Fraction Calculator with Step-by-Step Reduction
Add, subtract, multiply, and divide fractions online with Least Common Denominator (LCD), simplification steps, mixed numbers, and decimal conversion.
Mathematical Explanation
Ein Bruch stellt einen Teil eines Ganzen oder das Verhältnis zweier ganzer Zahlen dar. Er besteht aus dem Zähler (oben) und dem Nenner (unten).
Die 4 Grundrechenarten bei Brüchen:
- Addition & Subtraktion: Ungleichnamige Brüche müssen zuerst auf einen gemeinsamen Hauptnenner (das kleinste gemeinsame Vielfache kgV) erweitert werden. Anschließend werden die Zähler addiert bzw. subtrahiert.
- Multiplikation: Zähler wird mit Zähler und Nenner mit Nenner multipliziert:
(a/b) × (c/d) = (a×c)/(b×d). - Division: Man multipliziert mit dem Kehrwert des Divisors:
(a/b) ÷ (c/d) = (a/b) × (d/c) = (a×d)/(b×c).
Klammern & Rangfolge der Rechenarten (Punkt vor Strich):
Enthält ein Bruchterm mehrere Operationen oder Klammern wie (1/2 + 2/3) × (3/4 - 1/6), gelten die mathematischen Prioritätsregeln:
- Klammern zuerst: Ausdrücke in Klammern werden zuerst berechnet und zu einem Bruch zusammengefasst.
- Punkt vor Strich: Multiplikation (×) und Division (÷) binden stärker als Addition (+) und Subtraktion (-).
- Gleichrangige Operationen von links nach rechts: Mehrere Additionen/Subtraktionen oder Multiplikationen/Divisionen werden der Reihe nach von links nach rechts abgearbeitet.
Historical Origin & The Challenge Solved
Why was this mathematics developed, where does it come from, and what real-world problem did it solve?
1. Fair Distribution of Bread & Grain Rations
How do you divide 7 loaves of bread equally among 10 pyramid construction workers without anyone feeling cheated? Whole integers were insufficient for complex rationing and storehouse administration.
2. Unit Fractions & Common Denominators
Scribe Ahmes documented rational arithmetic in the Rhind Papyrus (1650 BCE), decomposing fractions into unit fractions (e.g. 7/10 = 1/2 + 1/5). Later mathematicians developed lowest common denominators to add and multiply arbitrary fractions.
3. Music Theory, Clock Ratios & Computer Science
Musical tuning systems (frequency ratios 3/2, 4/3), mechanical gears, and exact rational arithmetic in computer algebra systems rely on fraction theory.
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:
Material Cutting Without Waste
Woodworkers and machinists divide panels and beams into precise fractional parts to ensure perfect joint fits and minimize offcut scrap.
Dividing a 2 1/2 meter board into 4 equal segments equals (5/2) / 4 = 5/8 meter = 62.5 cm per piece.
Note Values & Time Signatures
Musicians and composers structure songs using note fractions (whole, half, quarter, eighth, sixteenth) and meter signatures like 3/4 or 6/8.
A 4/4 measure can be composed of one half note (1/2) and two quarter notes (2 * 1/4): 1/2 + 2/4 = 1.
Mixing Ratios & Serial Dilutions
Laboratories prepare chemical solutions with exact compound ratios by utilizing fractions for solute and solvent concentrations.
A 1:4 dilution ratio (1/5 concentrate and 4/5 water) requires exactly 1 liter of concentrate for a 5-liter total mixture.
Frequently Asked Questions (FAQ)
How do you add fractions with different denominators?
Find the Least Common Denominator (LCD), convert both fractions to equivalent fractions with this common denominator, and add only the numerators while keeping the denominator.
How do you divide two fractions?
Multiply the first fraction by the reciprocal (flip numerator and denominator) of the second fraction.