Maths question for a problem of limited info

This is a maths problem that I found slightly interesting and which I have actually encountered during in my current research;

Suppose you want to know what x is when you know

x = y / (a + b + c + d )

BUT you don't know the values of y or a or b or c or d (or x).
BUT you DO know the values of the fractions y/a and y/b and y/c and y/d.
So you DO know the values p and q and r and s where you know;

p = y/a
q = y/b
r = y/c
s = y/d

What is the formula for x in terms of the known p and q and r and s ?
In other words, what is required is;

x = RHS

where RHS is a formula that ONLY contains the variables p and q and r and s.

Would you naturally think no such formula exists (so you cannot know the value of x) because of insufficient information?

Originally posted by @humy
This is a maths problem that I found slightly interesting and which I have actually encountered during in my current research;

Suppose you want to know what x is when you know

x = y / (a + b + c + d )

BUT you don't know the values of y or a or b or c or d (or x).
BUT you DO know the values of the fractions y/a and y/b and y/c and y/d.
So you ...[text shortened]... no such formula exists (so you cannot know the value of x) because of insufficient information?

x can be written in terms of p, q, r, and s.

Originally posted by @joe-shmo
x can be written in terms of p, q, r, and s.

so what is the formula?

Originally posted by @humy
so what is the formula?

x = 1/(1/p + 1/q + 1/r + 1/s)

Originally posted by @joe-shmo
x = 1/(1/p + 1/q + 1/r + 1/s)

correct 🙂
But I confess it took me quite a while before I got this one but it is easy once you see it as the solution simply comes from;

x = y / (a + b + c + d )
= 1 / ( (a + b + c + d ) / y )
= 1 / ( a/y + b/y + c/y + d/y )
= 1/(1/(y/a) + 1/(y/b) + 1/(y/c) + 1/(y/d))
= 1/(1/p + 1/q + 1/r + 1/s)

But I think there is something strange about this one that I just bet would make many people find it hard to get!
Just explaining the problem seems to me to somehow make it sound a lot harder to solve and more complicated than what it actually is.
This is one of the tiny bits of maths I will explain in my book.

Originally posted by @joe-shmo
x = 1/(1/p + 1/q + 1/r + 1/s)

That looks suspiciously like the parallel resistance formula.

Originally posted by @sonhouse
That looks suspiciously like the parallel resistance formula.

It effectively is. I can’t say anything intelligent about humy’s application ( because I have no idea what it may be) but it too may be interpreted in general as parallel paths under a common potential if his model is correctly applied, and this is some result.

Although there is some resemblance in that OP formula to that for parallel resistance formula, that is purely maths coincidental and the application it is of has nothing to do with parallel resistance (or at least, with my limited imagination, I cannot see how the two could meaningfully relate).

In the particular application I used this OP result in my current research, in the OP, a and b and c and d are all possible unknown probability values for k of a categorical distribution having k number of categories except, and unlike what is implied by my OP, there isn't just merely 4 possible categories to this distribution but an infinite number of them that become progressively smaller so that the sum up to 1 (else this distribution OF a categorical distribution will not normalize).
But what you would know is what the relative proportions of y/a and y/b etc must be where y is the probability of the distribution having k=c categories where c is the total number of sampled categories so far (there may be more categories not yet sampled so actual unknown k may be greater than c).

There are other parameters I didn't mention in the OP but, with huge difficulty and this took my several days, the equation for y I eventually derived was

y = k! (k – 1)! / ( (k + m – 1)! ∑[n=0, ∞] (k + n – 1)! (k + n)! / ( n! (k + m + n – 1)! ) )

where k=c and where
m = the number of samplings of a/any category (whether of the same category or a different category).
k = number of categories of distribution
c = total number of sampled categories so far
y = probability of number of categories of distribution being the same as that of total c number of sampled categories so far

I just bet all the above info would do far more to inadvertently confuse you all rather than inform you! 🙄