How do I get 1/15 of something, only by divide with 2 or 3 and add the result back together?

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.

• Seems like, based on what you described, you could just split $1$ belt into $2$, those into $4$, those into $8$, and then those into $16$, and then from there merge any pair to obtain a total of $15$ belts. Not sure if that's what you meant by all this though, seems a little too easy. – Eevee Trainer May 5 at 21:45
• No I Dont want 15 belts. I want the ressources from one belt split in a way, that I have 1/15 fraction of that resource again on one belt. – Christopher May 5 at 21:47
• It's enough to do a 1/5, then split each output into threes. You can do a 1/5 splitter like this: imgur.com/a/8YecreR – Jane Doé May 5 at 22:11

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.
• @JaneDoé I can see how to get a $\frac 15$ splitter with feedback as a limiting case (feedback 1/6 into the input of a three way splitter), and then it is trivial to get $\frac 1{15}$. But $\frac 1{15}$ is not attained exactly in finite time . OP doesn't talk about limiting processes of this kind. – Mark Bennet May 5 at 22:24