$P$  invested ₹$5000$ per month for $6$ months of a year and $Q$ invested ₹$x$ per month for $8$ months of the year in a partnership business. The profit is shared in proportion to the total investment made in that year.

If at the end of that investment year, $Q$ receives $\frac{4}{9}$ of the total profit, what is the value of $x$ (in ₹)?

1. $2500$
2. $3000$
3. $4687$
4. $8437$

We know that, ${\color{Green}{\boxed{\text{Profit} = \text{Investment} \times \text{Time}}}}$

Let the total profit be $₹\;k.$

• Profit received by $\text{Q} = ₹\;\frac{4}{9}k$
• Profit received by $\text{P} = k – \frac{4}{9}k = ₹\;\frac{5}{9}k$

Now, $\dfrac{\text{Profit}_{\text{P}}}{\text{Profit}_{\text{Q}}} = \dfrac{\text{Investment}_{\text{P}} \times \text{Time}_{\text{P}}}{\text{Investment}_{\text{Q}}\times \text{Time}_{\text{Q}}}$

$\Rightarrow \dfrac{\frac{5k}{9}}{\frac{4k}{9}} = \dfrac{5000 \times 6}{x \times 8}$

$\Rightarrow \dfrac{5}{4} = \dfrac{5000 \times 3}{4x }$

$\Rightarrow 5x = 15000$

$\Rightarrow {\color{Blue}{\boxed{x = 3000}}}$

$\therefore$ The value of $x = ₹\;3000.$

Correct Answer $:\text{B}$

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