Set $A$ consists of disjoint subsets $a_1, a_2, ..., a_n$. Although you do not know the cardinality of set $A$ ($|A|$) 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 $\alpha_1 = \frac{|a_1|}{|A|}$. We denote the change in proportionality of $a_1$ using $\delta_{a_1}$