Proportionalität)
Gilt der Grundsatz »Je mehr Ausgangsmenge, desto mehr Zielmenge« (oder »Je weniger, desto weniger«), liegt ein proportionaler Dreisatz vor. Das Verhältnis beider Größen ist konstant:
2. Antiproportionaler Dreisatz (Umgekehrte Proportionalität)
Gilt »Je mehr Ausgangsgröße, desto weniger Zielgröße« (z. B. mehr Arbeiter benötigen für dieselbe Aufgabe weniger Stunden), handelt es sich um einen umgekehrt proportionalen Dreisatz. Hier ist das Produkt beider Größen konstant:
Historical Origin & The Challenge Solved
Why was this mathematics developed, where does it come from, and what real-world problem did it solve?
1. Barter Trade with Fluctuating Commodities & Prices
Merchants on the Silk Road faced daily calculations: if 12 bolts of damask silk cost 45 silver coins, how much do 28 bolts cost? They needed a foolproof method accessible to merchants without university training.
2. The "Golden Rule" via the Intermediate Unit 1
Indian scholars formalized the scheme as Trairasika (Rule of Three). By computing the unit rate (intermediate step 1) and scaling proportionally, the method became celebrated across Europe as the "Golden Rule".
3. Recipe Scaling, Medical Dosages & Chemistry
Stoichiometry in chemistry, weight-based pharmaceutical dosing in healthcare, and recipe adjustments in culinary arts represent everyday applications of the Rule of Three.
Real-World Applications & Practical Use Cases
Why do we learn this? Here is how this mathematics is applied in business, engineering, science, and everyday life:
Proportional Recipe Scaling
Chefs and bakers scale recipe ingredient amounts to any desired number of guests. If the guest count doubles, all ingredients scale proportionally.
If 300g of flour serves 4 people, serving 10 people requires exactly (300 / 4) * 10 = 750g of flour.
Inverse Workforce Scheduling
Project managers calculate how project timelines decrease when more workers are assigned (more workers = fewer required work days).
If 3 tilers take 8 days to finish a building, 6 tilers will finish the exact same job in (3 * 8) / 6 = 4 days.
Fuel Consumption & Travel Costs
Transport companies calculate fuel requirements and carbon footprints over variable route distances using average consumption rates.
If a truck consumes 28 liters per 100 km, driving 650 km requires exactly (28 / 100) * 650 = 182 liters.
Frequently Asked Questions (FAQ)
What is a direct proportion?
A direct proportion occurs when multiplying the first variable by a factor results in the second variable being multiplied by the same factor (e.g. quantity vs cost).
What is an inverse proportion?
An inverse proportion occurs when multiplying the first variable results in the second variable being divided by the same factor (e.g. speed vs travel time).