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.
On
When you say profit increases from $P\%$ to $Q\%$ do you mean something different from $\frac P{100}$ to $\frac Q{100}$ (of what-sales, for example-are you assuming that sales stay the same)? If not, A receives $\frac xy \frac Q{100}$ instead of $\frac xy \frac P{100}$. You have a problem of units-$\frac xy \frac Q{100}$ is unitless, but you pay A dollars.
On
Let us suppose that A had invested $X$ into the company. If A's income has increased by $I_a$, it means that $X$ times the ratio change in profits must be $I_a$.
Now, to calculate the ratio change in profits. The percent change is $Q-P$. Of course, this is out of $100$, so to find the actual ratio we need to divide by $100$. This tells us that the ratio change in profits is
$$\frac{Q-P}{100}$$
Putting it all together,
$$I_{a} = X \cdot \frac{Q-P}{100}$$
Solving for $X$,
$$X = \frac{100I_{a}}{Q-P}$$
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.