Torsha started a business with ₹ 2250 while Trisha invested ₹ 2880 after a few months. If the end-of-12-months (from Torsha's investment) profit is shared by Torsha and Trisha in the ratio 25: 24, for how many months did Trisha stay invested?

In a partnership business, the profit distribution ratio is equal to the product of the investment amount and its corresponding time period duration.
Let:

* Torsha's investment ($I_1$) $=2250$
* Torsha's time period ($T_1$) $=12\text{ months}$
* Trisha's investment ($I_2$) $=2880$
* Trisha's time period ($T_2$) $=x\text{ months}$

The given profit sharing ratio is:

$$\frac{\text{Profit of Torsha}}{\text{Profit of Trisha}}=\frac{25}{24}$$

Set up the investment-time equation:

$$\frac{I_1\times T_1}{I_2\times T_2}=\frac{25}{24}$$

$$\frac{2250\times 12}{2880\times x}=\frac{25}{24}$$

Simplify the expression:

$$\frac{225\times 12}{288\times x}=\frac{25}{24}$$

Since $288÷12=24$:

$$\frac{225}{24\times x}=\frac{25}{24}$$

Cancel out $24$ from both denominators:

$$\frac{225}{x}=25$$

$$x=\frac{225}{25}=9$$

Thus, Trisha stayed invested in the business for $9$ months.