I'm currently playing the game Satisfactory, where I need to balance the conveyor belts to ensure a 100% efficient factory.
To help me in this job I have Merger and Splitter. The Splitter can split belts into 2 or 3 conveyor belts and the Merger can join 2 or 3 belts into one.
Now I have a certain Input of N Ressources and want 1/15 of N Ressources at the End. Which is the amount of Splitter and Merger I need for this problem and how can I calculate, if it is even possible to achieve 1/15 or other fractures.
Hope somebody can help me with this problem.
Look at the denominators. If you merge (add) two streams with denominators which have only prime factors $2$ and $3$ the sum has only prime factors $2$ or $3$ (some factor may cancel eg $\frac 12+\frac 12=1$).
Likewise if you split a faction which has only prime factors $2$ or $3$ in the denominator into two or three equal pieces, the resulting fractions have only prime factors $2$ or $3$ in their denominators.
Therefore you can never get a fraction with $5$ in the denominator.
You can approximate $\frac 1{15}$ as closely as you like but can never get there exactly.