Generalized FiveThirtyEight Sequences Adapted from this FiveThirtyEight riddle.
Background
Examine the following infinite sequence:
3 3 3 2 3 3 3 2 3 3 3 2 3 3 2 3 3 3 2 ...

Let's say the sequence is 1-indexed. The ith number in the sequence determines how many 3s there are before the ith 2 and following any previous 2s. So since the sequence starts with a 3 the sequence must begin 3 3 3 2 and since there are three 3s at the beginning of the sequence the subsequence 3 3 3 2 must repeat itself three times. After that you reach 3 3 2 because the fourth number in the sequence is 2.
The FiveThirtyEight riddle asks for the limit of the ratios of threes to twos (which I won't spoil here) but you can also ask what the cumulative ratio is after index i. For example the ratio at i=4 is 3/1 = 3 and at i=15 it's 11/4 = 2.75.
Let's get general
Given numbers n and k we can make a similar sequence that starts with n and just like the original sequence described the number at index i determines how many ns show up before the ith k and following any previous ks.
Examples:
n=2, k=5 gives the sequence 2 2 5 2 2 5 2 2 2 2 2 5 2 2 5 ...
n=3, k=0 gives 3 3 3 0 3 3 3 0 3 3 3 0 0 3 3 3 0 ...
n=1, k=3 gives 1 3 1 1 1 3 1 3 1 3 1 3 1 1 1 3 1 ...
The Challenge
Write a function/program and with it do the following. Take as input:

a positive integer n
a nonnegative integer k ≠ n
a positive integer i &gt; n

The first two inputs n and k determine a sequence as described above and i is an index. I am using 1-indexing in the examples but you have the freedom to use 0- or 1-indexing. If 0-indexed then the restriction on i is i ≥ n.
With the three numbers output the ratio of ns to ks in the sequence up to and including the number at the index i. The format of the output can either be a decimal value with at least 5 digits of precision or an exact value as a ratio like 3524/837 or 3524:837.
In decimal form, the last digit can be rounded however you like. Trailing zeros and whitespace are allowed.
In either of the string forms the two numbers need to be normalized so that they are coprime. For example if the ratio was 22/4, 11/2 and 11:2 are acceptable but 22/4 is not.
Examples
n   k   i      output
2   4   15     2.75     or   11/4
6   0   666    5.1101   or   557:109
50  89  64     63       or   63:1
3   2   1000   2.7453   or   733/267
9   12  345    9.4545   or   104/11

This is code golf per language, so shortest code in each language is the winner.
Husk, 16 bytes

¤/#ωȯ↑⁰J¹`C∞²N¹²


Try it online!

Takes inputs in the same order as the test cases.
Outputs a rational number.
I feel like this has too many superscripts, but I don't know how to get rid of them...

Explanation

¤/#ωȯ↑⁰J¹`C∞²N¹²  Inputs are n, k, i.
             N    Starting with the natural numbers [1,2,3..
   ωȯ             do this until a fixed point is reached:
                    Argument is a list s.
           ∞²       Take the infinite list [n,n,n..
         `C         and split it to the lengths in s.
       J¹           Join the resulting blocks with k.
     ↑⁰             Take the first i elements.
                  Call the result x.
¤             ¹²  For each of n and k,
  #               count their number of occurrences in x
 /                and perform exact division on the results.