In a joint business, the ratio of the capitals of three friends was $\frac{1}{2}:\frac{1}{3}:\frac{1}{4}$. After $4$ months, the first friend withdrew half of his capital and $8$ months thereafter, $\text{Rs. }2024$ was recorded as their total profit. What was the share of profit of the first friend?

Let's convert the fractional ratio of capitals into integers by multiplying each term by the LCM of $2,3,$ and $4$, which is $12$:
$$\text{Ratio}=\left(\frac{1}{2}\times 12\right):\left(\frac{1}{3}\times 12\right):\left(\frac{1}{4}\times 12\right)=6:4:3$$

Let their initial capitals be $6k$, $4k$, and $3k$ respectively.
The total duration of the business investment is $4+8=12$ months.

Compute the effective capital-month product for each friend:

* First Friend: Invested $6k$ for $4$ months, then withdrew half (leaving $3k$) for the remaining $8$ months.

$${\text{Equivalent Capital}}_1=(6k\times 4)+(3k\times 8)=24k+24k=48k$$

* Second Friend: Invested $4k$ for the entire $12$ months.

$${\text{Equivalent Capital}}_2=4k\times 12=48k$$

* Third Friend: Invested $3k$ for the entire $12$ months.

$${\text{Equivalent Capital}}_3=3k\times 12=36k$$

The profit sharing ratio is:

$$\text{Ratio}=48k:48k:36k=4:4:3$$

The sum of the ratio parts is $4+4+3=11$.
The first friend's share of the total profit ($\text{Rs. }2024$) is:

$$\text{Share}=\frac{4}{11}\times 2024=4\times 184=\text{Rs. }736$$