$\text{P}$, $\text{Q}$ and $\text{R}$ enter into a partnership by investing their capitals in the ratio of $\frac{2}{5}:\frac{3}{4}:\frac{5}{8}$. After $4$ months, $\text{P}$ increased his capital by $50\%$, but $\text{Q}$ decreased his capital by $20\%$. What is the share of $\text{Q}$ in the total profit of $\text{₹}28,210$ at the end of a year?

Step 1: Simplify the initial ratio of capitals.
The given ratio of investments of $\text{P}$, $\text{Q}$ and $\text{R}$ is:

$$\frac{2}{5}:\frac{3}{4}:\frac{5}{8}$$

The LCM of the denominators $5$, $4$, and $8$ is $40$. Multiplying each term by $40$:

$$\text{P}:\text{Q}:\text{R}=\left(\frac{2}{5}\times 40\right):\left(\frac{3}{4}\times 40\right):\left(\frac{5}{8}\times 40\right)=16:30:25$$

Let their initial capitals be $16x$, $30x$, and $25x$ respectively.

Step 2: Calculate the effective capital for $12$ months (1 year).

*For $\text{P}$:** For the first $4$ months, capital is $16x$. Then increased by $50\%$, so new capital = $16x\times 1.5=24x$ for the remaining $8$ months.

$$\text{Total investment of P}=(16x\times 4)+(24x\times 8)=64x+192x=256x$$

*For $\text{Q}$:** For the first $4$ months, capital is $30x$. Then decreased by $20\%$, so new capital = $30x\times 0.8=24x$ for the remaining $8$ months.

$$\text{Total investment of Q}=(30x\times 4)+(24x\times 8)=120x+192x=312x$$

*For $\text{R}$:** Capital remains $25x$ for all $12$ months.

$$\text{Total investment of R}=25x\times 12=300x$$

Step 3: Find the ratio of profit distribution.

$$\text{P}:\text{Q}:\text{R}=256x:312x:300x$$

Dividing by $4x$:

$$\text{P}:\text{Q}:\text{R}=64:78:75$$

Step 4: Calculate $\text{Q}$ 's share in the total profit.

$$\text{Total ratio units}=64+78+75=217$$

$$\text{Share of Q}=\frac{78}{217}\times 28210$$

$$\text{Share of Q}=78\times 130=\text{₹}10,075$$