I am looking for a formula to calculate something like the title (perhaps) suggests. I am having problem formulating my question rigorously so let me give the example.

I have four fractions $\frac{a}{w},\frac{b}{x},\frac{c}{y},\frac{d}{z}$, where each constant is nonnegative and each nominator is not greater than its denominator, hence each fraction is in $[0,1]$.

I would like to obtain the sum of the nominators. In this case, I have $a+b+c+d$.

However, I want to consider other possibilities where I remove one, two, OR even three fractions, as long as I do not remove one specific fraction, say $a/w$. However, every time I remove a fraction, I must add its denominator to $w$ and rescale $a/w$. For instance, if I remove $\frac{c}{y}$, I will have

$$\frac{\frac{a}{w}\times(w+y)}{w+y},\frac{b}{x},\frac{d}{z}$$ and hence the sum of nominators is $[\frac{a}{w}\times(w+y)]+b+d$.

If I instead remove two fractions $\frac{c}{y}$ and $\frac{d}{z}$, I will have $$\frac{\frac{a}{w}\times(w+y+z)}{w+y+z},\frac{b}{x}$$ and the sum I am looking for is $[\frac{a}{w}\times(w+y+z)]+b$

My goal is to consider all these possibilites and find the maximum sum of the nominators.

I can calculate it easily using computer help if only one fraction is removed. Not that I can't calculate it when more fractions are removed, but I think this involves like $2^{n}$ calculations if there are $n+1$ fractions.

My guess is that this relates to weighted average but I have no idea beside doing brute-force stuff. I am working this simply using Ms. Excel where I put the fractions and calculate everything there. Is there any easier method? (please edit my tags if necessary)


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