Please explain how this ratio is being calculated A,B and C are partners of a company. A receives $\frac{x}{y}$ of profit. B and C share the remaining profit equally among them.
A's income increases by $I_a$ if overall profit increases from P% to Q%. How much A had invested in their company.
I know the answer: $\frac{I_a\cdot100}{P-Q}$.
This may be a very simple question, but I don't understand how it comes.

Let $A$ be the amount that Alicia has invested in the company. Let $\frac{x}{y}$ be the fraction of the company that she owns. So if $V$ is the total value of the company, then $A=\frac{x}{y}V$.
The old percentage profit was $P$. So the old profit was $\frac{P}{100}V$.
Alicia got the fraction $\frac{x}{y}$ of this, so Alicia's old profit was 
$$\frac{x}{y}\frac{P}{100}V=\frac{P}{100}\frac{x}{y}V=\frac{P}{100}A.$$
Similarly, Alicia's new profit is 
$$\frac{Q}{100}A,$$
so the change in profit is 
$$\frac{Q}{100}A-\frac{P}{100}A.$$
This is equal to $I_a$. So 
$$I_a=\frac{Q-P}{100}A,$$
and therefore 
$$A=\frac{100 I_a}{Q-P}.$$
Note that the fraction $\frac{x}{y}$ turned out to be irrelevant, as of course did the fact that there are other shareholders. 
