P and Q start a business with ₹$45,000$ and ₹$54,000$. After $8$ months R joined the business with ₹$36,000$. After $2$ years, what will be the amount of profit (in ₹) for Q, if all the three earned a total profit of ₹$1,47,600$?

The profit sharing ratio in a partnership is proportional to the product of the investment amount and the corresponding time duration.
The total duration of the business = $2$ years $=24$ months.

* P's investment duration: $24$ months
* Q's investment duration: $24$ months
* R's investment duration: Joins after 8 months $\to 24-8=16$ months

Let's compute the ratio of profits ($\text{P}:\text{Q}:\text{R}$):
$$\text{Ratio}=(45000\times 24):(54000\times 24):(36000\times 16)$$

Divide by 1000 to simplify:
$$\text{Ratio}=(45\times 24):(54\times 24):(36\times 16)$$

Now, divide each term by 12:
$$\text{Ratio}=(45\times 2):(54\times 2):(3\times 16)=90:108:48$$

Now, divide each term by 6:
$$\text{Ratio}=15:18:8$$

Sum of ratio components = $15+18+8=41$.
Total profit = ₹$1,47,600$.

The share of profit for Q is:
$$\text{Profit of Q}=\frac{18}{41}\times 147600$$
$$147600÷41=3600$$
$$\text{Profit of Q}=18\times 3600=\text{₹}64,800$$