# How to split a number up by ratios

I've looked everywhere for a solution, but the problem is I'm not a mathematician so have no frame of reference.

I'm trying to split a number up by ratio. For example, if you take the number 10:

• If the smallest partition is 3, the result is 3, 7
• If the smallest partition is 2, the result is 2, 3, 5
• If the smallest partition is 1, the result is 1, 2, 3, 4

Any help would be much appreciated.

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I don't know what you mean by "split up a number by ratio." I don't understand why, if the smallest part is 2, the result isn't 2,4,4 or 2,8. I don't understand why, if the smallest part is 1, the result isn't 1,4,5, or 1,3,3,3, or 1,9, or 1,2,7, or 1,2,2,2,3, or something else. What are you really trying to say? – Gerry Myerson May 29 '12 at 12:26
I see what you mean, the first sequence confuses things. I'll change it to 3, 7, i.e., the next number has to be bigger each time. Does that make things clearer? – habitullence May 29 '12 at 12:43
I still don't quite see it. You are given a base number and a 'smallest partition,' and you want to write the base number as a sum. The summands start with the 'smallest partition' and must get bigger at each step. That's what I see so far. But I don't know what your question is. And this isn't well-defined. For $13$, for example, with 'smallest partition $1$, I have both $1, 3, 4, 5$ and $1, 2, 4, 6$ I suppose what I want to say is: what's your question? And what's the source of the question? – mixedmath May 29 '12 at 13:00
Thanks @mixedmath, didn't think of that. I guess the second example would be better cause the increase is more evenly distributed. I'm work on creating ranks, e.g., in a computer game. – habitullence May 29 '12 at 13:13
So let's say I have 100 ranks in a system and a total of 1,000,000 XP. I want to create a series of ranks that are increasingly harder to obtain and gaining 1,000,000 XP means you're at the highest rank. Also the lowest rank threshold can be set to a number, e.g., 500. Does that make things clearer? – habitullence May 29 '12 at 13:19

From the comments, you want 100 numbers satisfying $$500=m_1\lt m_2\lt\cdots\lt m_{100}$$ and $$m_1+m_2+\cdots+m_{100}=1000000$$ and you want the numbers $m_{i+1}-m_i$ to be roughly equal. If they were exactly equal, say, $m_{i+1}-m_i=d$ for all $i$, you'd have a 100 term arithmetic progression, starting with 500, and summing to 1000000. You'd have $m_{100}=500+99d$, and you'd have $$50(1000+99d)=1000000$$ That's $1000+99d=20000$, $99d=19000$, $191\lt d\lt192$ if I've done my arithmetic right. So you could make some of the steps 191, and the rest of them 192.