Calculate new percentages of the member of a society when one member died.

Question

I have a question that may seem easy to solve but for me it is not so easy. Suppose I have a society with $4$ members:$A,B$ and $C$. $A$ owns the $50\%$ of the society, $B$ the $20\%$, $C$ the $20\%$ and finally $D$ owns the $10\%$. Suppose that $D$ dies and we want to divide its money between $A, B$ and $C$ according to their percentage. How can I do this? I need to redefine the new percentages without $D$, but how?

Answer 1

I'd say the obvious thing was to divide it according to their current ownership percentages.

Take $A$. $A$ owns $50$ out of the $90$ that $D$ does not own. Thus $A$ owns $\frac {50}{90}=.555\cdots$ of the part not owned by $D$. in this way, the rescaled percentages are $$\{\frac 59,\frac 29, \frac 29\}$$

Note that, of course, these add up to $1$.

Thus, $A$ should receive $\frac 59\times 10=5.\overline 5\%$ from $D$ and so on.