In a business, P invests $\text{Rs. }12,000$. After some days, Q joins the business with an investment of $\text{Rs. }16,000$. If P and Q receive an equal share of the profit $9\text{ months}$ after Q joined, find the total duration for which P's capital remained invested.

Let the total duration of P's investment be $t\text{ months}$.
As profits are checked $9\text{ months}$ after Q joined, Q's investment span is precisely $9\text{ months}$.

Equivalent month-wise capital of P = $12000\times t$
Equivalent month-wise capital of Q = $16000\times 9$

Because their final profits are equal, their monthly equivalent investments must align identically:

$$12000\times t=16000\times 9$$

$$12t=144$$

$$t=\frac{144}{12}=12\text{ months}$$