P and Q start a joint business. P invests $\text{Rs. }500$ for $9\text{ months}$ and Q invests a certain sum for $6\text{ months}$. If the total profit made is $\text{Rs. }69$ and Q's share of the profit is $\text{Rs. }46$, find the capital amount invested by Q.

Given criteria:
* Total combined profit = $\text{Rs. }69$
* Profit share of Q = $\text{Rs. }46$

Thus, the profit share of P = $\text{Total Profit}-\text{Profit of Q}$

$$\text{Profit of P}=69-46=\text{Rs. }23$$

Ratio of the profits of P and Q:

$$\text{Profit Ratio}=23:46=1:2$$

Let the principal capital invested by Q be $x$.
Ratio of their equivalent month-based capitals:

$$\text{Capital Ratio}=\frac{\text{Capital of P}\times \text{Time of P}}{\text{Capital of Q}\times \text{Time of Q}}=\frac{500\times 9}{x\times 6}=\frac{4500}{6x}$$

Since the ratio of capital must equal the ratio of profit shares:

$$\frac{4500}{6x}=\frac{1}{2}$$

$$6x=4500\times 2$$

$$6x=9000$$

$$x=\frac{9000}{6}=\text{Rs. }1500$$