Set $A$ consists of disjoint subsets $a_1, a_2, ..., a_n$. Although you do not know the cardinality of set $A$ (denoted by $|A|$) and the cardinality of each of the subset, you do know the proportion of each subset in terms of set $A$, that is you know $\frac{|a_1|}{|A|}, \frac{|a_2|}{|A|}, ... \frac{|a_n|}{|A|}$. You are allowed to add elements into subsets of $A$, and observe the changes in proportionality of the subsets. What is the cardinality of set $A$ before elements were added to its subsets?

This problem can be solved by modelling the process of adding $n$ elements to subset $a_1$. We denote the proportionality of $a_1$ as $\alpha_1$, that is $\alpha_1 = \frac{|a_1|}{|A|}$, and the change in proportionality of $a_1$ using $\delta_{a_1}$

We begin by writing down the process that leads to the proportionality change:

$$ \frac{|a_1| + n }{|A| + n} - \frac{|a_1|}{|A|} = \delta_{a_1} $$

By multiplying the equation with the denominators in the right hand side, we obtain:

$$ (|a_1| + n )(|A|) - (|a_1|)(|A| + n) = \delta_{a_1}(|A| + n)(|A|)$$

By expanding the brackets, we obtain:

$$ (|a_1|)(|A|) + (n)(|A|) - (|a_1|)(|A|) - (|a_1|)(n) = \delta_{a_1}|A|^2 + \delta_{a_1}(|A|)(n)