In a partnership business, the monthly investment by three friends for the first six months is in the ratio $3:4:5.$ After six months, they had to increase their monthly investments by $10 \%, 15 \%, \; \text{and} \; 20 \%,$ respectively, of their initial monthly investment. The new investment ratio was kept constant for the next six months.

What is the ratio of their shares in the total profit (in the same order) at the end of the year such that the share is proportional to their individual total investment over the year?

1. $22:23:24$
2. $22:33:50$
3. $33:46:60$
4. $63:86:110$

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 $:\text{D}$

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