I have two theories and I was pondering on what basis we are choosing one over the other. Let me take specific example and describe the two theories and any discussion in this regard would be helpful to me.
$\textbf{Question}$ There were two partners $A$ and $B$ who had invested $Rs. 50,000$ and $Rs. 80,000$ respectively. After four months $C$ joined and the total profit at the end of the year was $40000$. Of this, the share of $C$ was $15,000$. Then what was the amount invested by $C$?
$\textbf{Common solution suggested}$ Profit is directly proportional to the investment amount times the time. If $X_A$, $X_B$, and $X_C$ represent the shares of profit of $A, B$ and $C$ respectively, then $$X_A : X_B : X_C = 50000\times 12: 80000\times 12 : I_C\times 8,$$ where $I_C$ is the amount invested by $C$.
The calculations would finally lead to the answer $I_C = 1,17,000$.
$\textbf{Another Approach}$
The profit for year is $40000$ and hence profit for four months in $\frac{40000}{3}$. As $C$ was not yet joined, this profit can be thought of shared between $A$ and $B$. The profit shares come out to be $\frac{40000\times 5}{39}$ and $\frac{40000 \times 8}{39}$ between $A$ and $B$. Now at the end of fourth month, we may add these profits to the investments of $A$ and $B$ to calculate the new investments of $A$ and $B$ to be $\frac{50000\times 43}{39}$ and $\frac{80000 \times 43}{39}$, respectively. Now, $C$ has also invested, say some $Y$ amount. The total profit with these new investments is $\frac{40000\times 2}{3}$. Now, if we divide the shares of this new profit in the ratio of their new investments, I get a figure different from that got in the previous example. That is, I get $Y = \frac{12,90,000}{7}$.
What is wrong with this alternative approach. Why it shouldn't be preferred over the common sharing pattern described earlier?
In your alternative approach, you have added the profits of A and B to their investments which never happened.
Acc. to the question framed, profits are reaped only at the end of the year. Therefore, you shouldn't add their profits to their investment mid-way. Everything else seems fine.
Edit: Added Full Solution
After the first four months, A and B share $\frac{40000}{3}$ in the ratio 5:8 as you've mentioned.
Let see after this, at the year end C claims $15000$ as his share of profit. That is $\frac{15000}{\frac{80000}{3}}$ of the next 8 months profit, which simplifies to $\frac{9}{16}$ of 8th month proft. So, he would have invested $\frac{9}{16}$ of the total invested amount by A, B and C. So A and B must have invested the remaining $\frac{7}{16}$ of the total invested amount and we know together A and B invested 130000.
So, $$\frac{7x}{16} = 130000 \Rightarrow x = \frac{16 \cdot 130000}{7}$$
Amount Invested by C = $$\frac{9x}{16} \Rightarrow \frac{9}{16} \cdot \frac{16 \cdot 130000}{7}$$ $$\Rightarrow \frac{9 \cdot 130000}{7} = \frac{1170000}{7}$$