Let the three friends be $\text{A, B}, $ and $\text{C}.$
Given that, $\text{I}_{\text{A}}: \text{I}_{\text{B}}: \text{I}_{\text{C}} = 3:4:5\quad {\color{Blue}{\text{(For first 6 months)}}}$
Let,
- $\text{I}_{\text{A}} = 3x$
- $\text{I}_{\text{B}} = 4x$
- $\text{I}_{\text{C}} = 5x$
${\color{Teal}{\text{Investment for next 6 months}:}}$
- $\text{I}_{\text{A}} = 3x \times \frac{110}{100} = 3.3x$
- $\text{I}_{\text{B}} = 4x \times \frac{115}{100} = 4.6x$
- $\text{I}_{\text{C}} = 5x \times \frac{120}{100} = 6x$
We know that, ${\color{Green}{\text{Profit (P)} = \text{Investment (I)} \times \text{Time (T)}}}$
Now,
- $\text{P}_{\text{A}} = 3x \times 6 + 3.3x \times 6 = 37.8x$
- $\text{P}_{\text{B}} = 4x \times 6 + 4.6x \times 6 = 51.6x$
- $\text{P}_{\text{C}} = 5x \times 6 + 6x \times 6 = 66x$
Now, $\text{P}_{\text{A}}: \text{P}_{\text{B}} : \text{P}_{\text{C}} = 37.8x:51.6x:66x$
$\Rightarrow \text{P}_{\text{A}}: \text{P}_{\text{B}} : \text{P}_{\text{C}} = 37.8:51.6:66$
$\Rightarrow \text{P}_{\text{A}}: \text{P}_{\text{B}} : \text{P}_{\text{C}} = 378:516:660$
$\Rightarrow {\color{DarkBlue}{\boxed{\text{P}_{\text{A}}: \text{P}_{\text{B}} : \text{P}_{\text{C}} = 63:86:110}}}$
${\color{Magenta}{\textbf{Short Method:}}}$
$\begin{array} {llll} & \textbf{A} & \textbf{B} & \textbf{C} \\\hline \text{Time (month):} & 6 & 6 & 6 \\ \text{Investment:} & 300 & 400 & 500 \\\hline \text{Time (month):} & 6 & 6 & 6 \\ \text{Investment:} & 330 & 460 & 600 \\\hline \text{Total Time (Year):} & 1 & 1 & 1 \\ \text{Tota Investment:} & 630 & 860 & 1100 \end{array}$
$\therefore \text{P}_{\text{A}}: \text{P}_{\text{B}} : \text{P}_{\text{C}} = 630:860:1100$
$\Rightarrow {\color{Blue}{\boxed{\text{P}_{\text{A}}: \text{P}_{\text{B}} : \text{P}_{\text{C}} = 63:86:110}}}$
Correct Answer $:\text{D}$