Three partners, A, B and C shared profits in the ratio $2:3:4$. A new partner D joined who took half of the shares of each A and C. If D's share of profit now is $\text{Rs. }100$, find the total profit.

Let the initial profit shares of A, B, and C be $2x,3x,$ and $4x$ respectively.
The total profit $=2x+3x+4x=9x$.

D joins the business and takes half of the profit shares from both A and C.

$$\text{Share taken from A}=\frac{1}{2}\times 2x=x$$

$$\text{Share taken from C}=\frac{1}{2}\times 4x=2x$$

Thus, D's total profit share:

$$\text{D’s share}=x+2x=3x$$

Given that D's share of profit is $\text{Rs. }100$:

$$3x=100⟹x=\frac{100}{3}$$

Therefore, the total profit is:

$$\text{Total Profit}=9x=9\times \left(\frac{100}{3}\right)=3\times 100=\text{Rs. }300$$